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Roof Beam with Double Cantilever

Roof Beam with Double Cantilever

Roof Beam with Double Cantilever

(OP)
Here is an example of a roof beam similar to one I encountered in the last year of my practice.  Assuming the top and bottom flanges are laterally supported at points b and c and nowhere else, what is the buckling length of the beam?   

BA

RE: Roof Beam with Double Cantilever

I'll bite. 16'

RE: Roof Beam with Double Cantilever

(OP)
hokie,

If, instead of the point loads at the tips of the cantilevers, you tied a cable between points a and d and applied a tension, would the buckling length of the beam be 16'?
 

BA

RE: Roof Beam with Double Cantilever

I'm not any of the wiser; I get 16' as well, even with the hint. However the eccentricity magnification of the load would need to be accounted for.

An expert is a man who has made all the mistakes which can be made in a very narrow field

RE: Roof Beam with Double Cantilever

BA,

In my opinion if your designing the beam for bending, then I would take the effective length as 16'.

If on the other hand you want to design the beam as a column such as having a cable in tension between a and d then I would take the effective length as 28' assuming the beam/column did not have adequate lateral restraint of the whole section at b and c.

RE: Roof Beam with Double Cantilever

My mistake!

If designing the member as a column I would design for effective length of 16'for middle portion and 12'for cantilever section which would not be critical so back to 16'in my world.

 

RE: Roof Beam with Double Cantilever

Yep, still 16'.

RE: Roof Beam with Double Cantilever

(OP)
civeng80,

If the beam is completely unsupported, i.e. a free body, and is stressed with a cable between a and d, what is the buckling length as a column?

How do the lateral supports at b and c change the buckling other than to ensure that those two points remain stationary?  The buckling curve can be accommodated unchanged with only two points held in position.
 

BA

RE: Roof Beam with Double Cantilever

(OP)
hokie,

You are a very brave man.

BA

RE: Roof Beam with Double Cantilever

Thanks, BA.

RE: Roof Beam with Double Cantilever

BA,

If beam unsupported and cable tensioned between ends buckling length is 28'.

If b and c are points of restraint in both x and y directions then I would design as column 16'as above.

As a beam I would take the effective length as 16'.

BA is this a trick question? Also did you end up putting this cable in or is this a hypothetical question?

 

RE: Roof Beam with Double Cantilever

(OP)
civeng80,

It is not a trick question and it is not a hypothetical question.  It is a question which needed to be addressed in a particular structure in Edmonton, Alberta a few years ago.

Wood roof trusses spanned 28' parallel to the beam shown and dumped their load on spandrel beams which were supported at the ends of the cantilevers shown in elevation.  A large area of lower roof was also hung from the ends of the cantilevers, so that the beam in question had no load other than two point loads as shown in the diagram.  

Lateral bracing to the adjacent wood trusses was provided at the columns and at a few other points.  I questioned the detail used.  Instead of defending the bracing detail, the engineer of record responded that no bracing was necessary and produced calculations to defend his statement.

One of the points he made was that the torsional capacity of the two columns is an important consideration.  I felt that was a very good point and, since the columns were HSS (Hollow Structural Sections), they would tend to resist rotation about a vertical axis.

I have simplified the problem in this thread so that the torsional resistance of the columns is not a factor, i.e. there can be free rotation about a vertical axis at each support.  I do this because I believe it is essential to understand the fundamental behavior of the structure first, then make adjustments for the oddball conditions.

BA

RE: Roof Beam with Double Cantilever

Good points BA, but this example raises another question in my mind.  

When, if ever, should we tell an Architect, contractor, developer, or client that some design will not work for the sake of gaining structural redundancy.  I.E., we have to add something to make it work, when, in fact, it still would without the addition?

Mike McCann
MMC Engineering
Motto:  KISS
Motivation:  Don't ask

RE: Roof Beam with Double Cantilever

What is the answer if it is not 16'?

Mike,
this is dependent on the engineer and the engineers constraints, at the end of the day a redundancy of 1 is acceptable as long as the risk factor is appropriate. Otherwise how would you design wind towers/monopoles.
  

An expert is a man who has made all the mistakes which can be made in a very narrow field

RE: Roof Beam with Double Cantilever

I look at BA's problem this way:  the bending moment in the central section is the same regardless of the span, so the buckling tendency of the centre span just depends on the span dimension.  If the span is very short, buckling in the cantilevers will control.   

RE: Roof Beam with Double Cantilever

With the cable in place, I make the length 28'. with just two restraints, the strut can go into single curvature in either or both directions.

Michael.
Timing has a lot to do with the outcome of a rain dance.

RE: Roof Beam with Double Cantilever

(OP)
hokie & rowingengineer,

Suppose the load consists of an equal and opposite moment applied at points 'a' and 'd'.  Do you still believe the effective length of the beam is 16'?

I don't believe there is an "answer".  What we have is a difference of opinion.  In my view, the beam will buckle over a length of 28', not 16'.  This makes a considerable difference in the selection of a beam capable of doing the job.

BA

RE: Roof Beam with Double Cantilever

(OP)
civeng80,

On re-reading your comment, the cable is hypothetical.  So is the moment at each end.  The actual load is a gravity load at the end of each cantilever.

BA

RE: Roof Beam with Double Cantilever

Quote (BAretired):

Suppose the load consists of an equal and opposite moment applied at points 'a' and 'd'.  Do you still believe the effective length of the beam is 16'?
The buckling phenomenon you're checking here is lateral torsional buckling.  The section is braced from twist and displacement at b and c, so the buckling length is 16'.  I don't see room here for a difference of opinion.  This is the same as the case with vertical loads applied at the ends.

For the case with a cable pulling the ends a and d together, the buckling phenomenon here is flexural (Euler) buckling.  This is different than lateral torsional buckling.  Because the buckling length here is 28' does not mean 16' is not correct for lateral torsional buckling.

RE: Roof Beam with Double Cantilever

I will throw a spanner in the works.

What if you turn the problem upside down and treat the end loads like support reactions and the column reactions as point loads, do you still think it is 16'?

RE: Roof Beam with Double Cantilever

If the beam is still braced at b and c, where the point loads are located, yes, 16'.

RE: Roof Beam with Double Cantilever

(OP)
nutte,

I agree that we are talking about lateral torsional buckling but I have always considered that to be similar to Euler buckling, i.e. the compression flange buckles in a sine wave or similar curve from end to end.  It is not clear to me why the sine wave cannot fit smoothly when only two points along its length are fixed against translation.

The beam is fixed against rotation about its axis at b and c, but the ends of the cantilevers are free to rotate in a direction opposite to that of the span.

In the case of moment at a and d, no reactions are required at b and c.  If these supports are removed, the beam buckles over its entire length.  How does its buckled shape differ from that of the supported beam?

BA

RE: Roof Beam with Double Cantilever

If you remove the supports at b and c, where is the beam braced against twist and lateral displacement?  If it's braced at a and d, then your unbraced length is 28'.  This is different than the case originally presented, though.  In the original case, the twist at b and c is zero.  In this new case, the twist at a and d is zero, and something greater than zero at b and c.  That's why these two are different.

RE: Roof Beam with Double Cantilever

(OP)
nutte,

The hypothetical beam is floating in the air or is weightless and suspended by the top flange at any two points, say b and c.  Both flanges are laterally unbraced throughout the entire 28' length.

If equal and opposite moments are applied at a and d, what is the effective length of the beam to be considered for lateral torsional buckling?  How does the buckled shape differ from that of the beam laterally supported top and bottom at b and c?

BA

RE: Roof Beam with Double Cantilever

BA, you threw me a curve when you put the axial force (rope) into the mix. I assume in the discussion that followed, Euler buckling means elastic buckling. For me, that is when the beam tries to rotate through about 90 degrees and allow the tension flange to shrink back and the compression flange to stretch back to their original lengths. It is a low energy position.

Given the above, and going back to the beam as diagrammed, the effective length of the span is 16' and the cantilevers get 2x6'= 12'

I may be showing my age here, but 2 was once the multiplier for an unrestrained cantilever.

Michael.
Timing has a lot to do with the outcome of a rain dance.

RE: Roof Beam with Double Cantilever

We keep talking about different things.

This last scenario, with the beam floating in air, supported at the top flange at b and c, is just like a lifting beam an erector might use.  The appropriate buckling length for this scenario would entail a much longer discussion on its own.

I maintain that for the original scenario presented, with a beam braced against twist and lateral displacement at b and c, loaded with either vertical loads or applied moments at the ends, the unbraced length is 16', with little room for discussion.

RE: Roof Beam with Double Cantilever

I agree with nutte, within the terms of his last paragraph.

RE: Roof Beam with Double Cantilever

(OP)
Seems like I am a minority of one.

paddington,

The effective buckling length of a 6' cantilever is 2 x 6' = 12' when the root is fixed.  If the root is a rotational spring, it is longer.  The greater the flexibility, the greater the length. In the present case, the root's flexibility is determined by the stiffness of the span.

nutte,

It seems to me that the lifting beam analogy is a good one.  That is the way I believe the beam as sketched would behave.  

There should always be room for discussion but I guess there is no point in beating a dead horse.  

BA

RE: Roof Beam with Double Cantilever

(OP)
If we make the member a truss instead of a beam and consider only the case of equal and opposite end moments, the top chord is in pure tension.  The bottom chord is in pure compression from a to d and the web members are unstressed.  

If the moments are gradually increased until Euler buckling occurs, the bottom chord will buckle laterally in a half sine wave extending from a to d.  The buckling length will be 28', not 16'.

BA

RE: Roof Beam with Double Cantilever

BA,
Change the central span to 2' rather than 16'.  Same moment.  What will the buckling length be then?

RE: Roof Beam with Double Cantilever

BA

Your replacing the end cantilever with a moment at b and c.  Now at b and c the body is pinned laterally so that the effective length of the bottom chord is 16'i.e the distance between the restraints.

If you wish to check the cantiver portion for buckling then the effective length is 12'. So obviously the critical part of the member is between b and c.

 

RE: Roof Beam with Double Cantilever

To make my view even more clear, the lateral and rotational fixity called for nutte are definitive to give the unbraced length as 16 feet. Unbraced. Normally we don't think full rotational restraint proper is a general requirement of bracing, but maybe we must understand so is the case if we ask the compressed flange be restricted against LTB, i.e., tumbling and hence we may surmise for many cases, rotation.

If by whatever the means, stiffness of the column, maybe helped by orthogonal beams at the support points, we may assume the conditions above are met, we are talking of 16' unbraced length. The buckling length may be even less (K=0.5 for full bending fixity on weak axis)

But contrarily, if in spite of good vertical support and lateral restraint, there is no rotational restraint at all, as in the hanged beam later called for by BAretired, the buckling length would become the total length, for as he is assuming, a single curvature LTB shape can form.

Again it is shown the convenience of the modern methods of design that trough proper introduction of material and geometrical nonlinearities, and maybe initial imperfections, free the designer of these issues. Once this is accounted, K is 1 or less for every segment member.

RE: Roof Beam with Double Cantilever

The hanged beam wouldn't even have "good" lateral restraint except if we intently so provide. If lacking the lateral restraint it might require a bit of thought to assert definitively that K=1 with the length tip top tip is enough for the buckling length, even if the likely answer is yes because of the likely single curvature buckling shape between tips. But there maybe unsymmetrical tip loads' cases -specially accounting the horizontal action of the spandrels bringing the loads- that require further thought.

RE: Roof Beam with Double Cantilever

(OP)
hokie,

That's easy.  The buckling length would be 6 + 2 + 6 = 14'.  The location of the supports are not important because they don't do anything.  The buckling length would be the full length of the beam.

civeng80,

Quote:

Your replacing the end cantilever with a moment at b and c.  Now at b and c the body is pinned laterally so that the effective length of the bottom chord is 16'i.e the distance between the restraints.

You are mistaken.  A column of length L with axial load applied at each end and laterally supported at two intermediate points has an effective length of L no matter where the intermediate points are.

Quote:

If you wish to check the cantiver portion for buckling then the effective length is 12'. So obviously the critical part of the member is between b and c.

Again, you are mistaken.  As I told paddington earlier, the effective length of a 6' long cantilever is 12' ONLY IF the support is fully fixed.  If the support is a rotational hinge, the effective length is greater than 12'.

ishvaag,

I don't understand your argument, so I won't attempt to refute it, but you can rest assured that I believe you to be mistaken.

BA

RE: Roof Beam with Double Cantilever

BAretired, in my last posts I point to the fact of KL be the relevant magnitude for the calculation of the buckling stress (LTB as well). I won't say there may not be room for discussion about many things, particularly as to the relevance of the quality of the restraints at bracing points, that maybe not always is stated with clarity enough in the codes, a proof of which is the present discussion.

For example, at AISC 360-05

Lb = length between points that are either braced against lateral displacement of compression flange or braced against twist of the cross section, in.(mm)

So it is clear that it wants the compression flange at bracing points not move outside the web midplane. OK, then take your same brought cantilever in your last post, even fixed at root; what is Lb? at the other end there's no such fixity in all cases. I mean, codes give guide but not always. And so you resource to literature or maybe commentary etc (I haven't looked at) and say, well twice or more the length of the cantilever. But that, even if it is not named so in the code, is a K buckling factor approach, akin to the column fixed at the base and free to displace atop. And it is this Lb (actually an implicit KL) that the code wants as input for the check of the fixed at root cantilever.

Examining the LTB of some compressed flange for which Lb is equal to the length between the physical restraints, the implicit K equals 1 because relative displacement outside the web midplane as per the definition in the code is prevented and then K=1 is applicable. But, if the restriction was enough to provide full fixity at the bracing point in Mx, My and Mz, an implicit favourable K=0.5 could be considered, that, well, the code may not acknowledge by its calibration, method, consistency or intent, but as you can see by my explanation here has as much basis as doubling the actual length of the fixed cantilever for the checks as Lb. It is exactly the same thing, if you can find proper to double the length of the cantilever for Lb, you might be tempted to halve it for full moment fixity for inner segments.

So I maybe may have mislead someone about the straightforward application of some code, but hope not about the intricacies of LTB.

On the other hand I feel not at all wrong in reminding that the method making resource of material and geometrical nonlinearities of the current code (and since LRFD 93 on at least) relieves the designer of much of these considerations, by automatically producing segment solicitations to be always checked at K=1 or less. After the relaxation brought by the P-Deltas and deltas, the nodes are bracing points.

RE: Roof Beam with Double Cantilever

(OP)
ishvaaag,

Quote:

To make my view even more clear, the lateral and rotational fixity called for nutte are definitive to give the unbraced length as 16 feet.

I do not agree that the lateral and rotational fixity specified give an unbraced length of 16'.  Moreover, I think it is a dangerous assumption.

Quote:

If by whatever the means, stiffness of the column, maybe helped by orthogonal beams at the support points, we may assume the conditions above are met, we are talking of 16' unbraced length. The buckling length may be even less (K=0.5 for full bending fixity on weak axis)

If joints b and c are fixed against rotation about a vertical axis, then I agree with effective length of 12' for each cantilever and 16' for the span.  However, these points are not fixed against rotation in the example given.

Quote:

But contrarily, if in spite of good vertical support and lateral restraint, there is no rotational restraint at all, as in the hanged beam later called for by BAretired, the buckling length would become the total length, for as he is assuming, a single curvature LTB shape can form.

In this paragraph, you seem to be agreeing with me.  Please confirm if I am interpreting this correctly.

BA

RE: Roof Beam with Double Cantilever

(OP)
ishvaaag,

Sorry, I was busy typing when your latest message came through.

If lateral bracing is located at all four points, a, b, c and d then I agree that 16' would be a conservative estimate of the unbraced length (or the buckling length).  It is probably more like 0.75*16 = 12' because of the stiffness of the two cantilever sections.

If the columns at b and c are infinitely stiff in torsion, the K value for the span would be 0.5, and the buckling length would be only 8' for the span and 12' for the cantilevers.

However, if points b and c are the only braced points and the beam is free to rotate about a vertical axis at those points, then for constant moment from a to d, the buckling length should be taken as the full length of the beam.

Perhaps you and I concur on this point.

BA

RE: Roof Beam with Double Cantilever

Yes, I concurr that for symmetrical loading the full length would be a good assumption of the length to port to the checks, if the sections free to rotate about vertical axes within the web. But I took (tanslated to spanish) twist in nutte's statement as fixity precisely about such rotation (whereas it refers to torsional restraint, as the code, was wrong in that) and so there was no doubt that in such amovility Lb 16'.

I must point however that even if this seems to me rational, the code seems not particularly well suited to the consideration of braced lengths other than those physical between members (and this includes from the 2 factor for cantilever to any reduction for inner segments of beams between braces). For as you see for the definition given, nutte's statement is exactly what demanded for the code: it has lateral restraint and torsional restraint, in between 16', and hence by literal code, Lb=16'.

When further applying Cb factor for more precise critical moment or moment strength, also, I doubt would be reasonably consistent with the kind of solicitations present in the whole length of the beam plus cantilevers ... and application to subsegments might be adequate but of necessary investigation, for the commentary says "moment diagrams within the unbraced segment", for our interpretation the whole beam tip to tip if free to rotate at column points.

The P-Delta and deltas plus material nonlinearities pursue the deformation to stable limit position if available, and then all issues respect to the restriction to rotation in the axis of the column are cleared. plus the facto of that we are liberated of any estimations of Ks if so we want, (become 1 for every segment modeled) for hand estimation of the adequate K corresponding to the restriction to rotation on its own axis provided by the column may become from risky to adventurous, or require substructure analyses that support any K to factor unto our buckling length.

RE: Roof Beam with Double Cantilever

(OP)
I am not familiar with other codes, but the CAN/CSA S16-01 (the Canadian standard) seems to assume that the ends of members will be laterally braced and that there may be 1, 2, 3, 4 or more equally spaced braces in between (Article 9.2.6.2).  The code does not consider the possibility of a cantilever at one or both ends of the member.

If the member to be braced is in compliance with the code assumption, I do not believe we would be arguing the issue now.  I am merely attempting to point out that, in the event of a cantilever at one or both ends of the beam, the code does not apply.

BA

RE: Roof Beam with Double Cantilever

(OP)
Let me modify that last statement to:

In the event of a cantilever, unbraced at the tip, at one or both ends of a beam, S16-01 bracing requirements do not apply.

BA

RE: Roof Beam with Double Cantilever

BA,
If you don't think 2 ft for the centre span is short enough to provide a "fixed" support, what about 1 ft?  My point is that for a very short centre span with lateral restraint, the ends are truly cantilevered.

RE: Roof Beam with Double Cantilever

(OP)
hokie,

I agree that one foot is shorter than two feet.  It really doesn't matter.  Let's say the span is 0'.  In that case, the buckling length is 2*6' = 12' and the ends are truly cantilevered.  I have been trying to say this in this and other threads.   

BA

RE: Roof Beam with Double Cantilever

I still think your are confusing buckling length with buckled shape.  But I guess we will have to agree to disagree.

RE: Roof Beam with Double Cantilever

BA

I think the other important point is that critcal (Buckling) flange for LTB in a cantilever is the tension flange whereas for the middle portion it is the compression flange.

 

RE: Roof Beam with Double Cantilever

(OP)
hokie,

I don't believe I am confusing buckling length with buckled shape.  If I am, I want to know how.  I need something a little more convincing than what you have provided to date.

Why must we agree to disagree?  I am not a disagreeable person and I have come to know that you are not either.  This subject is important.  Let us thrash it out until we come to a conclusion.

If I am wrong, I want to know in what respect I am wrong.  If you are wrong, I would like you to acknowledge it so that the junior engineers participating in this forum can gain a better understanding of how mature engineers are capable of determining how stuff works.

BA

RE: Roof Beam with Double Cantilever

(OP)
civeng80,

Quote:

I think the other important point is that critical (Buckling) flange for LTB in a cantilever is the tension flange whereas for the middle portion it is the compression flange.

I have read that in the literature too, but I am damned if I am able to understand it.  Perhaps you would be good enough to explain it to me.

In most situations, the top flange of a Gerber beam is laterally braced by the floor or roof  joists, so that bracing of the tension flange is not an issue.  In situations where the top flange is not braced, perhaps additional measures are required to ensure stability.  The design engineer is free to specify any bracing he deems necessary.

A prudent engineer will take such measures as he deems necessary in design to ensure that his structure is sufficient to carry all possible load combinations with an adequate factor of safety.

A more difficult problem, of course, is the assessment of existing structures designed by others, particularly when the design is in dispute.  In this case, the engineer must evaluate precisely what has been specified and what has been built...not always an easy task.

BA

RE: Roof Beam with Double Cantilever

BA,
I'll try, but it may take a bit of time to cogitate and agagitate before I get back to you.

RE: Roof Beam with Double Cantilever



BA

I think your main problem lies with the fact that their is a difference between column (Euler) Buckling and Lateral torsional buckling.

I must admit that without looking at the fairly complex derivation (I think its derived in Timoshenko Strength of Materials Part 2) why in a cantilever the tension flange is critical rather than a compression flange remains one of those nagging little problems in structural mechanics.  This reinforces the fact that LTB is not quite the same as Euler buckling and indeed without looking at any references I dont think Euler buckling was even used in deriving LTB in a beam.

The fact is that the tension flange is the critical flange in a cantilever, so that again I would take the effective beam length as 16'.

 

RE: Roof Beam with Double Cantilever

civeng80, of course LTB and weak axis buckling are different, and solved in both cases by the proper equations of equilibrium.

This said, however, it doesn't mean that buckling out of the weak axis plane of the flange has not a say in LTB, when interpreted in the ordinary way as out of plane buckling of the compressed flange.

I attach copy of a page of

Flexural-Torsional Buckling of Structures
N.S. Trahair
CRC Press, 2000

that pertains to its eigth chapeter, dealing with restrained beams where it is clearly stated that it has been observed

"that the effects of unequal flange end restraints at the ends of the beam should be approximated by using the solution obtained for braced columns with unequal end restraints."

Hence the over twice length for cantilevers and so on as brought by BAretired is technically a satisfactory view.

Those wishing even more accuracy will need mathematics to a level at which most of us are not conversant enough. Just a look to the Trahair's book -mainly an outlay of the problems and solutions- will be sufficient proof.

RE: Roof Beam with Double Cantilever

thanks ish

I know there is a lot of material on LTB and Trahair is an authority on that.

Can you explain why in a cantilever the critical flange is the tension flange rather than the compression flange so that BA and I may be enlightened?

I've wondered about this for a long time.

RE: Roof Beam with Double Cantilever

The critical flange in a cantilever is the tension flange because it translates further than the compression in bending. Testing has shown that tension bracing is more effective than compression bracing on cantilevers.

Hokie you should take a look at the work by Essa and Kennedy, they did some testing on cantilevers with a back span (point loads), which in my mind is the same as the example. They showed the effective length is governed by the lowest buckling capacity of the back span or cantilever, not combined. Something about the member with the highest buckling capacity restraining the lower buckling capacity beam.  
 

An expert is a man who has made all the mistakes which can be made in a very narrow field

RE: Roof Beam with Double Cantilever

Quite likely in ordinary terms the explanation is what rowinengineer gives. For mathematical support of the statement, it would require examination of the context in what made by Timoshenko or whomever expounded the behaviour that way, for normally one finds oneself looking for a critical moment or load more than a qualitative description of the behaviour or some less all-encompassing parameter in the buckling problem.

I attach a photo, again from Trahair's for the case.

RE: Roof Beam with Double Cantilever

To answer the logic on this you may need to look at it from the opposite direction.

Why is the tension flange not critical for a simple span?

The answer is that the tension acts like a cable tending to straighten the tension flange.

In a cantilever you have no such straightening action and therefore no addition stability.

RE: Roof Beam with Double Cantilever

BA,
Here I go again.  No fancy stuff, just intuit.  No argument about whether top or bottom flange of cantilever buckles first (depends on position of load and bracing condition at end).  And no consideration of how bending gradient affects buckling.

Take your original beam...6,16,6.  Now make the ends a and d the supports, with the 70k loads at b and c.  The supports and load positions both brace the beam.  Same moments, same deflected shape.  16' unbraced or buckling length.  Agree?

A cantilever's buckling length is twice the length only in the case where the end is unbraced laterally, e.g. a flagpole.  If the end is braced laterally, the buckling length is just the length.  Agree?

Another way of looking at it...I think there is no difference in the buckling behaviour between a cantilever which is fixed at the support and one where rotation is allowed, as in your example with the backspan.  If you just consider what happens to the cantilever relative to a tangent at the support, it is the same in both cases.  Agree?

Disagree?

  

RE: Roof Beam with Double Cantilever

(OP)
hokie,

Quote:

Take your original beam...6,16,6.  Now make the ends a and d the supports, with the 70k loads at b and c.  The supports and load positions both brace the beam.  Same moments, same deflected shape.  16' unbraced or buckling length.  Agree?

I agree if the supports at a and d prevent lateral displacement.  I don't agree if the supports at a and d provide only a vertical reaction.

Quote:

A cantilever's buckling length is twice the length only in the case where the end is unbraced laterally, e.g. a flagpole.  If the end is braced laterally, the buckling length is just the length.  Agree?

I agree.

Quote:

Another way of looking at it...I think there is no difference in the buckling behaviour between a cantilever which is fixed at the support and one where rotation is allowed, as in your example with the backspan.  If you just consider what happens to the cantilever relative to a tangent at the support, it is the same in both cases.  Agree?

I agree.  But the compression at the end of the cantilever adds moment to the backspan about its minor axis which magnifies the lateral deflection of the backspan.
 

BA

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