Structural Analysis- small deformation

[Hassan2]

Structural Analysis-"small deformation"

Hi all,

Assume a cantilever beam fixed to a wall. We let the beam bend under its own weight. In practice the bending could be significant and as the bar bends, the distance between the tip of the bar and the wall decreases.

Now my question are:

1. Can this bending be modeled numerically ( finite element method) using linear elasticity?

2. They say linear elasticity is for small deformation. Does it mean that a large bending can't be modeled by methods based on linear elasticity?

The figure shows the result from my code which is based on linear elasticity. It's not what I expected as the bar stretches as it bends so that the distance between the tip and the wall remains the same.

P.S: Does anyone know of any active forum on numerical methods such as finite element ? I'm particularly interested in structural analysis as well as electromegnetics as my research is about coupled magneto-mechanical problems.

 

[kjohnson]

1. Can this bending be modeled numerically ( finite element method) using linear elasticity?

2. They say linear elasticity is for small deformation. Does it mean that a large bending can't be modeled by methods based on linear elasticity?

1. Yes you can use finite element method to model the problem.

2. Linear elasticity problems can involve large deformation, as long as the stresses and strains within the material do not exceed the linear region. For example a very thin beam can be bent into a tight radius easily while the stress in the beam remains small and thus in the linear range.

But, because of the large deformation you must take into account this change in geometry because as the beam deforms it affects the loads. This is usually not taken into account in basic mechanics of materials engineering formulas, but FEA should have no problem.

 

[AlephZero]

Linear elasticity applies to small strains not small displacements. If the structure rotates (like a bending cantilever) you can have small strains with finite (large) rotations.

You can model this by including an extra terms in the stiffness which represent the work done against the internal stresses in the structure. This is called the "stress stiffness" or "geometric stiffness" because it represents finite sized changes in the geometry of the model. This is a standard option in most "serious" FE software but the implementation is much more complicated than simple lnear FE.

The best websites I know on structural FE analysis are http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/Home.html for linear analysis and
http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d/Home.html for nonlinear.
Carlos Felippa has been a "guru" in this field for at least 30 years already.

 

[Hassan2]

Thank you both. I think a naive method is to simulate the large deformation in several steps and to calculate the new stiffness matrix for each step.

 

[AlephZero]

I think a naive method is to simulate the large deformation in several steps and to calculate the new stiffness matrix for each step.

If you try it, and you will find out you are wrong, unless you take into account the internal stress distribution for the new shape at each step. (This stresses are zero for the initial shape, which is why they are not in the formulation for a linear analysis).

A nice test problem is a thin beam with a moment (not a shear force) applied to the free end. This gives a constant bending moment along the beam and therefore constant curvature. So a correct nonlinear analysis should be able to bend the cantilever into a full circle.

 

[Hassan2]

Yes, even common sense tells me to consider the internal stress. Is this what you previously referred to as stress stiffness? I hope I can find a material about it in the recommended website.

 

[bda23]

P.S: Does anyone know of any active forum on numerical methods such as finite element ? I'm particularly interested in structural analysis as well as electromegnetics as my research is about coupled magneto-mechanical problems.

That's actually a really good question. I'd also be interested to know about such a forum on FEM and other numerical methods. Unfortunately I don't know of any really useful ones online, but do you reckon a section on numerical methods could be started within this forum? I really don't know whether enough demand could be mobilized, though...

 

[AlephZero]

Yes, even common sense tells me to consider the internal stress. Is this what you previously referred to as stress stiffness? I hope I can find a material about it in the recommended website.

The "nonlinear" course note is getting into the details by about Chapter 7 - but I would recommend starting from the beginning!

 

[Hassan2]

That's actually a really good question. I'd also be interested to know about such a forum on FEM and other numerical methods. Unfortunately I don't know of any really useful ones online, but do you reckon a section on numerical methods could be started within this forum? I really don't know whether enough demand could be mobilized, though...

Numerical methods are essential tools in engineering fields and I think such a section would be very helpful. Without the help of experienced ones , learning and implementing numerical methods is difficult as it's a combination of physics, mathematics and computer programming.
As for the demand, I'm not sure. I'm rather new to the forum and have no historical records of threads related to numerical methods.