# How to model elastic support in FEM?

I need a 8 m x 4 m x 0.6 m RC plinth (mechanical requirement) which needs to be built onto loose gravel. Obviously I am going to over excavate, back fill with selected material and bed my plinth on that.

The plinth accommodates 2 x 65 m3 water tanks, placed symmetrically. The tanks are $\phi$3 m, 9 m high and weigh 796 kN when full. The wind load on each creates a 206 kNm moment and horizontal force of 38 kN.

I need to design the plinth for bending and check the soil for shear failure. I assume the max. bearing capacity of the soil to be 150 kPa.

I want to simulate this using FEM. How do I address the support conditions. I understand that I should use springs as supports, but how would I get the equivalent spring constants if my spring is soil?

• What is the purpose of the FEA? Is this a learning exercise, or do you just want to design the tank foundation and think that FEA is the way to go? Your answer to this comment is crucial as it differentiates between putting you in a world of hurt (and much cost) or a rather simple RC design... – Paul Uszak Dec 4 '15 at 21:52
• This is a real design, for a real project. My design suite has a module that employs FEA for beams on elastic supports, i.e. ground beams. This module does meshing and modelling the springs in the background. But I wish to understand the FEA modelling side so I can use the 3D modelling software to do the same. – SlydeRule Dec 5 '15 at 5:27

Before jumping into elastic support conditions, you have to realise that it's not a ground beam. For three reasons:

1. With a minor axis width to depth ratio of 6ish, it's not far away from a square. If you stick a wide support somewhere near the ends of the section, the load will simply arch over the supports in pure compression. You'd almost be able to build it in brickwork!

2. Ground beams by definition span between supports. Your plinth is continuously supported by the fill. It's a raft if anything.

3. Also, how will you apply the moment? Water does't transfer bending moment.

The approximations suggested by Wasabi of the ground elasticity will totally negate your efforts at an accurate assessment of stresses within both the supports and the plinth.

If you're going to use an FEA analysis, you'll be looking at modelling the plinth as several layers of concrete cubes, sat on several more layers of cubes of fill, all within a surrounding matrix of cubes representing the gravel. If you're designing a skyscraper have at it, otherwise the soil investigations and modelling effort required would make this one of the most over designed structures I've ever seen. Do the following:

1. Treat the plinth as infinitely stiff

2. Check that the allowable bearing pressure isn't exceeded by either the dead weigh with a uniform ground pressure, or a triangular ground pressure due to the moment when the tanks are empty, or both.

3. Design the plinth for bending and shear due to the same load patterns

4. Ignore the lateral 38. The plinth won't slide due to self weight and friction.

You could do this in an afternoon after getting back from the pub. Flippancy aside, I've done similar (your approach) for a multimillion pound project. Don't. It's a pig.

What you need is the modulus of subgrade reaction (MSR) of the soil. This is a measure of soil deflection under a given pressure, so the unit is in (for example) kPa/m, or equivalently, as I'm more accustomed to see it, kN/m3.

MSR is obtained via plate-bearing tests, but there are also quite a few different tables that give some "typical values" for different soil compositions (obtained via plate-bearing tests). That being said, these tables can present huge ranges for a given soil type. I've found some tables here (page 37), here (Table C.3 on pages 1397-1398) and here (Table 1.2 page 16).

Technically this does not mean, however, that you can simply get the MSR and multiply it by the tributary area of each element since MSR isn't a soil property but a shorthand for the interaction between the soil and the foundation (different foundation sizes and shapes imply in different MSR values). There are equations you can use to find the corrected MSR, references to which can be found here (section 5.2.2 starting on page 128).

That being said, I have seen some people (including books) simply adopt the default MSR, so do that at your own risk. Foundations aren't my specialty, so I can't even say how significant the correction equation is.