Yep. I think so.
I think the reason undelays below the fact that the premises used to derive the Euler’s critical load fail for small slender ratios.
Plasticity accentuates even more the Ncr reduction.
Iy in Stahlbau examples is wrong, 2 * 40 * 2.4 * 21.2^2+(40 * 1.2^3)/12 = 86298 != 92692 cm^4. That gives Pki,y = 14782 kN, not 15877. I guess difference can be attributed to modelization issues not to a more complex behaviour like inelastic effects. Euler buckling formula is based on Berouilli’s beam theory and 3D FEM is even beyond Timoshenko’s beam theory in the representation of elastic behaviour. Then simplified boundary conditions can make the beam a bit stiffer in the bending plane…
There is an error in the developement of the formula but the result is “almost” right.
Exact value is :
Iy= 92.784 cm4
Pki,y= 15.893KN
I still think that the discrepancy is attributable to the modelization and not to a more complex behaviour like inelastic buckling.
i have done the Eigenvalue Buckling Analysis with a commercial 3D Frame Analysis Software.
with 1 beam element and the properties of the beam cross-section in the paper.
for Iz (25.606,0) buckling factor is 4406,0 kN = 0,49 %
for Iy (92.785,0) buckling factor is 15430,0 kN = 2,91 %
than i have divide the beam in 15 parts:
for Iz (25.606,0) buckling factor is 4371,0 kN = 0,25 %
for Iy (92.785,0) buckling factor is 15259,0 kN = 3,99 %
wbr
Excellent paper of the evolution of the different methods to evaluate the Buckling load, from Euler to the actual methods included in modern design codes like Eurocode3.
Some graphical references may help to explain the discrepancies shown in our results between the “simplistic” Euler formulation and FEM approach.
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Hello Disla,
thank your for the paper.I’ll check these out.
I have two issues or definitions:
Euler’s critical load, depending on:
- Young’s modulus,
- second moment of area
- length of column
that’s it. nothing more.
And the second issue or definition ís
load bearing capacity, depending
- stress
- Euler’s critical load
- and so on
for the formula of Euler’s critical load
it makes no difference between S235 or S460.
but for the load bearing capacity it is a variable
For me, 100 % is the Euler’s critical load, not the fem result,
like you propose it or use it.
i have a connection with a person from university and we discuss these issue.
I’ll give you an update if a have a better understanding or information.
update:
the paper is very interesting and with a lot of information.
I’ll check and try make the line, where stress is an issue!?
i have a common issue,
these is
- plate buckling (includes shell buckling with rotation structure like cylinder and so)
- lateral buckling (these issue)
- torsional buckling
- lateral torsional buckling
wbr and thnx for your support!
Hello Dichtstoff ,
I think your split of the problem is the correct one.
Euler’s critical load formula is simple and depending only on three variables (Young’s modulus, second moment of area and length of column) but we can’t forget there are 10 previous assumptions underneath (wiki) that should be satisfied before we can apply it. We certainly could use it but the result may deviate from reality.
I agree load bearing capacity is a second step but highlights the importance of delimiting the slenderness.
In the particular case of EC3 the nondimensional slenderness must be inside a specific range before evaluating the load bearing capacity.
It warranties the applicability of the formulas.
Nondimensional Slenderness < 0.2 Buckling effects may be ignored
Nondimensional Slenderness > 2 intolerable slenderness in the main elements (This value differs depending on the code)
Nondimensional Slenderness > 3 intolerable slenderness even in bracing elements (end of the Buckling Curves for EC3)
(NOTE) In our case the slenderness associated to the second buckling mode is way below the recommended Lambdap shown on your picture.
Let me know if you progress with this and find the particular origin of that small deviation (4%). I would say it could be related with the shear that Euler’s theory discards.
Regards
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just found this, might be interesting as well
a video of lateral buckling test, with data: I Beam - Lateral Torsional Buckling Test - YouTube
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Btw. do you know if there’s a difference between flexural torsional buckling and lateral torsional buckling ? Or are those just 2 names for the same thing ? The nomenclature in my country is really different.
probably is the same. I never used flexural (torsional) buckling. In the literature in English I think lateral is the usual term (is the one used by Timoshenko in his book on elastic stability) however I can’t say for sure, English is neither my mother tongue. In Aerospace literature the usual term is lateral buckling (lateral-torsional if combined), also might be is the usual term in other fields not familiar to me.
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To my understanding, mostly it’s related to unsymmetrical in load and/or section shapes of compression zones. Torsional and lateral buckling is a different things. When some text denote lateral torsional buckling it has meaning a combination of both.
I think good answers starts with the right questions.
Let’s say we want to validate our FE solver and find the right way to set up our BC when searching for the buckling modes of beams (Shells,…).
1-Question: ¿ Is it appropriate to compare our Finite Element Method results with the Euler Formula for Buckling analysis of Beams?
2-Question: In case it is, ¿Must both methods ideally agree no matter what or should we compare only inside a specific range of geometry, Poisson’s ratio, Slenderness, Cross section,……?
3-Question: In case there is an acceptable range, ¿can we use beam , shells or solids indistinctly to solve the problems?
I throw myself into the pool anticipating my answer:
From my point of view, FEM can only compare Euler’s buckling formula in a limited range in which compressive stress is very small compared to bending stress and for specific slenderness Ratios. To state the right procedure and validate the solver I would firstly choose beams (shells) inside the appropriate range. Once the solver and set up is validated I would go outside the range with more confidence. I don’t think the buckling mode on strong axis of our beam is inside the range where both methods can be compared, and the deviation is completely normal.
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hello,
here is a result of a commercial software that uses only “pure” line elements. the result is nearly exact with only 1 element, but exact with 11 elements.
there is another software (from the same company), that use line, plane and volume elements, and then the result of the line element is same like in ccx:
pi^2 x 21000 x 92785 / 1100^2 = 15893.2
Let me know if you progress with this and find the particular origin of that small deviation (4%). I would say it could be related with the shear that Euler’s theory discards.
yes, these could be the reason, Bernoulli-Euler Beam Theory vs Timoshenko beam Theory!?
but I’m not sure if these explains the different between the both software results.
the beam elements theory are equeal.
wbr
I think we have a lot of points in common. Let me know if you agree.
1-Answer: Yes
2- Answer: No. We should compare only inside a specific range.
3-Answer: No. Euler buckling load is derived according to the Euler-Bernoulli beam theory where shear deformation is neglected. Solving buckling problems with shell or solid formulations may provide different results as they take shear into consideration.
Note: Your example for the commercial solver using just pure beam elements doesn’t show the second buckling mode (Pure torsion) 10.488. It is a shear mode.
¿Which beam formulation is using that commercial software?. ¿Is it a Euler-Bernoulli beam? It would make sense it skipped that mode.
the reason for the two different result is the shear stiffness of members.
there is a option for using shear stiffness of members.
thnx a lot for your support
wbr dichtstoff
I think that what you get with more simple methods is not taking into account that the Euler buckling mode in not the critical for this beam, so CCX provides you a solution that is Euler with some torsional effect (small), not a pure bending mode, therefore giving a lower value. In that sense CCX is more accurate. If same calculation is repeated with a slender beam with thicker cross section where the Euler buckling mode is the first buckling mode, then both solutions will be much closer, when the cross section is closer to a thin walled beam you don’t get pure Euler buckling modes and hence the result. Also Euler and torsional buckling are close enough so you don’t get a pure eigenvalue but some mixed mode. It is a usual approach to check for interaction when buckling modes are close.
HI JuanP74,
The three first buckling factors on the compressed beam under consideration are:
| Buckling mode | Calculix | Δ (Stahlbau III) | Stahlbau III/ Euler |
|---|---|---|---|
| 1-Lateral buckling weak axis. Pure bending. | 4,371 | 0.32% | 4,385 |
| 2-Torsional buckling . Pure shear/Torsion | 10,489 | 0.59% | 10,427 |
| 3-Lateral buckling Strong axis. “Pure” bending. | 15,239 | 4.02% | 15,877 |
1-The first agree well with the Euler’s predicted Buckling mode. The beam bends easily around the weak axis without web shear development. (0.32% agreement)
2-The second mode is a pure shear mode. The beam element (and Euler theory) can’t see it. Shell and solid elements capture it properly.
3-The third mode, bending around the strong axis, requires a significant development of shear on the web. This is the one under debate and its 4% deviation according to the second Euler Buckling mode value.
I would say your proposal is perfectly compatible with what dichtstoff and I stated above if “shear” and “torsion” are considered related.
Regarding your video, thanks for the link , it is very interesting but there is something I don’t like, and it is very common when talking about the subject.
Lateral torsion fenomena is commonly associated with the upper flange in compression who tends to buckle and the lower flange in tension who tries to stabilize.
You can easily find lateral buckling on a simple plate without flanges.
I think that the shear component on the web plays a significant role on all this subject.
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maybe I misunderstood you because of this:
Let me know if you progress with this and find the particular origin of that small deviation (4%). I would say it could be related with the shear that Euler’s theory discards.
yes, these could be the reason, Bernoulli-Euler Beam Theory vs Timoshenko beam Theory!?
bernoulli and timoshenko beam theories do not differ in torsional shearing but shear due to bending, so I thought you were talking about shear from bending…I think the way to check the origin of the discrepancy is to analyze a beam that has very separate buckling modes so they cannot interact.
regarding your remark about the video you are right, you can have lateral-torsional buckling if the beam is just a vertical web. I apologise for that, as I said, not my field of knowledge.