Tallyho!, to all and Pppilot,
I was eagerly awaiting the evolution of this thread but class had begun then I had to leave for a personal trip to FRG- RDU-ATL so I spent the weekend, in a clapped out Seneca,only Stormscope and an old GPS, in solid IMC--- killer turbulence two diversions--with 3 horrified friends and one nervous pilot

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So, I haven't had time to reply...I've thoroughly read all of the replies and I just have a few pedagogical comments that I've repeated a million times in class, concerning DPs and gradients....now, I have not worked for "most major airlines", but the 135 operations with which I've been involved, all absolutely required that the AEO Departure path was looked after.., by the pilot---the limiting weight and speeds for the OEI TODR and flight path already having been done---here's an outline assuming the most primitive and basic of operations---because different types/AFMs/ companies--all do it in the same fundamental manner.
There are two basic documents required or can be used.
1. the AFM--- Departure Climb Gradients, all engines operating---but beware the charts will only cover from the gear up point, but the gradients [if nonstandard] that have to be maintained above the obstacle clearance surface [OCS] that begins at the DER and a properly planned TO [i.e OEI performance] will have a 15% distance margin---that will allow an aircraft to meet the AEO perf. segments anyways.
2. the actual procedure will have a table in the TPP that has a climb gradient chart if TAS is computed and the required gradient is known the the ROC for a given gradient are then shown [don't be too precise here], this will serve as a double check
---you should find that the AEO segments will rarely if ever be found to limit TOW, but they can and will at times---so don't skip this when higher than standard gradients are required!
Some comments on SIDS and ODPs there are three conditions all previouosly mentioned above by others---
1. A higher than normal gradients has been established for environmental reasons, noise abatements and such---as previously stated---so the few moments of computation allows the ground people to consider pilots to be friendly neighbors
2. Limiting obstacles aways out from the DER which affect the OCS sometimes require that a higher than standard gradient, because the gradient required to over the OCS had to account for some limiting obstacle. meaning a 48 '/nm climb has to be maintained
over the OCS...
---SO, if the OCS surface is already at a gradient of say 200'/nm then in order to clear it, legally, one needs a 400'/nm climb gradient.---so here AEO performance may take on a more serious meaning!
3. the last case is a little special ---and is due to specifically Low Close-In Obstacles
because such a scenario would mean that a very high climb gradient would have to be guaranteed but would be unnecessary later down the line, so Obstacle Departures are published and are annotated with the word [Obstacle] the textual description gives pilots the location and height of the obstacle and allows for
Visual clearance of the obstacle. but climb gradients are also established---or if not clearly stated you may assume standard conditions--
I understand your frustration, because the EFATO case is so thoroughly considered [and rightly so!] as it is usually more critical--very little considers the possibility of the AEO flight path---not just the net flight path---
I hope I've helped a bit
PA