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Beam in compression
4

Beam in compression

Beam in compression

(OP)
Hi All,

let me preface I am far from being a structural engineer, which will explain the triviality of my post.

How to calculate deflections for a horizontal beam, both ends clamped, under distributed force (own weight) and compression?

I am looking at the basics, but as you can not superimpose the Euler  beam theory solution to a compressive case my repertoire is almost over.

There must be more than one approach for such a common case, I am hoping in your patience!

Thank you and all the best

RE: Beam in compression

a column is a "beam" loaded in compression, easily solved by Euler.

a beam is a beam (der) loaded with transverse loads, easily solved by equations of equilibrium.

a "beam column" is the combination of the two.  you're right you can't superimpose them ... the bending deflections due to the transverse loads exagerate the column loading.  "advanced" structures books, like Bruhn, solve this structure ... it isn't any more complicated than a redundant beam.

clear as mud ?

RE: Beam in compression

A solution is published in Roark's Formulas for Stress and Strain.

RE: Beam in compression

Do you not have an analysis software?

RE: Beam in compression

2
SEIT,

That is not an appropriate suggestion.  If he doesn't know how to do it manually, he shouldn't be using software to solve his problem.

RE: Beam in compression

A bit off topic, but, how is using software any less than using Roark's??

Personally, I would think a novice has a better chance of screwing up using Roark's than use something like SAP or RAM.

Just because you can write it on a piece of paper doesn't mean you know know it, ya know?  

RE: Beam in compression

I suspect very few engineers actually know how to truly solve this problem (beam-column).

We use codes and approximations to come up with safe solutions . But to understand the TRUE behavior of the member requires a solution to a differential equation unique to each problem. In the US, at least, this is not even taught until grad school. And this is only "easy" with "well-behaved" materials. With concrete, good luck understanding the "true" behavior.

I mention this because hokie66's comment seemed to have a self-righteous undertone. If this is not the case, I apologize in advance. I just think we should recognize the limit of our knowledge. Software, as long as one understands it, is a very powerful tool to aid engineers in mundane tasks.

RE: Beam in compression

NO NO NO!  NEVER Design by BLACK-BOX methods what you cannot solve to within 30% by HAND!!!

Sorry frv, love your postings, as well as respect your opinion from numerous other threads, but I have to counter-comment on your comment.  I doubt there was anything self-righteous in Hokie's comment... And even if it was, that wouldn't change the fact that our transplanted American friend is ABSOLUTELY RIGHT.  I do not accept that there is anything inappropriate about telling someone that they should not use software to solve a problem they aren't able to do by hand.

And it's true that beam-column solutions are complex, often requiring higher level analysis methods, however even then, with all of our high-end wizardry, we still CANNOT know the TRUE state of stresses.  Wooten's third law still applies my friend.

With utmost respect and with the hopes that no one misunderstands my passion as being arrogance or disrespect,

YS

B.Eng (Carleton)
Working in New Zealand, thinking of my snow covered home...

RE: Beam in compression

youngstructural-

I gave you a star for disagreeing with me :). Primarily because you were so nice about it. Is it the Canadian thing?

I don't misunderstand. In fact, you have a valid point.

Ironically, two of the posters I have come to consider most insightful are you and StructuralEIT. I say "ironically" because it would seem that at the core, you two disagree about this issue. Our current political climate in the US notwithstanding, people can disagree, yet respect one another.

I also respect Hokie66's opinion. It is obvious he has years of experience and is well versed in many aspects of what we do. If my post came off as anything other than respectful disagreement, it was inadvertent, and I again apologize for it.

I wasn't suggesting that someone with NO knowledge of structural behavior attempt to model this. However, I do think that if you know how to properly model something, software can be an invaluable tool. Therein lies the problem. How do you define "know how to properly model"? I think this is where our disagreement lies.

BTW..  I think 30% is excessive. If your hand calculation is 30% off of a computer generated one, you probably didn't catch the second-order effects.

 

RE: Beam in compression

Thanks frv; I was concerned I would be misinterpreted, and am greatly relieved that you have understood the heart of my concern.

Yes, I suppose being polite is something that many Canadian pride themselves upon.  It tends to be a faut-pas to fail to be polite, however we have our fair share of rude & obnoxious people!

I suppose StructuralEIT and I are disagreeing here, althoug I too normally find his (her?) posts insightful;  At the end of the day, for this case, you're probably right that we'll have to agree to disagree.  However, that said, I think a lawyer would easily chop a designer up on the stand if they blindly followed a computer analysis and were not able to check the results by hand.

With regard to the level of accuracy for a check, 30% is my red flag, do it again until it checks out point.  It's the no-go point for my analysis before it goes to the design phase.  I normally find I can estimate structures to withing 15%, and am only really happy if my back of the envelope is out less than 10%.

Cheers,
And thanks for the star,

YS

B.Eng (Carleton)
Working in New Zealand, thinking of my snow covered home...

RE: Beam in compression

(OP)
All,

many thanks to your suggestions.

I found the formula in Roark's book but I am reluctant tu use a formula which I do not the derivation of, hence significance of its assumptions.

The only software I could use is ANSYS, but I would be ashamed of using it for this problem..

Right, the beam column, I could certainly solve a 4th order differential equation numerically (as the EI is varying) but what about buckling?

Yesterday evening I went back to princioples and I am pretty confident that I have found the critical load for a beam in compresion and with one concentrated load in the middle (the maths was easier than the distributed load).

I will assume conservatively this case (will certainly give me a lower critical load) and start from here.

Many thanks to all

ATB

Favollo

RE: Beam in compression

I was speaking to a colleague today about using design packages, and in my opinion, if the software is going to perform all the calculations that I was going to perform by hand in a fraction of a second, then why not use the design package.

As to the question, if I was to solve this problem by hand, I would first solve my midspan delfection for vertical loads (by emperical formulas) then superimpose the additional deflections that result from P-delta effects.

As for the reduction in buckling capacity that results from the combined presence of moment and axial compression.  Unfortunately, I'm not to crash hot on my beam-column theory and like a true engineer, I work by the fudge factors set out in the code a reduce the member moment capacity in proportion to how hard the section is working in compression.   

RE: Beam in compression

YS

Slightly off topic.  Just out of curiosity what are Wootens First and Second Laws?

RE: Beam in compression

A forum such as this is not suitable for discussing Wooten's first and second laws.

RE: Beam in compression

Well, there has been quite a bit of discussion since I last logged on.  I certainly didn't mean to imply that anyone should blindly accept output from a computer.
All I was trying to get at is that if you have a difficult problem in which you at least understand the behavior and can tell (at least qualitatively) that the program is doing the right thing that I wouldn't see anything wrong with it.
As an example for the OP, I might try using a minimum potential energy solution assuming a deflected shape.  That can still be time consuming (and may not accurately capture the second order effect).  
I should also say if I can't analyze the problem completely accurately I would use a conservative analysis and compare that to the computer output knowing that my results are conservative.


Favollo-
You can use a minimum potential energy solution to get pretty close to the buckling load for a fixed end column with a lateral load applied.  It will take some work/time to put the equations (for equilibruim) together and solve, but is much, much easier than using a 4th order differential equation and will get you very close (probably within 3% - 5%.  Did I mention that it's A LOT easier than going the 4th order DE route?!  

RE: Beam in compression

Favollo-

In your original post, you described yourself as "far from being a structural engineer", and asked for help to calculate "the deflections for a horizontal beam, both ends clamped, under distributed force (own weight) and compression."

In your second post, you are solving for the buckling load using 4th order differential equations on a section with variable EI.

Whew! That's quite a jump.  Now it sounds like you're intending to design a beam-column.  If that's the case, and you are truly far from being a structural engineer, then you probably ought to find a structural engineer to design it for you.

On the question of Roark vs. computer solution, NS4U raises a valid question, i.e., is using a formula from Roark any different than using someone else's software?  One difference I see is that, when using a software package or a purchased spreadsheet, there is a tendency to shift some of the responsibility for the correctness of the result to the software, even though the software license agreement disclaims any responsibility.  When one calculates by hand, it's clear where the responsibility lies, so I think, we are more fastidious, resulting in fewer errors.  Also, when using Roark, the formulas are there for scrutiny, the text preceding the tables gives insight to the solution, and references to the development of the formulas are also given.  That quite a bit different than an input manual or a spreadsheet with cryptic formulas.

 

RE: Beam in compression

One question I have, how much compression is there?  If it is as low as 10 to 15% of the total compressive stress, maybe a simple beam calculation would be sufficient.

RE: Beam in compression

Favollo,

As graybeach has mentioned, the effect will usually be negligible if the axial load is less than 15% of the capacity.

If this is the case then additional deflections from the axial load can usually be ignored.

At higher axial loads the situation is more complex. At first order there is additional moment from the eccentricity of the axial load from the line of force, this causes additional eccentrity which causes additional (2nd order) moments and so on until it reaches equillibrium.

If this is a critical component, then I would definately reccommend that you get it designed by a structural engineer.  

RE: Beam in compression

(OP)
Graybeach, Csd72,

the design I am getting at would be loaded at 13.7% the compressive capacity. Still the moment due to eccentricity is more or lss 30% of the moment due to transversal loads, which looked too much.

I have not had time to look at the energy method yet as suggetsed by Structural/EIT, will give it a go later.


Miecz,

when I meant stating I am far from being a structural engineer I intended I do not have any experience of codes, subtitles, empirical formula, fudge factors, basically real world.

I can easily solve numerically a 4th order equation, but would not tell anybody to live under my beam as any ODE is a simplification. What if section are not plane, what if membrane loads are significant, what if linear kinematics is not sufficiently accurate?
That is why I wanted to hear your opinion.

The beam is not a critical component at all, its failure would result only in me being fired!

Many thanks sincerely to all for sharing some knowlegde, very appreciated.

Take Care

Favollo

 

RE: Beam in compression

If you have both Roark's and software use one to check the other, without being ashamed.
I certainly wouldn't touch a differential equation, 4th order or otherwise.
I've been given tools, they are widely accepted as being accurate therefore I use them.
Of course an experienced, critical look at the results is required.

RE: Beam in compression

Just thought i'd pop a thought in...

The use of software to solve situations such as beam columns requires a non-linear approach, and is greatly more complicated than a simple linear run.

 

RE: Beam in compression

RAM Advanse does a second order iterative, non-linear analysis.

RE: Beam in compression

but you will need several nodes along the length of the member as it only captures second order effect from nodal displacements, it doesn't capture second order effects from displacements between nodes.

RE: Beam in compression

Ussuri:  No one knows!  Tomfh is dead right in saying that this forum is not an appropriate place for Wooten's first and second laws, even if we don't know what they are.  Let's just say that Wooten's first and second law are more appropriately developed/proven between two consenting adults.

Sometimes I wonder if we don't have a major problem in our industry caused, for the most part, by software.  I have heard of engineers relying blindly on software and ignoring other evidential input as the project progressed.  Memory fails which, but there was a stadium case study at University which saw a partially erected roof frame system collapse DESPITE the fact that the engineers were warned by a number of project team members (foreman, steel erector, project manager) that the steel was deflecting too much...  And ignored it.  It seems that they were relying on a computer model that told them to expect the deflections.  They just didn't check that the connections could handle the rotation.  They couldn't.

Now, many engineers in my class pointed out that they missed a step of the design process, and that could have been missed in a hand based analysis as well.... BUT, our professor was quite right (in my opinion) when he pointed out that these mistakes are made easier by the laudanum-like effect of computer graphics presenting you with what looks like a ready-made, complete, solution inclusive of your stresses.

Apologies for the long post!

Cheers,

YS
 

B.Eng (Carleton)
Working in New Zealand, thinking of my snow covered home...

RE: Beam in compression

You have hit upon a gripe that has been going on for some time across manu industries.
One thing that stands out in your post was the engineers ignoring others vocal judgement. To do so requires either total confidence or just stupidity and arrogance. It seems the post has taken its own path rather than the op's intentions, however, my opinion is that you should always have an idea as to what you should expect in both numbers and behaviour if your going to do any software analysis (FE, Complicated mathcad, excel with macros etc). Otherwise how do you know if it reflects reality. If any of the engineers who work for me cannot do it by hand, i wont let them do it by black-box approach.

RE: Beam in compression

Hand calculation on small scale structures enhance the understanding of complicate matters. Computer analysis fasten the process, and make some impossible to been possible. At the end, common sense, or so called engineering judgement rules.

RE: Beam in compression

(OP)
Hi there,

this conversation is utterly interesting.

Without fear of being unpolite, I must though admit I am very surprised.
It has been suggested the beam colum should be designed by  as tructural engineers.
A few of you, has made their contributions and suggestions range from Roark's formula to non linear analysis, going through various intermediate proposals.

Is it really so difficult that no two of you, among which there are without doubt many experienced and skillful engineers without being sarcastic, have proposed the same idea?

A column with a distributed load and a compressive force? Non linear analysis?


Hope no one misunderstands how much I appreciate your contribution and truly respect any of you.

Best Regards

Muzialis

RE: Beam in compression

Muzialis,

Yes, we do tend to waffle on and stray from the original question.  The other factor is that people understood your question differently.  Perhaps if you gave the actual problem, including the section you are investigating along with the applied loads and support conditions, you would get a more consistent answer.

RE: Beam in compression

(OP)
Hokie66,

you are too right.
I will give some detail with the results I have found so far.

The beam column is loaded with a trnasversal load of 0.33 N/mm, span 8000 m, clamped both end, 4*10^5 N compressive force.

At the momemt I did some calcs for a section with I = 1.532*10^7 mm^4, stell modulus 200 GPa, yield 355 MPa.

Percentage compressive capacity used = 28%

Deflection without compressive force = 1.044 mm

Deflection with compresive force = 1.42 mm
(Found solving the 4th order Euler Beam equaition with EI constant, just to test the numerical accuracy).

Euler Buckling Load = 1.89 * 10^ 6 N

It seems to me the buckling load for the bema column is the same as for the beam, but I have top check this later on.

As soon as I can I will try the energy method.
Might try non linear finite element buckling analysis if I get time and will surely let you know for your imformtation.

Thank al ot to everybody, owe you a drink.

ATB

Favollo

 

RE: Beam in compression

Muzialis,

When you say "clamped", I assume you mean fixed at both ends.  That way, I get 1.15 mm deflection with only bending.  Are you sure the supports are fixed?  With simple supports, the deflection would be 5 times that much.

Based on your numbers, this member is bent about the strong axis.  I think the only buckling concern is in the other direction, and without knowing what type of member this is (wide flange, RHS, CHS), it is hard to know whether buckling will control.

RE: Beam in compression

(OP)
Hoki66,

thanks a lot.

The connection is a point I am struggling with.
the intention is to have it fixed through welding to a support, but there might be some deflection of the support as well.
Anyhow, in theory it is fixed.

The section is a squre hollow section, 150 x 1150 x 8.

My deepest gratitude for your comments.

All the Best

Favollo

RE: Beam in compression

Yes, fixing the beam could be difficult unless you have a massive column.  There will probably be some rotation.

By the way, my tables for a 150 x 150 x 8 SHS give the moment of inertia as 14.1 x 10^6.  Did you compute it?  If so, the radiused corners reduce this somewhat.

You only have about 14 MPa in bending stress, so this is a column with a bit of bending.

410 kN is too much for this section if the ends are pinned and there is no intermediate bracing, so you have to fix the ends for it to have a chance.  I would tend to use a larger section.  

RE: Beam in compression

I am skeptical about you having 2 fixed ends to this beam. supporting members will have some give in them allowing for a small amount of rotation.

I have been doing structural design for well over a decade, and I have never come across a situation where both ends of a beams are truly fixed.

I would try to err on the conservative side and maybe treat it as fixed at one end only.

Fixed both ends is something that only truly occurs in textbooks.

RE: Beam in compression

(OP)
Hokie66, Csd72,

both advices accepted, with one pinned connection I have to incresae trhe section up to 180 * 180 * 10 in order to saty in the region of 1 mm deflection.

Hokie66, yes I did calculate EI myself, did not think about the rounded corners..

Have a nice morning ( I am Europe based!)

Favollo

RE: Beam in compression

250SHS is more appropriate.

You will see a lot more than 1mm deflection with 150 or 180 SHS.

RE: Beam in compression

(OP)
Tomhf,

I do appreciate your comments but it was rather cryptic.

Which method did you use then?

Thanks al ot agin to all of you.

Take care

Favollo

RE: Beam in compression

Favollo,
your figure above for deflection in presence of compressive force seems reasonably correct to me: by using a sheet in the first site below and figures and boundary conditions per your post above at 19:18, I get 1.35 mm with compression and 1.08 without.
Also be careful with buckling: you are allowed to take a fixed-supported scheme only if the two ends are not allowed to displace laterally one relative to the other.

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RE: Beam in compression

favollo,

I assumed Span/1000 initial camber and d/2 load eccentricity.

RE: Beam in compression

When dealing with beams that subject to axial load, I always try to understand the source of the axial load and how the load is applied (from one end, both ends, in the middle). Also, what is the anticipated movement of the end supports with respect to each other (move in the same direction, opposite directions?) If the supports are been pushed closer to each other, doesn't axial shortening need to be addressed also? Just to throw some mud into this mix to see how it (this topic) develops.     

RE: Beam in compression

(OP)
Tomhf,

thanks for letting us know.

kslee1000,

you are very right but axial shortening is not too much of a concern given the application.
The end supports will tend to rotate in opposite senses.
the axial load is applied form both ends as a live load, in the sense that will always remain aligned with the beam (if the beam stays perpendiculor to its sipport).

I have as awell some more results for the both ends fixed situation, just to start with.

Roark's formula (Table 10,shear, slope, moment for beams with simultaneously applied compressive load and transversal load gives a dipslacement at mid span of 1.5 mm, rather close to what said so far.
moments and stresses match nicely as well.

And for curiosity's sake, a non linear  analysis with ANSYS using linear beam elements, 400 nodes, gives a deflection of 1.48.Timoshenko elements decrease the deflection to 1.45 mm.The buckling analysis is still running now, will check results tomorrow...

Many thanks to all the contibuters and have a nice weekend.

Regards

Favollo.
 

RE: Beam in compression

Favollo,
Is your allowable deflection for a 8 meter long beam 1mm?  I am kind of metrically challenged, but isn't that L/8000?  What is your application that has such a tight deflection limit?

RE: Beam in compression

Favollo,

While using Roark's case for fixed ends gives 1.5mm (I get 1.45mm), the case for pinned ends gives 37.6mm (did I do that right?).  I'd give close consideration to how rigid your supports are.

RE: Beam in compression

(OP)
Graybeach,

1 mm as a tolerance is not a limit imposed by anybodu, it is just the result that come out first and for some reason I am trying to stick to it.

The fcat is that all I know about elasticity was learnt at school, and if you apply a linear model is stricltly speaking correct only if deflrction are infinitesimal, which is clearly  a theoretical abstraction.

I do not have a clue about what a structural engineer considers acceptable, 10,5 or 1% a characteristic length.

The only criteria is not allowing buckling.

 

RE: Beam in compression

Favollo:

     Deflections are one of the most obscure and "gray" areas of Structural Engineering.  This is one of the areas where Structural Engineering is much more art than science.

     My best advice to you:  Consider your permissible deflection in terms of the length of your beam.  Then consider what you area limiting the deflection against, and choose an appropriate limit from the list below:

1)  You want to prevent people from seeing the deflections, and
     i) your line of sight is along the invert: SPAN/360
     ii) your line of sight is accross the invert: SPAN/180

2)  You want to prevent cracking of brittle cladding (glass, stucco): SPAN/500

3)  You want to minimize any possible vibration problems: SPAN/600 (VERY conservative)

Please note that a proper vibration check is needed (stiffness and source of the excitation are key to a proper check) if you want to be confident of minimizing vibration, however a good short hand is to check the following against EXPECTED (not factored or specified) loads:

Frequency (Hz) = 18/SQRT(Total Deflections in mm)

A result of 8 or greater is desirable, with 15 or greater being quite reliable.

Hope that helps,
Cheers,

YS

B.Eng (Carleton)
Working in New Zealand, thinking of my snow covered home...

RE: Beam in compression

Further to Youngstructurals post, in the small deflection theory of elastic bending, the limiting factor is the radius of curvature of the beam rather than related to the thickness of the beam, or span factor. Read timoshenko i recommend. You can also have large displacement but perfectly elstic bending of a beam depending upon such factors as material etc.
From a pure static strength point of view, the deflection will not be the critical factor, but may become so due to other considerations.
 

RE: Beam in compression

I think in this case the focus on deflection has become a red herring.  What Favolo has is a compressive strut, with bending only due to self weight.  The member required is dependent on the end conditions and whether or not there is any intermediate bracing.  It is similar to a horizontal strut in a braced roof plane.  I doubt that fixity should really be assumed.  The member should be designed as an 8 metre long column with pinned ends.  Many engineers ignore the self weight bending in these cases, and it only makes a small difference in member size.

RE: Beam in compression

(OP)
Youngstructural,

your post was enlighting.
I have never thought you people would limit deflections in order for people not to notice them, with even ad hoc factors subject on view angle. Dead crafty.

40818, I see your point which is of course evidenttly right.
This story about the displacements is coming from a post of mine which was rather unprecise, I want to limit the curvature not the deflection per se as well.The two being related explains my poor expression.

Anyhow, the all point of this conversation, by the way the most interesting, was how to calculate the section. Now the focus is on the design criteria, while it seemed to me any agreement on the calculaion method itself, leave the criteria, was far from reached.

However, assuming the calculation is fine, my criteria is simply avoid buckling, whatsoevr the displacements.
In order to use the calculation I propsed the curvature has to bee as small as possible though.

also in analysing buckling linearly deflections are undetermined, so there is no way of using a defklection as a criteria, if you alredy beyond the loss of stability.

I do apologize for the most verbous post and I hope to hear still from all of you.

Have a nice evening

ATB

Favollo

 

RE: Beam in compression

Favollo, it is true that the responses have gone all over the place, making for an interesting conversation.  This shows that your post is not really trivial.  Anyway, part of the art of structural engineering is knowing how far to go.  Instead of spending many hours trying to get the deflection absolutely correct, an experienced structural engineer might make some simplifying conservative assumptions and see how the deflection comes out.  If it turns out to less than the allowable, and as long as the assumptions were valid, you are done.

RE: Beam in compression

One more thing.  The checks to avoid buckling in a beam column involve, in ASD, calculating the stresses and checking them against an allowable that is calculated based on the properties of the section and the relative amounts of bending and compression, or the equivalent procedure in LRFD.  In the US, we do this based on the AISC code.  You must have something similar in your country.

RE: Beam in compression

(OP)
Graybeach,

needless to say I totally agree with the opinions presented  in your posts.

I am also sure in England there is a code equivalent to AISC, but I would not know where to look.

Anyhow the beam has been now designed considering the worst case between Roark's formulas and the Euler Beam column ODE.
Pinned ends have been considered for safety's sake.

The beam is being manufactured at the moment and will be used for one off event middle of July.

I will let this post's aficionados how events progressed...

Thanks to all and have a nice evening

Favollo

RE: Beam in compression

(OP)
Apsix,

you are kind in your irony, I would have accepted being called a fool rather than a brave.

I will certainly check the link at the first occasion.

Do not call the police, it is not a structural application and no damge to people could occur anyhow!

Kindest Regards

Favollo

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