Minimal Surface shape with gravity

[DuckAmuck]

Minimal surfaces are sort of the "shortest path" but in terms of surface shapes.
So I figured I could characterize the shape of a hammock by adding the influence of gravity, much like you can get the shape of a catenary cable (y=cosh(x)).

The equation of motion I get from the Lagrangian is:
$$z_x^2 + z_y^2 + 1 = z ( z_{xx} (z_y^2 + 1) + z_{yy} (z_x^2 +1) - 2 z_x z_y z_{xy} )$$
where z is the height of a point on the surface mapped to (x,y).

Of course, this is likely to have non-unique solutions just like other minimal surfaces.
One of the solutions I found is a cone:
$$z = \sqrt{x^2 + y^2}$$
What does *not* work as a solution is a "2-d catenary", which is what I initially suspected as solution
$$z = cosh(x)cosh(y)$$
Anyone else attempt this kind of problem? What were your findings? I'm basically just plugging things into the equation of motion and seeing if they work.

 

[Nidum]

You can do these types of problem for sheets of material with real physical properties using standard engineering analysis methods .

No initially flat sheet with edge restraints can distort into any other shape without some stretching of the material .

 

[DuckAmuck]

Would you mind linking me to a good source for this kind of engineering analysis methods?

You can do these types of problem for sheets of material with real physical properties using standard engineering analysis methods .

No initially flat sheet with edge restraints can distort into any other shape without some stretching of the material .

 

[Nidum]

https://en.wikipedia.org/wiki/Plate_theory

Timoshenko - Theory of Plates and Shells