A hydraulic prop works through a relief valve set to a fixed pressure. As the roof pushes down, the prop first behaves like a stiff spring, then once the set pressure is reached, the valve opens and lets the prop shorten while the load stays fixed at that set value. So the ideal curve should be an elastic ramp followed by a flat, constant line, with no rise, dip or drop after that point.
P: rises, peaks, then the load drops steadily as deformation grows. This is what happens when a support loses its carrying capacity after failure, not what a working hydraulic prop should do.
Q: rises only to a low load and flattens there. A relief valve set this low would let roof convergence build up with very little resistance, which defeats the purpose of the support.
R: rises close to its peak and then eases down to a somewhat lower plateau. This describes a valve or seal that is not holding pressure cleanly, so the resisting load slowly bleeds off instead of staying fixed.
S: rises to the highest load and then holds it dead flat for the rest of the deformation, exactly the constant resisting force a relief valve system is supposed to deliver.
Let's summarize:
The ideal hydraulic prop should show an elastic rise followed by a load that stays fully constant while the prop yields.
Curves that drop (P) or settle to a lower value (Q, R) all describe some loss of resisting capacity, which is not ideal behaviour.
So curve S is the ideal load-deformation characteristic of a hydraulic prop.
