in a long and warm 1% slope line (therefore it expands): what shoes do you use? straight shoes or inclined shoes?

- in the case of inclined shoes: the dilation of the line is in the inclined direction with respect to the horizontal and therefore tends to lift the shoe. but line can lower under the weight and the bottom of shoe rest on beam.

- in the case of using classic straight shoes: the base of the shoe is aligned with the pipe so it is inclined with respect to the horizontal: with the axial expansion the shoes can be stuck and blocked on the underlying beam. the solution may be to insert a rod between the shoe beam.

what solution do you usually use?

in caesar the shoe is "point" but in reality it has a lenght that stop rotations. how do you model them and what real shoe do you use?

It's generally not practical to fabricate a shoe with a narrow slope, but it is practical to increase the height of the shoe as the pipe gets farther away from the starting elevation.

Assuming you go flat:

For stress purposes, assume that your stresses are under-representing the fact that shoes are sitting flat on steel.

How much? Run a test. Route 5 straight spans (no slope is ok), and instead of supports, use displacements. 0" in the vertical and 1% slope's equivalent rotation at each support.

Run a WNC and WNC+D1 case. While it's not fully accurate, but it'll at least get you into the ballpark as to what it does to your pipe.

Assuming you go rod, I do expect there will be a net force of a long line moving from the high end to the low end, especially if your pipe is rigid, and your spans are too short.

The more stiff the pipe, the closer the supports are, the closer this approaches to a solid object sitting on a solid incline.

However, the less stiff the pipe, the further apart the supports are, the closer this approaches a catenary powerline. Powerlines don't slide downhill, after all.

As usual, it's not one-size fits all, and that's not to say the solid object on the solid incline effect isn't acceptable - it just means more loads on the low end, that may or may not be acceptable and addressable.