Analytical Solution versus FEA solution.. 1

[ONEPOINT]

I haverecently performed a FEA on a 100 mmm long cantilevered beam with a rectangular cross-section(Width 10mm and Height 20 mm). A load of 10N was applied downward at the free-end and the constraint was applied on the other end(fixed surface).The material was A2014.

I have used this model in order to explore the maximum/minimum stress values, displacements and strain energy. I have gotten the results for a Multi Pass Adaptive run with 5% convergence.

Now, I need to compare these FEA results with the analytical solution.

The question that I need to answer is if the analytical solution is "exact"? and Why?
I would appreciate any technical suggestion in regards with this comparasion.

Regards,
OnePoint

 

[rb1957]

onepoint,

the analytical solution is exact in as much as the underlying assumptions (small displacements, plane sections remain plane, elastic stress/strain) are obeyed. look over the derivation of the analytical solution, you should find more.

the FE solution is approximate in that it relies on calculations done at discrete points (rather than a continuum). as the formulation of the elements includes assumptions (possibly most significant is linear strain distribution). if you understand the limitations of the elements you'll be able to design loads such that the FE and the analytical give the same answer.

 

[Kwan]

I've run a solid mesh cantilever beam similar to Onepoint's. The results of the run showed a higher stress level a few elements away from the "fixed surface".

I worked out the issue with my beam. Fully fixing nodes at the surface will produce results different than expected. The nodes must be constrained in the direction of the applied load. Then pick a node to constrain the other directions.

 

[JStephen]

Way back in college, we did a lab that involved bending a bar about 1" x 1/8" the easy way. I did not know, and found out later, about an approximation in beam theory that affected the results. Specifically, beam theory assumes that all parts of the beam are free to deflect laterally. In a wide flat bar, this is not the case, as the compression elements are tied to the tension elements. As I recall, it makes a difference of (1-nu^2) in the deflection result (does not affect stress).

Now, in your case, you've got the bar turned the other direction, and this effect would be much smaller, but it is not necessarily all gone, either. Just one of the approximations used in the "exact" solution.

 

[GBor]

There are MANY things to consider when modeling a beam. Did you use beam elements? Or did you build the beam out of plates? If beam, did you include shear areas or not (may affect the comparison)?

I've done similar models MANY times for ISO certification (Internation Standards) and have received answers well within 1%, but you have to make sure that the analytical solution to which you are comparing is precisely the model that you've built. Linear analysis for small deformation, "0" shear area for non-Timoshenko beam equations (Timoshenko is the one that showed for deep beams that the assumption that "a line perpendicular to the flanges of a beam remains perpendicular" is invalid -- beams are shear-deformable)...a "simple" beam is not so simple until you understand what is going in to the analytical solution.

As far as "exact"...seems like a sound engineering term, but even the most well-known equations are still based on empiracle data (observations). There is still some "scatter" among the data, so the reality of an "exact" solution is unachievable, technically.