Finite difference and finite element analysis of beam-columns

[LRJ]

I appreciate that the finite element method (FEM) has considerable advantages over the finite difference method (FDM) when analysing 2D and 3D problems. However, for 1D beam-column problems I am not aware of any publications considering or comparing the differences between the methods.

What are the thoughts of engineers here regarding the two methods? Moreover, does one method produce more accurate results than the other method? If so, why?

My inclination is that FEM will provide more accurate results given the assumptions needed for FDM such as 'imaginary elements' for boundary conditions and use of the central difference theorem to determine slopes. That said, there are assumptions with FEM regarding shape functions, integration points, etc. but all of these seem to me to provide more accurate answers. Or am I just deluding myself with the illusion of complexity?

 

[gravityandinertia]

I would have to review a little bit, but I don't believe there would be much difference between FEM and FDM in 1D. A beam element is not much different than a difference in a beam.

 

[IDS]

Simple beam theory, ignoring shear deflections and second order effects, results in a cubic curve for deflections of a straight beam subject to a series of point loads (i.e. the shear force is uniform or stepped along the beam). Both FEA and FD will result in exactly the same cubic curve, if the beam is sub-divided at each applied load, and so will give exactly the same result.

I don't know how finite difference programs handle shear deflections, but in principle I don't see why they couldn't be handled with the same accuracy as in finite element programs.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[LRJ]

Thanks for the answers.

I agree that in theory both methods could compute the same answer although FEM is probably more efficient at this computation than FDM. My primary concerns with FDM are the 'imaginary elements' needed for boundary conditions and also the use of the central difference theorem; you'd need a lot of elements to be truly accurate - I'd expect FEM to get closer to the true answer with relatively fewer elements. I wondered whether there are any publications to support this conclusion.

 

[IDS]

I wondered whether there are any publications to support this conclusion.

Here is a publication with a different conclusion:
Continuous beam analysis with cubic splines




Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[LRJ]

Excuse my ignorance, but how does that relate to a comparison between FEM and FDM?

 

[rb1957]

(without looking at the link) I suspect it says that FEA does a good job modelling beam columns.

I thought FDA was just a poor man's FEA ... it makes very similar assumptions about the structural response to load but typically has many fewer elements (ok, the only time I've seen FDA was a 25 grid model embedded in an excel s/sheet ... co-workers were impressed, I wasn't ... "that's like a 25 grid FEA for a structure several feet long, with integral stiffeners, and complex boundary conditions ..."); so it isn't a hard limit of FDA.

in these days over extremely cheap FEA, why use FDA ?

another day in paradise, or is paradise one day closer ?

 

[IDS]

Excuse my ignorance, but how does that relate to a comparison between FEM and FDM?

It calculates the shear forces, moments and deflections in a continuous beam by fitting deflections to a cubic spline. The cubic spline formulation uses a numerical method to calculate slope and curvature at the end of each segment, which is the same process used in finite difference analysis.

It then compares the results to those from a finite element program (Strand7), and finds that the results are identical, within machine accuracy.

That is to be expected because beam elements in FEA programs work much the same way, except that mathematical method is more computationally expensive for a beam with many elements.

Referring back to your original post, the "imaginary elements" in finite difference analysis are simply a way of specifying the curvature and/or slope at the ends of a beam, and the central difference theorem gives "exact" results for a cubic curve, so neither will affect the accuracy of the results.

As for reasons for using the finite difference method, for small beam problems there probably is no real advantage, other than the interest in finding that a standard cubic spline formulation can be used to exactly match results from standard beam theory, but for large and complex non-linear problems I believe it can be quicker than FEA, but I have never used it myself in that context.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[LRJ]

Thanks for the explanations. That makes sense now.