It will be neither a parabola nor a catenary. The fixed end is vertical but the free end is not. It will approach vertical but will never reach it. The bending moment will be greater towards the fixed end because it supporting more "free" wire. The equation for a cantilever will not apply because they assume the deflection is small - yours is huge.

So the resulting shape cannot be symmetrical and cannot be a parabola or catenary.

It sounds like you're trying to solve a real-world problem. Are you wanting a formula for the shape it makes or to answer a question like "if the wire were 1mm thick, how long would it need to be to just touch the table"?

My guess is that there's no formula for the length - like there's no formula for the circumference of an ellipse. But I think it would be easy enough to solve it numerically - I'd just simulate it. Of course, you'd need the young's modulus and the density - I expect the values you find on the web would give you an answer accurate to +/-20%.

If you're wanting a practical answer, I'd try it with a thin wire then scale it up. IIRC, the stiffness of a round wire is proportional to dia^4. The mass is proportional to dia^2. So I'd expect the length required to be proportional to dia^2.

It will be neither a parabola nor a catenary. The fixed end is vertical but the free end is not. It will approach vertical but will never reach it. The bending moment will be greater towards the fixed end because it supporting more "free" wire. The equation for a cantilever will not apply because they assume the deflection is small - yours is huge.

So the resulting shape cannot be symmetrical and cannot be a parabola or catenary.

It sounds like you're trying to solve a real-world problem. Are you wanting a formula for the shape it makes or to answer a question like "if the wire were 1mm thick, how long would it need to be to just touch the table"?

My guess is that there's no formula for the length - like there's no formula for the circumference of an ellipse. But I think it would be easy enough to solve it numerically - I'd just simulate it. Of course, you'd need the young's modulus and the density - I expect the values you find on the web would give you an answer accurate to +/-20%.

If you're wanting a practical answer, I'd try it with a thin wire then scale it up. IIRC, the stiffness of a round wire is proportional to dia^4. The mass is proportional to length*dia^2. As it gets thicker it needs to be longer so its mass does up. So I'd expect the length required to be proportional to dia.