What theories solve the problem of weight dispersion to minimize damage?

I assume there's some simple link to give me to answer this, but I don't know the terminology for it and so haven't been able to solve this.

The use case it to avoid having my TV put divots into my IKEA KALLAX.

Here's what my intuition tells me:

1. The best shape to distribute the weight is a conical frustum, with the larger circle on the KALLAX, and the TV legs sitting on the smaller one.
2. The frustum should have increasing density as its diameter increases. (The density at the bottom should be high to "avoid bending", and low at the top to minimize added weight.)

Are the intuitions correct? I have the sense that a dense substance would distribute weight much better than a less dense one, but why?

My current solution is to use dense foam, which is an incredibly crude approximation of the frustum I'm talking about. How can I tell if it's useful?

Again, my intuition is telling me that there's a falloff curve such that, while nowhere along the foam, is the dispersed weight going to be zero, there's going to be a threshold such that there's no point in having the foam be as wide as it is without trimming.

• Since you already put an aesthetically rather unpleasing (in my opinion :P) foam on top of the shelf, would you be willing to rest the TV legs on some large steel washers with soft padding underneath? This is all about distributing the force of the TV legs on a larger area of the Kallax top board, look up surface pressure to read a bit about it. – OpticalResonator Aug 25 '20 at 7:52
• It looks fine at normal light levels, but sure, I'd do what you're asking if someone could validate why it's a good solution. "Surface pressure" didn't turn up anything good for me. – Jessy Aug 25 '20 at 8:15
• I think it's a clumsy translation on my end, contact mechanics maybe comes closer. – OpticalResonator Aug 25 '20 at 8:35
• @Jessy, a better way is use Self Stick Felt Furniture Pads – Mahendra Gunawardena Aug 25 '20 at 12:00
• I think as long as the spacer is strong enough to withstand the TV's weight, it doesn't really matter what shape it is. – user253751 Aug 25 '20 at 16:56

Assume that your TV weighs $$100 kg$$, then each leg exerts a force of $$0.25 \cdot 100 kg \cdot 9.81 m/s^2 = 245.25 N$$ on the top board of the shelf. The contact area of the foot I can only guess, but let's be pessimistic and approximate it as a patch of $$5 mm \times 5mm = 25 mm^2$$. Pressure is calculated as $$\text{Pressure} = \frac{\text{Force}}{\text{Area}},$$ so every leg exerts a surface pressure of $$245.25N/25mm^2 = 9.81 N/mm^2$$ on the shelf top.
It's a bit difficult to find how much stress wood, especially the thin top layer of IKEA furniture, can handle. For pine wood we can find values of approx. $$2 N/mm^2$$, so let's take this as conservative goal. Then the TV legs on their own would indeed damage the top of the shelf.
If you now take a large M20 washer, with an inner diameter of $$22 mm$$ and an outer diameter of $$60 mm$$, you get a significantly larger area, at about $$2450 mm^2$$. The surface pressure is now at about $$0,1 N/mm^2$$, something the top of the shelf should be able to handle.