# bending behaviour of a cantilever beam modeled by shell elements

Hello Everyone,

I have a question and I really appreciate any comment:

I have modeled a cantilever thin beam under transverse bending(load is applied at the end of the beam in the direction of thickness of the beam) with Belytschko-Lin-Tsay shell elements.I am using only one shell through the thickness.This element uses 5 integration points through the thickness and uses reduced integration(1 point) for in-plane+hourglass control...The problem is that the stiffness which I get from the FE Analysis with an explicit FE software,has 25% error compared to a Bernoulli-Euler beam analytical solution....The interesting point is that when I apply in-plane bending to the same beam(load is applied in the direction of the width of the beam),I get a very good agreement with analytical solution....

Is it related to the element which I am using?I thought maybe it is not able to capture the exact transverse bending behavior of the beam,but I cannot figure out why??

Any input is really appreciated

Setareh

## Comments

### I am still waiting for any

I am still waiting for any input.....

thanx a lot:)

### Let me assume you are doing

Let me assume you are doing a linear analysis. If both the width and the thickness are smaller enough comparing to the length, you should get comparable results between a beam theory and a plate/shell theory for the behavior you are looking at. If the thickness is much smaller than both the width and the length and the width is comparable to the length. You beam theory will fail as you cannot model it as a beam anymore.

If you want to dive into more technical details, your analytical beam formulas require unixal stress assumption: all the stress components in the cross-sectional plane vanish, including the normal stress along the thickness. These assumptions are realistic only if both the width and the thickness are much smaller than the length. Your plate/shell assumes normal stress along the thickness direction equal to zero. However, your normal stress along with width can be nonzero. This is valid if the thickness if much smaller than the width and length.

### Dear Wenbin Yu,thanx a

Dear Wenbin Yu,

thanx a lot for your comment. You are right, but the width and thickness of the beam are small comparing to its length..

But I think I have now understood why I get some 25% error in comparison to the Bernoulli beam theory. It seems that Bernoulli-Euler beam analytical solution is only valid for small displacements,while in my FE simulation,the beam goes under rather large displacement although it remains in the elastic region (deflection at the end of the beam is 5% of the length of the beam),so there would be an error w.r.t analytical solution!

I hope my understanding is right!

Thanks again for your clarification and comment!

Setareh

### Dear Wenbin Yu,thanx a

Dear Wenbin Yu,

thanx a lot for your comment. You are right, but the width and thickness of the beam are small comparing to its length..

But I think I have now understood why I get some 25% error in comparison to the Bernoulli beam theory. It seems that Bernoulli-Euler beam analytical solution is only valid for small displacements,while in my FE simulation,the beam goes under rather large displacement although it remains in the elastic region (deflection at the end of the beam is 5% of the length of the beam),so there would be an error w.r.t analytical solution!

I hope my understanding is right!

Thanks again for your clarification and comment!

Setareh

### Several things you need to

Several things you need to do to verify whether you figured out the real reason:

1. Verify whether you used linear element or nonlinear element in your finite element simulation. It really does not matter how large the deflection looks, if you used linear element in your FEA, and your beam is slender, you should be able to get a decent match.

2. Verify whether you FEA results changes with refinement of your mesh. Take the converged results as the true solution.

3. If you used nonlinear FEA, you can compare it nonlinear solution for large deflection beam. For a cantilever under tip transverse load, you can find the analytical formulas for the nonliner solution in Timoshenko's book. Gere, J. M., Timoshenko, S. P., 1990. Mechanics of Materials, 3rd Edition. PWS-Kent, Boston, Massachusetts. Chapter 7, deflection of beams.