Step 1: Recall how a hydraulic prop is meant to behave.
A hydraulic prop is a yielding roof support. It is set against the roof and floor with an initial setting load, and as the roof converges further, the prop is designed to keep resisting at a steady, constant load rather than either breaking or flattening out at a low load.
This constant resisting load, reached after the initial elastic rise, protects the roof strata from sudden overload while still allowing some safe convergence.
Step 2: Read the four curves against this idea.
All four curves rise in a straight, elastic line at first as the prop takes up load, then they behave differently once the load builds up.
Curve P rises to a peak and then the load falls away as deformation continues, this is the failure pattern of a prop that cannot sustain its load once it is overstressed.
Curve Q rises only to a fairly low load and then stays flat there, this represents a prop that yields too early at an insufficient resisting force.
Curve R rises close to the peak, then gradually softens and settles to a lower plateau, this is closer to the ideal but still loses some of its resisting capacity as it yields.
Curve S rises to the highest load of the four and then stays perfectly flat at that load as deformation continues, with no drop and no early softening.
Step 3: Match the ideal behaviour to a curve.
The ideal hydraulic prop should reach its rated, set load and then hold exactly that load constant while it yields, so the strata pressure on the prop never exceeds the design value and never falls short of it either.
Curve S is the only one of the four that keeps a fully constant load once the elastic rise is over, so it is the ideal characteristic.
Final Answer:
The ideal load-deformation curve of a hydraulic prop is S.
\[ \boxed{S} \]
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