fcFEM - FEA from start to finish

[bernd]

amazing harry!

[HarryvL]

Thanks Bernd. We need to start thinking/debating how best to make all of this available in FeeCAD. I think it could be of practical value. In the mean time I will continue adding to the functionality.

[bernd]

For sure! My idea was to have a look at this after the merge of the concrete branch. BUT this still has not happened ... But we are close. I finished tests for material quantity units. It is in master. Try to add m as unit to youngs modulus un material editor, when started from FEM material task panel. On ok it should give you a message, wrong unit ... This needs to be adapted to the reonforced material, than the example, than merge, than fcFEM.

At least we gone have a plan. There was some famous Danish with cools plans too, AFAIK.

[HarryvL]

Great. Integration into FC is going to be a big project with many options, choices and decisions; so it will need some debate and framing. I will therefore open a new topic for that.

[bernd]

very good idea ...

BTW: the famous Danish one with the cool plans ... https://en.wikipedia.org/wiki/Olsen_Gang

[HarryvL]

Some progress on implementing a Mohr Coulomb (MC) model, here depicted in principal stress space as an angular cone alongside the smooth Drucker Prager (DP) one. Stress states inside the cones are elastic, on the cones plastic and outside the cones inadmissible. This shows that the Drucker Prager model is less conservative for general stress states if fitted to Mohr Coulomb by matching uniaxial compression strength - something that I discussed in the first trench analysis:

Yield Criteria.png (108.64 KiB) Viewed 1895 times

Here the load-deflection curve for the first trench example (https://forum.freecadweb.org/viewtopic. ... 90#p296204).

fcFEM_MC_Excavation_c=10_pih=0_psi=0_L_u.png (31.78 KiB) Viewed 1895 times

Now the load factor at collapse is ~0.5 as per theory, without the need for adjusting the yield stress to get there, as explained here:

A hand calculation assuming a straight slip surface at 45 degrees shows that collapse occurs at a load factor of 0.5.

So why this 16% error in the analysis? Well part is always related to lack of mesh refinement. However previous analyses showed that even for coarse meshes rather accurate collapse loads can be predicted.

The answer is in the translation of undrained shear strength to yield stress. The von Mises stress (S_vm) for pure shear (Tau) equals S_vm = Sqrt(3) * Tau. So a von Mises material with yield stress Sy has a shear strength of of Sqrt(3) * Sy and not 2 * Sy as is the case with Tresca (= Mohr Coulomb with Phi=0) material. So by simply doubling the undrained shear strength to get a yield stress, we introduce as much as a 2/Sqrt(3) ~ 1.155 (or 15.5%) error in the collapse load.

This simple case study shows that the macro accurately predicts collapse, but that care should be taken with translation of measured soil parameters into model parameters

And here the results for the test case of a plate with hole:

fcFEM_Plate_with_Hole_MC_L_u.png (28.98 KiB) Viewed 1895 times

fcFEM_Plate_with_Hole_MC_Deformed_Mesh.png (56.16 KiB) Viewed 1895 times

fcFEM_Plate_with_Hole_MC_Deformed_Mesh_Top.png (50.28 KiB) Viewed 1895 times

Again, the load factor at collapse is in line with theory (0.5) and the deformation is credible with a shear band at 45 degrees.

I have also coded a tension cut-off on principal stress, similar to what I did for the Drucker Prager Model, but this needs further testing.

Next I will spend some time on coding an idea I have for estimating crack width in reinforced concrete.

[HarryvL]

I have also coded a tension cut-off on principal stress, similar to what I did for the Drucker Prager Model, but this needs further testing.

I have done as much testing as I can and am able to reproduce the beam failure load of 2.37 (see: https://forum.freecadweb.org/viewtopic. ... 60#p308289) accurately:

fcFEM_Concrete_Beam_MC_L_u_16.png (35.23 KiB) Viewed 1793 times

fcFEM_Concrete_Beam_MC_L_u_17.png (31.98 KiB) Viewed 1793 times

However, as you can see at that point the automatic load control mechanism decides to unload the beam - AI at work

It should be borne in mind that although non-associated plasticity (Psi = Phi -30) much better represents tests on granular materials (like soil, rock and concrete) and avoids severe over-prediction of collapse loads in constrained situations, it is much harder to control in numerical analysis.

If you are interested to know more then I can recommend the following article (http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf), which describes the phenomenon from a practical engineering point of view.

[bernd]

great stuff and a cool article.

[HarryvL]

... and for completeness the analysis result for associated material (Psi = Phi):

fcFEM_Concrete_Beam_MC_L_u_21.png (29.29 KiB) Viewed 1779 times

fcFEM_Concrete_Beam_MC_L_u_20.png (25.64 KiB) Viewed 1779 times

the fact that the load multiplier at collapse (i.e. 2.43) is only slightly above the theoretical value of 2.37 is because in this case the compression failure of the concrete is nearly (laterally) unconstrained. The only lateral stiffness preventing the lateral expansion inherent in the volume strain at collapse is due to the light lateral reinforcement. A top view of the beam at collapse (exaggerated by a factor 10,000) clearly shows the lateral expansion in the plastic hinge.

fcFEM_Concrete_Beam_MC_Psi=Phi_Deformation_1.png (24.31 KiB) Viewed 1779 times

fcFEM_Concrete_Beam_MC_Psi=Phi_Deformation_2.png (23.78 KiB) Viewed 1779 times

In massive concrete this lateral expansion would lead to an increase in internal pressure and that -combined with friction- would lead to an unrealistic increase in collapse load.

The top view reveals another phenomenon that may explain why this analysis is so difficult to control (even for associated plasticity). The mid-section of the beam completely plastifies, which means that several failure mechanisms co-exist. The left side of the beam sways laterally and rotates around its axis, compared to the right side of the beam.

Anyway, all-in-all a tough test case for fcFEM (and other packages, if we would get a chance to test those)

[HarryvL]

The reason for the excessive reserve strength is the fact that the maximum reinforcement ratio required at the bottom of the beam is applied over the full height. When I complete the functionality for varying the reinforcement ratio by integration point I will rerun the analysis with the minimum required reinforcement following from the elastic analysis.

Well I completed the functionality to use the reinforcement ratios from an elastic analysis as a starting point for the collapse analysis and it indeed gives a much smaller reserve strength.

fcFEM_Concrete_Beam_MC_rc_auto_L_u_5.png (30.87 KiB) Viewed 1722 times

fcFEM_Concrete_Beam_MC_rc_auto_L_u_6.png (26.19 KiB) Viewed 1722 times

Having said that, it would be impractical to reinforce the beam like this in reality and as a future development I would suggest that we try to create functionality to specify reinforcement ratio by zones or layers, e.g. by allowing the user to defined (curved) reinforcement surfaces. The direction of reinforcement within each layer can then be defined as a family of embedded curves. The applications for such functionality would be limitless; e.g. shell roofs, circular tanks, bridge box girders, etc.:

Concrete Shell.jpg (19.86 KiB) Viewed 1722 times

The minimal reinforcement from an elastic analysis can lead to failure in unexpected places. The beam now fails at a location of high shear and relatively low bending moment:

fcFEM_Concrete_Beam_MC_rc_auto_deformation.png (27.2 KiB) Viewed 1722 times

Clearly this would not happen in practice as anchoring requirements of the reinforcement bars would imply that the heavy reinforcement at the mid section can only be gradually reduced towards the supports, leading to more reinforcement than strictly required in zones closer to the supports. As the reinforcement in the present analysis is taken as the theoretical minimum for each location, the beam fails away from the mid section.