You need to decide whether your end supports are rigid enough to justify treating the rail as 'encastre'. By this I mean are they strong enough to resist the rotation of the end of the rail when it is loaded. I suspect that it would be fairer to treat the ends of the rail as being simply supported. In which case, the deflection is given by P(L^3)/48EI This will give a deflection four times greater than Irving's.

I would treat Irving's solution as the best possible case and mine as the worst possible case. The true answer would lie somewhere inbetween.