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# How to constrain a closed ring beam element for 90, 180 degree symmetr

## How to constrain a closed ring beam element for 90, 180 degree symmetr

(OP)
How to constrain a closed ring beam element for 90, 180 degree symmetry cases under external pressure.

I need help to define the following simple constrain beam case

I got a "circular  closed ring" with a square profile 25x25 mm and an mean radius of 518 mm, material steel, the ring is has a external pressure of 100 N/mm2 as example.

I would like to simulate this case using a FEA software, as a 1D beam element.

But I would like to use 3 different constrains sets (case A: 90 degree, case B: 180 degree symmetry) and case C: the full 360 degree model.

My intention is to compare results for buckling pressure values, (so I took this 3 cases to evaluate non symmetrical buckling cases -90 and 180 degrees- with regards to the 360 full model.

My question is how to set the constrains for this 3 cases.

Any help will be welcome

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

For a symmetric structure with symmetric loading, you must apply symmetric boundary conditions at the edges of your model.  This means you restrain the out-of-plane translation and the 2 out-of-plane rotations.  (This allows motion in the 2 in-plane translations and 1 in-plane rotation.)  With only symmetrical loading (external pressure), I don't think you'll get information about non-symmetrical cases.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

You should get the other 2 symmetric cases just by running the 360 degree case.

Tata

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

(OP)
I get solution with case A and Case B, but the 360 degree case, How do I do?.. I am fixing translation and rotation for X, Y, Z in just one point, I get the same stress in all 3 cases, but only deflexion in case A and B are the same, for case C not,,, so when I do a buckling analysis I get diferents answers for case C.
How do I need to model the constrains in case C to get the right deflexions?

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

does your symmetric boundary condition for case B = case C ?

what do the deflected shapes look like ... plot a whole circumference ... case C, case B (replicated at the symetric boundary), case A.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Now I see your problem with the 360-degree model.  Properly restraining rigid body motion with a ring structure can be tricky.  But your solution with all d.o.f. at one point seems to be OK since you did get the same stress result (and I assume the reaction forces were near zero).  The only difference in displacement would be a rigid body component.  (It's not clear to me how a rigid body motion would affect a buckling calculation.)  But if you want to get displacements for case C that also match cases A and B, you can apply restraints that are similar.  For example, restraining X at 0 and 180 deg, Y at 90 deg, and Z at 0, 90, and 180 deg would take care of the rigid body restraints and should give you symmetric displacement results.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Don't apply any restraints to case C. The first 6 mode shapes will be zero for the 3 translations and rotations. The 7th shape will be the 'first' buckling mode shape.

Tata

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

(OP)
CORUS: Well, the only way for a model to run without constrains is for modal analysis only, so I will agreed that the first 6 modes will give you the 6 rigid movements, but for a Buckling analysis with not constrains I don't see how to do it (and I will guest is not possible).

I am looking to do a buckling analysis for a more complex shape but I would like to verify first with a simple model the results from pro/mechanica so I would like to verify the models constrains to compare with theory (P= 3.E.I/r3).

I have been playing with constrains options, BUT still I don't get the right constrains set to allow a single ring under external pressure to behave as theory say.

Any other suggestion hoe to constrains this model?

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Mark,

"Buckling analysis with not constrains I don't see how to do it (and I will guest is not possible)."

You are quite right. Although modal and buckling analyses are both eigen problems, there is no equivalent to "free-free" analysis with a buckling problem, the structure requires sufficient restraints to prevent rigid body motion. Also a buckling analysis can produce both positive and negative eigen values. Ignore the negative values, they are mathematically valid solutions, but do not represent any real physical solution.

Why don't you model 3D shells or solids instead of 1D beam elements? How is the "real" structure that you are modelling actually supported?

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

(OP)
Hello John
I am trying to model a fuselage section, basically simple assumed as a one support ring and a cylindrical shell skin under external pressure. (like a submarine)....

I want to see the stress in the ring, as you could image, the ring is free in the space but supported by the shell skin..

Basically I want to prove my model with a basic ring model, but I cannot get the way to model such simple structure (the ring) and compare with theory.. when I apply the buckling equation 3EI/R3 I get a buckling pressure that do not correspond with what I get in my FEA.

I believe this equation is only for in-plane shapes deformations.. but my FEA is given me in-plane and out plane deformations,, those values for the in plane shapes modes do not mach the theory one.. so what to do.

Dimple model:
Square section 25x25mm
E=193053 N/mm2 poisson=0.3
Inertia: 32552.08 mm4

Theory buckling pressure: 150.8226 N/mm

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Have you compared the theoretical buckling pressure against your applied pressure multiplied by the eigen value that the FEA gives you?

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Please can we clarify some basics?

You quote the buckling equation where pressure = 3EI/r^3. In this equation should I be Inertia per unit length?

You state that the material is steel with E=193053N/mm^2. Surely these cannot both be correct?

You quote a theoretical buckling pressure of 150.83N/mm. Shouldn't pressure be in units of N/mm^2, and how do you get the value of 150.83N/mm with the input data you show?

The Timoshenko equation will be the elastic critical buckling pressure. Is your FE analysis an eigenvalue buckling analysis, and not some sort of nonlinear collapse load?

Is Pro/Mechanica a suitable tool for this task, with an adequate range of analysis options and restraints?

If you wan't to emulate the Timoshenko equation will you not need to provide restraints out of plane?

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

E = 193 GPa is reasonable for steel, = 28E6psi

there's definitely a dimension thing happening 3EI/R^3 doesn't give you pressure.  my guess, without checking calcs, is that gives your running load = pressure*width, which is why the answer is N/mm, but i get 18.85 N/mm ... suspiciously 1/8 of 150.83 ??

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

See http://www.environmentaltechniques.com/rr2_06.pdf

which shows the derivation of this formula.

A 180 degree model should suffice to get the first buckling mode.

Note, in the first post "and an mean radius of 518 mm" was quoted, now it is 1000 mm.

As Crisb has asked "Is your FE analysis an eigenvalue buckling analysis, and not some sort of nonlinear collapse load?"

"I want to see the stress in the ring" - An Eigen solution will not give you an actual stress value. You get an Eigen value (which equals critical load when multiplied by applied load) and an Eigen Mode shape which is always dimensionless whether you are doing modal or buckling analysis. You can of course from the mode shape derive a stress "pattern" (as many solvers do), but not a value at any one point that you can quote.

But, a static non-linear geometry solution could give you a collapse load and stresses as you increment up the applied loading.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

(OP)
Thanks all.

Sorry if I mix up some numbers.
Let fix the case for future reference to R= 1000 mm .
Square section: 25x25 mm
I: 32552 mm4
Yes, material is steel E=193053 N/mm
Load case: unitary external force 1 N/mm (since my benchmark model is beam element)

Timoshenko buckling pressure: 18.85 N.

My FEA is simple lineal buckling calculation, no exotics cases (large deformation, non lineal cases etc) are being taking into account.

I just one to see from my FEA a buckling factor of 18.85, which I don't get it.. my FEA shows values of  81 as a buckling load factor.

John, I agreed with you, a modal analysis will not give you stress, I just do it to verify the modal shapes and to check any rigid motion in my constrain assumptions.

I run a static analysis and from those results I run a buckling analysis.

I would like to see, if somebody has a FEA tool to verify this simple case I am posting, so we can verify constrains set etc.

I am wondering, if the Timoshenko buckling pressure for a ring (3*E*I/R3) is just for "in plane" deformation cases, my model give me in plane and out planes buckling cases, so I guest this equation was developed from a 2D analysis (strain plane analysis), so if is so, how to get the out plane buckling calculation from hand calculations? For me to compare with my results... (thanks for the paper, I had it before).

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Mark, what software are you using and the element type in that software?

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

I ran this and got the lowest mode with a load factor of 25.1. To get this I had to use isobeam elements which model the beam curvature within each element. Using straight beam elements appeared to give spurious results, however small the mesh. To avoid spurious negative modes I found it better to apply the pressure externally.
Maybe this is a tough test for analysis methods and element formulation?

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

(OP)
software is mechanica from proe WF4.
element a beam element with 25x25 section.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

(OP)
crisb,
Could you let me know what are your constrains for your model.
did you modeled as a full 360 Degree model?.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

180 degree model in Adina. Semi circle restrained out of plane. One end point fully restrained translation and rotation, other point also fully restrained except free to move towards first point. Relevant mode shapes similar to above paper. 4 node elements with 50mm mesh, LF 25.3 with 2 node elements 100mm mesh.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

mark
first, "pressure" is N/mm^2, not N nor N/mm.  Timoshenko is calc'ing pressure per unit width (i believe), leaving it as a running load (N/mm) is fine as it matches the load you're applying.

2nd, how are you applying a radial pressure load to a beam element ?  maybe ProE allows this; in a model i'm working on now I wanted to do the same thing and settled for adding dummy skin elements that would take the pressure (normal to the panel = radial) and apply it to the beam elements.

3rd, "I run a static analysis and from those results I run a buckling analysis" ... what does the 2nd part of the sentence mean ?  what "buckling analysis" are you running with the FEA stresses ?

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

(OP)

Thanks rb1957
To be more clear:
1) Load is: newtons per unit length ( N/mm)..not pressure (let try not confuse with my original case post..let run the simple case as described before).

2) Before to run a skin model for load distribution, let set the model for as simple ring as described before.

3) to run a buckling analysis, a static analysis need to be performed (at least with pro/mechanical), so the buckling load you get is times the load you set in your static analysis.

CRIBS:
180 degree model...OK
Semi circle restrained out of plane.. you mean you no out of plane movement and rotatrion for your semi circle...so you pick all your semi circle and set rotations for x,y,z fix.?

One end point fully restrained translation and rotation... you mean point fully fix

other point also fully restrained except free to move towards first point...you mean only translation movement towards firt point allowed and rotation in X,Y,Z are fix?

Theory: 18.85 you get 25. 3.

I try to do it and I don't get the same, could you let me know what are the frequency of your model ( just for me to verify) and what is the stress for your static analysis?.. does make sense the deformation shape of your static analysis for this load?.

How can you get those symmetrical shapes for buckling when you got a fully fix point?.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Mark:
Semi circle restrained out of plane...y translation only.
One end point fully fixed....yes
Other end point fully fixed except z translation.
Theory 18.85 I get 25.1 (smallest mesh attempted).
First mode shape attached.
I think but am not certain that the element type could have a lot to do with the results, am doubtful that standard straight beam elements will give you good results.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

My model is getting the critical buckling pressure very close to 4EI/r^3, so there is probably a boundary condition problem that I haven't figured out!

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Cosmos verification manual problem B5 is very similar to this problem, and is showing the theoretical solution as 4EI/r^3. Any comments?
http://www.stresscalc.ru/cosmos/BasicSystem_2.pdf

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Cris, the cosmos example is for 90 degrees only.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

This is all very interesting.

Firstly considering only the in-plane buckling:

A few years ago I did an investigation of restrained buckling of pipes (I have attached a paper I did on it, although it only mentions unrestrained buckling in passing). In the course of that I compared Euler's solution with FEA solutions, using Strand7.  I found that the "linear buckling" solver in Strand7 gave a solution very close to 4EI/R^3, rather than the theoretical solution of 3EI/R^3, but if I did a non-linear analysis, applying a very small additional point load at one node to initiate buckling, I got a buckling load close to 3EI/R^3.

I have just re-run this analysis with the section properties quoted above, and get the same results; i.e.:

Non-linear analysis buckling load = approx 18.9 N/mm
Linear buckling load = 25.16 N/mm (4EI/R^3 = 25.14 N/mm)

The FEA model was a full circle modelled with 80 beam elements, with global freedom conditions set to "2D beam", i.e. Z displacement and bending about X and Y restrained. The top and bottom nodes were fully restrained except for Y translation, and the mid height nodes were restrained in the Y direction. A quarter circle model with the same restraints would have been equivalent.

I never worked out why the linear buckling analysis gave different results to the buckling equation and the non-linear analysis, and I still don't know, but it interesting that the COSMOSM results are similar, and they quote a theoretical value of 4EI/R^3. I'm not familiar with the Donnell Approximation.

If I allow out of plane deformation, from the linear buckling analysis I get two buckling modes at 15.3 N/mm, then the third is the in-plane buckling mode at 25.16 N/mm.  With the non-linear analysis I get out of plane buckling at about 8.8 N/mm.  At the moment I'd recommend going with the 3D non-linear results, remembering that a small out of plane force or initial deformation is required to initiate buckling in the analysis.

If anyone can shed any light on the difference between the linear buckling analyses and the non-linear analyses, I'd be very interested.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Looking at the thread in the Mech Eng forum:

if this is for a stiffner to a cylindrical pipe subject to external pressure, the pipe will provide significant restraint to out of plane buckling, so taking the fully unrestrained case would be very conservative.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

IDS: I am wondering if the difference could be the application of load and the 3* is if the load is normal to the deformed shape and 4* is in the original undeformed direction. Presumably your nonlinear analysis is a traditional nonlinear analysis and not an eigenvalue buckling analysis. This is obviously now becoming a highly theoretical topic, see following paper as an example of different factors quoted over the years:
Johnhors: The first mode shope is doubly symmetric so should be identical to the 90 degree mode.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

(OP)
I don't get these buckling load ( 209 !!!!).
and I habe been playing with all BC possible.

and including 3D solid model.

I tried to do it and I don't get the same, could you let me know what are the frequency of your model ( just for me to verify) and what is the stress for your static analysis?.. does make sense the deformation shape of your static analysis for this load?.

for the out plane buckling I get around 59 ( which comes as the 1 buckling mode)...the 209 is the buckling for the 1 inplane mode shape.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

I haven't run natural frequency or static analyses. I don't think its helpful for us to compare freqencies, as this is introducing another variable (mass or density). The static stress can easily be checked from simple hand calculations.

The usefulness of simple hand calculations as a sanity check has been "stressed" to other posters many times recently but seems to fall on deaf ears.

Please check carefully your geometry, section properties, and modulus of elasticity, and perhaps post a picture of the first mode showing elements, loadings, and boundary conditions for the in plane mode.

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Mark - your numbers keep changing!  Could you post a sketch of your simplest model showing dimensions and end-restraints, and summarising the buckling loads you get for different modes and methods of calculation.

crisb - do you have a simple hand calculation method for out of plane buckling of a circular ring?  If so, could you post it.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

#### Quote:

I am wondering if the difference could be the application of load and the 3* is if the load is normal to the deformed shape and 4* is in the original undeformed direction.

That seems to be the case.

If I do the non-linear analysis with point loads directed towards the centre of the circle at each node I get a buckling load of about 25.3 N/mm, which is in good agreement with the linear buckling analysis and the 4EI/R^3.  The earlier analysis used distributed beam loads, applied perpendicular to the beam, so the load direction would be adjusted at each iteration.

As you surmised, my non-linear analysis is simply a static analysis including geometric non-linear effects, with the load incremented in small steps until the response becomes clearly non-linear.  The linear buckling analysis is an eigenvalue analysis.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

### RE: How to constrain a closed ring beam element for 90, 180 degree symmetr

Doug: Equation 146 in the following reference appears to give an expression for the out of plane buckling pressure on a ring, which seems to be attributed to Wah. I just got it from Googling and am not commenting on it's accuracy.
http://archives.njit.edu/vol01/etd/1970s/1979/njit-etd1979-005/njit-etd1979-005.pdf

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