Continuing previous ....

Determining tonnage ratings are a tricky business, and actual experience is often needed to adjust theoretical figures. However, the basic physics of railroading still apply.

As far as taking a run at a grade - there are what they call "momentum grades" that assume a train can take a run at it. I have never operated on a momentum grade, at least in compliance with the rules. The problem with a momentum grade is that the speed limit must allow taking a run at it, and the grade must be short enough that by the time the train's speed drops to 8-10 mph you have started to crest the grade. Unless part of the train is on a downgrade either helping to push or pull, the train generally has to be shorter than the length of the grade. It always amazes me when an engineer takes a run at a grade (usually a shortline with a 10 mph speed limit) when their momentum dies out within the first mile of a several mile grade. It is actually far more efficient to double the grade and leave the remaining cars on the flat or in a sag where they are easy to secure with a few handbrakes rather than in the middle of a hill you weren't ever going to make anyway.

When you start talking about heavy grades or heavy trains you have to consider drawbar limits. Not only do you have the tractive effort of the engines pulling on a train, while going up a hill you have gravity pulling the train in the opposite direction. The older coupler knuckles were designed to fail at about 250,000 pounds, so the drawbar forces had to be limited to this. There also had to be an additional factor added in the event the train had to be started on a hill since the static resistance is higher than rolling resistance and there is no reliable formula for determining it. For example on its Tillamook branch with up to 3% and plenty of curves SP limited trains to pulling no more than 3,000 tons. Trains heavier than that had to cut in helpers which had their own limitations again being restricted to not pulling more than 3,000 tons or pushing more than 250,000 pounds of tractive effort to keep from jackknifing the train ahead of them. Modern HT couplers are designed for 360,000 pounds of pull but there are still limits. A 10,000 ton train needs over 500,000 pounds of tractive effort to go up a 2.2% grade, so it is necessary to put manned or DPU helpers either mid-train or on the rear to limit the coupler forces. This is one of the primary purposes of Distributed Power.

Taking your proposed example:

Normal rolling resistance varies due to numerous considerations and is calculated using the updated Davis Formula, but to simplify things here the nominal figure of 5 pounds/ton is used which is probably generous enough except for maybe the power.

The effects of the grade (due to our friend gravity) are easy to calculate at 20 pounds per ton per percent of grade.

I've seen several figures claimed for curve resistance, so I am using 1 pound per ton per degree of curve as a nominal figure.

Remember that when calculating curve and grade resistance that it is actually the average of the entire train length, but short trains are affected more by variations.

Using your example, the worst case scenario is a 3.75% grade on a constant 15 degree curve. This works out to 5 + (3.75 x 20) + 15 = 95 pounds per ton. 20 modern cars would be almost a 1/4 mile in length so I doubt that then entire train would be hung up in the curve on that grade at the same time, so I will use 90 pounds per ton as the guesstimate here.

The desired 3,500 ton train would require up to 315,000 pounds of tractive effort to move up the hill, plus another 36,000 pounds of tractive effort to move a 400 ton engine consist resulting in the need to supply a minimum of 351,000 pounds of tractive effort. 3,500 trailing tons is probably the limit of the trailing tonnage you would want to handle to stay within the coupler limits. Several years ago a BNSF BARSAN train was climbing Miramar hill into San Diego when they derailed some empty auto racks on the head end. Seems they had lost a rear DPU engine which ended up stringlining the train due to excessive in train forces.

A 6,000 horsepower 12 powered axle consist (either a GP40-slug-GP40 or 2 x SD40 or 3 x GP38) would generate about 180,000 ponds of tractive effort at 10 mph. Substituting 3 x GP39 or 3 x GP40 wouldn't do much good unless the track speed was greater than 10 mph, and even then they wouldn't haul any more tonnage just the same a little faster. 180,000 / 90 = 2,000 tons - 400 ton engine consist = 1,600 trailing tons or 11 loads @ 143 tons. Calculated at 75 pounds per ton as an average grade would be 2,000 trailing tons at 10 mph (14 loads) which would drag down to maybe 8 mph in the worst curve and grade. Regardless a 3,500 ton train would have to double the hill.

Back to the 3,500 ton train with AC units - EMD's website states the continuous tractive effort for an SD70Ace is 155,000 pounds so even a pair of full 6 motor Aces or P6s wouldn't create the required 351,000 pounds of tractive effort for the worst case example. They would be able to handle 3,500 trailing tons if the train resistance was held to 79 tons per ton. Alternatively the tonnage rating would be about 3,300 tons at 8 mph. I'm not sure how a SD70Ace is able to obtain the listed 189,000 pounds of starting tractive effort but my guess it is software controlled and who knows how the software actually works? Remember we are discussing the six motor rating here and I am guessing that a P4 would only have 2/3 the rating requiring three P4s to create the same tractive effort as two Aces/P6s.

If Tacoma Rail wants to move 3,500 tons up the hill I would suggest using both P4s and a pair of SD40-2s. This should provide about 438,000 pounds of tractive effort and move 4,000 tons up the hill. This would clearly require a rear DPU or helper to stay within coupler limits. So the reality is it is probably most cost efficient to continue doubling the hill despite the expensive modern technology. Had they leased a pair of Aces or P6s instead then the Genset kicker would have allowed them to move 3,500 tons at 9 mph (at least on paper).

One of the greatest misconceptions is that modern engines can deliver their full traction rating at all times regardless of rail conditions. I've had a GP38 with two cars on flat track slipping due to the number of caterpillars on the rail. Modern adhesion systems have pushed the 25% nominal rating as high as 45%, but poor rail conditions are poor rail conditions so it is possible that a P4 on bad rail that is rated at 45% adhesion starting and 35% continuous could conceivably be limited to much less. If the 35% rating is limited to 30% due to poor rail this results in a 15% loss of tractive effort and tonnage capacity. There are two things that cannot be avoided - the low joints and curves. Low joints will cause a weight transfer between axles which can cause slipping at drag speeds. Likewise a curve, even with steerable trucks, requires the inside wheel to slide to keep up or the outside wheel to slip to slow down which becomes more dramatic as the curvature increases.

Besides being sure the sanders are working properly, the only other trick engineers will use is to apply 10 pounds of independent brake cylinder pressure. This is usually just enough to put the brake shoes against the wheels and try to scrub the contaminants off the wheels and keep them from spinning freely. I have seen an air diagram where GE actually did this as part of the wheel slip circuit though I don't think it ever became common. Unfortunately this only works if all the axles in your consist are being actively powered so it is not something you could use on a P4 or C4 with their unpowered axles.

An EMD P4 or GE C4 would be one of the worse engines I could imagine for being expected to pull a heavy grade with poor rail conditions down on their knees at low speed. I don't see any advantage to running a DPU unless the tonnage exceeds the drawbar and curve limits.

In comparison, an SD9 could usually handle 5 loads on poor rail and curves on grades up to 3%. Loaded cars varied between 220k and 263k cars. Using an average load of 120 tons or 600 tons per unit the train resistance calculates out to about 67 pounds per ton at 10 mph.