1D beam modelling

[amarfayyad]

Hi Kevin, Thank you very much for your response. However, I think the deflection should be time-dependent under constant force. I have attached a simple calculation as proof and you can see the same in the 2D beam tutorial in which the beam is subjected to a constant load. Please correct me if I am wrong. However, the 1D beam model is giving a constant deflection and I am not sure why. I am under the impression that something is wrong with my sif file or with the 1D beam solver. For the circle case, the error I get is

ERROR:: CheckLinearSolverOptions: MUMPS solver has not been installed.

However, I don't get any error for the beam case just a seemingly erroneous result. If you could help with the beam case, that would be very much appreciated.
- Amar

[kevinarden]

The deflection is only time-dependent if it is ramped and not instantaneous. Some codes automatically ramp loads by default and other codes automatically default to instantaneous application of load. Some codes allow you to pick which one. Elmer is instantaneous, and you can only change it, if you define the load as a function of time.

If you do not have MUMPS, than you can change the direct solver to banded, or umfpack. Or change to an iterative solver.

[amarfayyad]

Hi,
Thank you very much for your response, Kevin. Solved the MUMPS problem. However, the deflection will be time-dependent either the force is time-dependent or constant because of inertia. If the code doesn't produce the same, I think we have to figure out why it doesn't. I have given a simple mathematical proof in my previous post about the same or think of a mass-spring system under gravity. We can also observe the time-dependent displacement under constant load in the Elmer 2D beam tutorial as shown in (a) of the attached figure. Elmer 1D beam model in my experience, acts quasi-static/quasi-steady. So, when there is a time-dependent force or prescribed time-dependent displacement, it predicts finite displacement else the displacement goes to zero. This is similar to evaluating a static equation every instant and not solving the dynamic equation. (b) in the attached figure shows the response of one node with respect to the prescribed motion of another node and (c) shows the response if the system is given an initial displacement. Unlike as shown in (c), the transient solver should have given an oscillation that stays forever in the absence of damping. Think of a diving board situation, if it's given an initial displacement, it should vibrate forever in the absence of damping. To make sure, this is not due to numerical damping, I tried different displacements. I quickly checked the source code to see that inertia terms are modeled. However, I am not sure, if it's a mistake in my sif file. So, if someone could help me with the sif file to model a 1D cantilevered beam that gives a time-dependent displacement for a given initial displacement/ constant force/ force that is absent after some time (e.g; F= 1000N for 2 seconds and then zero for rest of the time), it would be very much appreciated. The mesh and sif files are attached. Thanks a lot for your time and consideration.
- Amar

[kevinarden]

Your mathematical proof assumes the load is a function of time The computer code is not making that assumption. The applied load in computer codes are either instantaneous or it is ramped with time. Elmer assumes all loads are instantaneous, unless you provide the time function. Contrast this with ABAQUS which assumes all loads are ramped in a transient analysis, unless you say otherwise.

Since you are using direct solution, the computer is solving F=Kx, F is force, K is Stiffness, X is displacement, since the entire load is applied at time 0 (instantaneous) x=F/k therefore at time 0 the displacement is the entire displacement, and since F does not vary over time the displacement at every time step is x=F/k. F is constant, K is constant, therefore x is constant

If you want the load, even the gravity load, to ramp up with time, you have to tell Elmer it is ramped with time. You can make the load any function of time, but if you want to see the displacement due to gravity to increase with time, you must supply the load as a function of time.
Once you do then F(t)=KX(t) is solved, and x(t)=F(t)/K. F(t) is variable, K is constant, and therefore x(t) is is a function of F(t), but the usere mus define F(t), or else it is assumed to be constant.

An example SIF is attached.

case.sif

(2.18 KiB) Downloaded 28 times

[amarfayyad]

Hi,
Thanks again for your prompt reply Kevin. In the proof, the force is assumed to be constant and not time-dependent. The constant force should give a time-dependent displacement is what it proves.
It seems like with comparison of ramped/instantaneou/F=kx, you are talking about a static case and not dynamic case which should include inertia terms (mass*accelaration). I agree with your explanation on static case.
What I am trying to say is more like, if I pull the free end of a diving board or ruler with one end fixed, to have an initial displacement (Which replciates a cantilevered beam with initial conditon of displacement as in the case file I had uploaded) and leave it suddenly, in the absence of damping, it should vibrate forever. The frequency of this free vibration should provide insight into its natural frequencies. You can see 2D beam model doing it in Elmer and not 1D beam model. I am trying to see if someone can help me in this regard. Thanks.
- Amar

[mika]

There seems to be a problem that the beam solver has not automatically set the order of time derivatives. Therefore one has needed an explicit command

Code: Select all

Time Derivative Order = 2

in the sif file. I however updated the code as

https://github.com/ElmerCSC/elmerfem/co ... 2c4b2e02b6

so the correct value is now given automatically. As this is a second-order system with respect to time derivatives, setting

Code: Select all

Timestepping method = Implicit Euler

is not consistent. This might not make any harm when the time derivative order is set correctly. The only supported option for the second-order systems is

Code: Select all

Timestepping method = Bossak

Could you try running your cases with the time derivative order set correctly. If there is something else missing, then I need to make a second look.

-- Mika

[kevinarden]

Sorry I misunderstood your question. Of course if you define a time dependent load like pulling on the free end of a cantilever and let it go, it will vibrate. However, you must define that time dependent load and the code will solve F(t)=ma(t)+vC(t)+kx(t).

I thought this problem was solved in
https://www.elmerfem.org/forum/viewtopi ... cle#p27791

Perhaps there is more problems

[kevinarden]

Mika's suggestions, solve the problem

vib.PNG

(240.94 KiB) Not downloaded yet

case.sif

(2.13 KiB) Downloaded 46 times

[mika]

As another test, I now checked that if the initial deflection is such that the displacement component U 2 depends linearly on the coordinate X in the absense of external loads, U 2 at the free end starts to oscillate like this

end-u2.png

(26.85 KiB) Not downloaded yet

Probably a refined mesh would be needed, so that all eigenmodes which are needed in a reasonbly accurate eigenmode expansion of the initial displacement could be represented by the finite element approximation. The sif to obtain these results is attached.

[amarfayyad]

Hi,
Thank you very much for your prompt responses, Mika and Kevin. Even though the response is slightly improved, unfortunately, the problem is not solved yet. I used/modified suggestions/sif file to run transient cases for 1D beam. The response is heavily damped for the right area parameters (for a given dimension, the right cross-sectional area, a right moment of inertia, and right torsional rigidity assuming a rectangular section) and manages to get the less damped or sometimes seemingly right response for wrong area parameters ( Mis matched moment of inertia/cross-sectional area/torsional rigidity for given dimensions). Figure (a) shows the response from the right area parameters and Figure (b) from the wrong one. The respective sif files are also attached. Even though, the attached sif files are for the initially displaced system, the same trend can be obtained for the case in which force is ramped up and made zero. Both these cases are free vibrations and were supposed to vibrate forever with a magnitude of the initial displacement as there was no damping. Can you kindly let me know how this can be solved? Thanks again.
-Amar