Roof Beam with Double Cantilever

[BAretired]

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

 

[hokie66]

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.

 

[BAretired]

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

 

[hokie66]

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

 

[civeng80]

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.

 

[BAretired]

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

 

[BAretired]

civeng80,

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

 

[hokie66]

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

 

[civeng80]

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'.

 

[ishvaaag]

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.

 

[civeng80]

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.

 

[rowingengineer]

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

 

[ishvaaag]

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.

 

[csd72]

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.

 

[hokie66]

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?

 

[BAretired]

hokie,

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.

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.

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