Dear Michael,

Based on your description, it might depend.

Normally, if bending effects are the main effect, you might consider 1D beam elements based on the Bernoulli or Timoshenko theory. For this case, it is assumed that the length of the cantilever is significantly higher than the other two dimensions at the cross-section level. The option for Timoshenko or Bernoulli formulation depends if shear forces are an important issue in your problem or not, respectively.

However, if the the length of the cantilever is about the same 'order' (let's say) of the cross-section dimension, you might need to opt for 2D elements. In this case, the utilization of shell elements seems the most robust option (based on your description) since it is able to compute the displacements in X,Y and Z as well as rotations tetaX and tetaY.

All in all, and in my opinion, the geometry of your cantilever will dictate the best option. Based on my experience, you might model almost everything with shell elements however, this is not the most intelligent/smart approach.

In order to get more information about this, I suggest you to see in the Youtube: 'Lec 19 | MIT Finite Element Procedures for Solids and Structures, Nonlinear Analysis'.

Best regards,

HFMS

I'm using Matlab following a book "Matlab Guide to finite elements" by Peter Kattan. Looking at the pdf you sent I'm using the same set of equations. I think perhaps it may be because I'm doing testing with a horizontal cantilever and a vertical load on the end node. Calculating the displacements in a single step. If I load in small steps, recalculating the stiffness matrix at each step then I do start to get a small displacement in x. Is that the correct way to do things?

Does anyone know what sort of strains you'd expect this method to be accurate for? I was doing some measurements with a steel rule and seemed like I got deviations for even pretty small deflections (epsilon ~ few%).

I'm sure this is all pretty obvious to an engineer but I'm just trying to get some understanding of the methodology

Dear Michael,

Two things:

1) About loading in small steps:

This only is needed if you have a non-linear behavior of the cantilever material. You mentioned testing in a steel rule and I suppose that you are not expecting to produce permanent deformations in that rule (hopefully :). In this case, the problem is linear and therefore, the step loading approach is meaningless.

2) About the expected strains magnitude:

I would say that for a steel rule in elastic domain you would get something less that 1000x10-6 (and this is a very high level already). About the accuracy of the method, it all depends on the value that you fix for the modulus of elasticity, E, and the area moment, I.

2.1) Modulus of elasticity, E:

Indeed, this is the trickiest part of experiment, based on my experience. Just an example, in concrete structures you might have a coefficient of variation that might reach 20% for the modulus of elasticity, which is a lot (!) for the case of accuracy in the final results is the main criteria of analysis. For steel, the dispersion is usually lower, and you might have better results, i.e., lower errors.

2.2) Area moment, I

Another thing that might be very important is the bar dimensions (mainly the section height) because of the calculation of the area moment, I, that is proportional to the h^3! So, if you don't model with high precision the section height, you might introduce significant bias in the expected results.

Ex: steel ruler section height = 3 mm (the real value, suppose). If you model with a value of 2.9 mm you are introducing an error of (3^3-2.9^3)/3^3 = 10%!

If you take this two parameters correctly in your analysis, I believe that your results might be what you expect.

BR,

HFMS

Hi,

Your question cannot be answered without knowing the dimensions, material characteristics, and load. If the cantilever is stiff, you will practically see deflection in the direction of the applied load only. One way to check your model is to create one with small curvature and compare the results with the closed form solution of a comparable straight cantilever.

Dear Michael,

You have problem with large deflections of the cantilever beam. The easiest way to model such a problem is to take the co-rotational finite elements into consideration. You can find excellent formulation from Professor Yaw. You can download it from his homepage, i.e. http://people.wallawalla.edu/~louie.yaw/co_rotational%20papers.html

Best regards

http://people.wallawalla.edu/~louie.yaw/co_rotational%20papers.html

Dear Michael,

Your problem is related to the nonlinearity of your case. Helder is wrong saying that it is important only when plastic deformation occurs.

In basic formulation (probably used by you), FEM is linear. It means, that if you apply Y load in your case, you will get ONLY Y displacement, regardless value of load (so yes, linear FEM is in some sense "decoupling" Degrees of Freedom).

This is because of linear FEM assumption that deformation of the structure is small, and is not changing significantly the geometry of the beam.

If you want to capture change of the movement direction for increasing load, you have to use NONLINEAR solver, but not because of so called material nonlinearity (like plasticity mentioned by Helder) but because of so called geometric nonlinearity (mentioned above change of direction of displacement during deflection).

Nonlinear solver will do automatically something similar to your trial - will solve problem in many small steps (but with automatic control of the step length). In every step, the stiffness matrix is updated to resemble change in geometry. Every single step is linear, but full procedure in nonlinear.

Your elements are probably OK, problem is in solving procedure (linear - one step). Correct is nonlinear (multiple steps, called in FEM increments)

You can simulate the nonlinear procedure in simply way, just increase your load from zero to max in 10-100 steps. It is not efficient, but will work.

By the way, the name of element is strange, Plane Frame Element?

It is (should be?) probably so called Plain Stress Element. Don't use Plain Stain Element (it for completely different thing).

Surely it is not shell element, because shell has 6 Degrees of Freedom per node (works in 3D), not 3 (XYR) like yours.

I hope its simply enough and helpful

Artur

Dear Artur and Michael,

This is a good discussion.

About what you said, you are absolutely right. If large deformations are expected, so you will have also to take into account geometric non-linearity besides possible material non-linearity (if the case).

What is not clear, at least for me, is if this is indeed the case of the Michael's problem. What is the expected X displacement if we take as reference value the Y displacement (0.1%, 1%, 10%, etc.)? I think the problem should be tackled in this perspective before any considerations for the FE analysis approach. Without this first, we cannot say in a straightforward manner the correct answer. Of course that what Artur mentioned is absolutely correct, however the geometric non-linearity approach might be meaningless if we are in the domain of small deformations (i.e., the ratio between the maximum deflection and span length is very very low).

Even so, I should have mentioned that my answers are based on my experience on concrete structures, as which you might well know, small deformations are normally expected (the ratio of the deflections vs. span length is very very reduced) and normally geometric non-linearity might be disregarded, if compared with the displacements' magnitude due to bending effects. And in this context, we normally use a non-linear approach for the materials' behavior. This was not clear in my previous answers.

BR,

Helder

Helder,

my answer was based on the words

...only seem to get deflection in Y but with the other end tethered in the real world my cantilever should move in X and Y."

For my its enough. If Michael EXPECTED for real beam the X displacement, it means, that it probably is non-negligible, ergo - geometrical nonlinearity is involved.

Of course, this base is fragile, if Michael has completely no experience with elastic structures, but second evidence is a reaching some X displacement in a few small steps. This effect is not dependent on the Michael's lack of experience.

Michael,

pay attention, that for significant deformations in nonlinear analysis, there are two kinds of the load behavior:

1. the load direction can follow rotation of beam's end and will rotate too

2. the load direction remains constant (typical for gravity load)

Matlab is good for simply parametric study. If you want to carry out an analysis of a little more complicated structures, you probably will have to use professional FEM software or cooperate. It will save a lot of your time, because in FEM there is a lot of problems like described by you...

I think the nonlinear vs linear analysis part of the discussion is covered.

So I'll just add a little extra: if you use beam finite elements to model your curved beam, you should remember to update the cross section properties (e.g., bending stiffness) to reflect the curvature. The elastic axis of a cross section of a curved beam is placed differently from that of a straight beam. You may want to take a look at the free code by Wenbin Yu, GEBT (Geometrically Exact Beam Theory).

Good luck!

Hi,

In my opinion, curved geometry is best modelled by using a fine discretization of straight elements. If the cross section is small enough compared to the length of the whole beam, you can use simple beam elements to discretize. If the cross section is not small, then the assumption of plane sections will not hold and you need to do 2D modelling with 2D elements: either membrane elements (if all behavior is in plane) or a shell element if you expect out of plane behavior.

"Small enough" is a subjective term so some experimentation is necesary. In my eyes maybe a 20 to 1 ratio between length and section height would warrant using the 1D model.

Nothing better than writing your own code though.

Regards,

J

Hi

Following on from the last point.... there is a nice summary of an analysis of a curved beam using plane stress (i.e. membrane) elements here: https://confluence.cornell.edu/display/SIMULATION/Bending+of+a+Curved+Beam

Regards

Bryan

If you're working in the range of elasticity i.e. linearity then you might have to use the rectangular shell element. It is advantageous over the triangular shell element because

triangular elements have been found to have convergence problem and show singular behavior

over certain meshes (Reddy, 2006). The elemental equation of the rectangular shell

element can be derived by combing the 4 nodes linear rectangular element for plane elasticity (membrane element) having 8 degrees of freedom with the bending element derived using the classical plate theory that has 12 degrees of freedom. Hence the element for this analysis will have 5 degrees of freedom per node making a total of 20 degree of freedom per element. My publication will be of help since it is also worked out in MATLAB. 'Finite Element Analysis Of Von-Mises Stress and in- plane displacements In Ellipsoidal and Circular Cylinderical Petroleum Tankers'

It seems as some contributors suggested that the solution to my question was simply to do the loading in several smaller steps. At each force increment you recalculate the global stiffness matrix. Even my very simple model worked very well when compared with an experiment. Thanks to all for their suggestions.

Michael,

One thing you also might try is to use three note frame element that is more accurate capturing the frame curvature. Also in your case the element length might be a factor. In all cases regardless you do linear or nonlinear analysis you still have to get deformation in X and Y direction. All suggested contributions help to improve the accuracy but not having deflection in X direction might be a result of something in the BC.