Simple Bending Theory

[rabi24]

Hi all,

To test out a FEA software, I modeled a cantilever beam 6m long and 0.7m deep with a concentrated load at the tip of the beam. I compared the stresses at the support from FEA to those obtained by simple bending theory (Euler-Bernoulli), I got a difference of about 15% (FEA is larger). Mesh was fine.

I was wondering if that was a departure from simple bending theory or just a numerical approximation error? Seems quite large to be an approximation error.

Thanks,

Joe

 

[KootK]

I vote for a departure from simple bending theory. If I had to guess, I'd say that your support is restraining poisson's ratio effects locally.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[bookowski]

Since you're checking bending I assume you are looking at M/S stress by hand calc?

What if you refine the mesh substantially near and at the support and also don't restrain vertical at any nodes except the bottom, i.e. all connection nodes are restrained in x and base node is x & y (or z if z is your vertical).

 

[IDS]

Is this a 2D analysis with plane stress elements? If so, even a coarse mesh should give a close agreement if you compare XX stresses with the theoretical value. If you compare principal tensile stresses then the FEA will be higher where there is a vertical restraint because it is combining flexural and shear stresses, but even in this case if you restrain only the base node vertically, the top node PTS should be close to theoretical flexural stress (within about 1%).

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rabi24]

bookowski, yeah M/S by hand calcs. Without refining the mesh and just releasing the vertical restraints except the bottom, I get very close stresses to the theoretical values. Thanks

IDS, Yes it is a 2D analysis with plane stress elements.

If I restrain only the base node vertically, the results at the top are very close to the theoretical (1.5%). So it was a departure from simple bending theory, and the extra stresses at the support are due to the addition of shear stresses. Spot on.

Thanks guys! Much appreciated!

 

[JStephen]

If you compare the stresses partway out along the beam, how do they compare? I'm wondering if you aren't getting some funky effects from your choice of boundary conditions where it is "fixed". Do you know that maximum stress is parallel to the beam, or are you picking up components in the transverse direction as well?

The resulting loads on the support should all total up to the proper moment.

Could you have self-weight of the beam that's making a difference?

There will be a certain amount of extra deflection due to shear, and a wide flat beam can have reduced deflection due to Poisson's effects, but neither would affect stress.

 

[drawoh]

rabi24,

Solution by double integration is inaccurate if there is a lot of bending. According to double integration, the end of the beam travels straight down. That's not what it does in reality.

For most practical analyses, the deflections are small. Double integration is convenient, and accurate enough.

--
JHG

 

[BAretired]

What happens if you try checking a simple beam with span of 12m with vertical load acting at midspan? If P was the load at the end of the cantilever, you should use 2P at midspan of the simple span. Do you still find a 15% difference?

BA

 

[rabi24]

JStephen, at midway along the beam the stresses are almost identical. Does that mean it can't be shear stresses at the support causing the difference? I have modeled the fix end by fixing all nodes at the support in all directions. By parallel to the beam you mean normal stresses? Yes I know they are.

The self-weight is included in both theoretical calcs and model.

 

[rabi24]

drawoh,

so you're predicting a numerical error? This is kinda like a practical analysis tho, where loads and spans are pretty reasonable, deflections are small but a 15% difference in stress is surely not accurate enough, don't you think?

 

[IDS]

JStephen, at midway along the beam the stresses are almost identical. Does that mean it can't be shear stresses at the support causing the difference? I have modeled the fix end by fixing all nodes at the support in all directions. By parallel to the beam you mean normal stresses? Yes I know they are.

Halfway along the beam the shear stress distribution will be approximately parabolic with zero shear at the top and bottom face.

At the support the behaviour is no longer predominantly flexural, so you wouldn't expect the stresses to match those from flexural theory.

Regarding the suggestion that the problem is due to inaccuracies in the double integration process, that would only affect the calculated deflections, and assuming the FEA was linear elastic, the computer makes the same assumptions (small deflections) anyway.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/