Disconect A/THR in engine failure in cruise
 

Why we need to disconnect the A/THR with engine failure in cruise, on AIRBUS FBW?

Depending on strategy we select the speed, and set OPEN DESCEND , so it uses the elevator to to maintain the speed.

Beside the reason , to avoid Thrust reduction in the operative engine, what are the other reasons?

So the thrust on the other engine doesn't come back in the drift down decent.

You disconnect the autothrust then push the levers into MCT to obtain max continuous from the remaining engine. As you say the elevators control the pitch to maintain your chosen speed, depending on your strategy. MCT gives you the lowest rate of descent for the selected speed, with autothrust on it would, as you said, go to idle. So MCT increases your range, compared to idle descent, as you drift down to the one engine inoperative ceiling. There is a note in the QRH to engage autothrust again when the descent rate is small. If you were over your chosen divert airport, if indeed you decided to divert, then an idle / autothrust on descent would reduced your descent time - I suppose the QRH is designed for the worst case where you need to maximise range.

No you don't. First put it to MCT then disconnect. When the ROD drops to less than 500ft/mt. Set and pull VS -500 and put ATHR ON to maintain minimum ROD in compliance with performance computation.

For the OEI descent, whichever strategy, the intention is to keep the remaining engine operating at MCT for the duration of the descent until approaching your chosen (and achievable) ceiling. Shallowing the dive / drift-down as much as possible.

Step 1 - Both TLs to MCT (without the need to identify which one has failed).

Later on, you will be selecting OP DES for the reasons already stated. In the OEI condition, the A/THR range extends up to the MCT detent, thus it would command THR IDLE which is undesirable. To prevent this from happening:

Step 2 - A/THR: OFF.

a) AVIATE is done. Next
b) NAVIGATE
- action the briefed contingency procedure e.g. drift-down escape route, track offset if applicable. Next
c) COMMUNICATE
- speak to the other crew member verifying both are on the same page and paragraph
- alert ATC i.a.w. regional procedures (Emergency, RVSM, MNPS). Next
d) MANAGE SYSTEMS
- follow the appropriate engine drill

I am not aware of any reasons to disengage A/THR apart from keeping the MCT with OP DES. That one is good enough, innit?

Once approaching the OEI ceiling the flight path will shallow asymptotically which becomes impractical, so the technique is to select V/S -500 (increasing descent rate). With the extra available energy due to MCT the speed would start to increase. That is undesirable from FF viewpoint (as well), so A/THR is armed and engaged to allow conventional speed control for the chosen target.