How finite element analysis differs from mathematic derivation in beam bending?

[olski1]

So I just started learning to use the finite element package abaqus for modelling beam tip deflection under different loading conditions. I think I understand the theory behind it but was wondering if some one could answer a few questions about it to further my understanding.

Firstly, how do more elements in a simple beam with linear displacement deformation change the results compared to the classical beam bending theory results?

secondly, what would be the difference if i used a quadratic or cubic displacement function defining the deformation behaviour compared to the bending theory calculations while also increasing the number of elements?

Lastly, what would happen if i used a uniformly distributed load for the above cases?

I believe that more elements leads to closer agreement between theory and computer results as the derivation is a integral. however, I am not sure how the different deformation distributions (cubic and quadratic) affect the agreement from the theoretical results. Or for that matter how uniformly distributed loads are not perfectly represented by the modelling either.

These were just some points in the conceptual part of the book that like always have no direct answer. My beam bending theory has always been a little bit sketch, and I thought this would be a good time to rectify that.

 

[Mech_Engineer]

Firstly, how do more elements in a simple beam with linear displacement deformation change the results compared to the classical beam bending theory results?

It depends on if you're talking about 1-D, 2-D, or 3-D modeling. Are you just talking about "beam elements" which are geometrically just a line? Each element will give you results at the nodes, and then interpolated results between the nodes. The more nodes, the less interpolation has to be done. I think for most FEA codes the 1-D beam element is basically an implementation of simple beam bending theory.

secondly, what would be the difference if i used a quadratic or cubic displacement function defining the deformation behaviour compared to the bending theory calculations while also increasing the number of elements?

I'm not sure what you mean by "quadratic or cubic displacement function," are you referring to the interpolation function in the element, or something else? Nonlinear material properties?

 

[olski1]

I'm not sure what you mean by "quadratic or cubic displacement function," are you referring to the interpolation function in the element, or something else? Nonlinear material properties?

I suppose I mean the deformation properties of materials. For example, if I had results from computer modelling that showed better agreement with the theoretical results when using cubic displacement function over linear for 5 and 10 elements over a 1m beam what could I infer from that?

 

[Mech_Engineer]

It's possible to model plastic deformation of a model using a material's a full stress-strain curve in most FEA codes. I'm not sure you can "infer" anything useful from plastic deformation of a 1-D beam model through; the stresses calculated will be off from the real case where geometry and how the beam is attached at the boundary conditions will widely affect the solution.