Calculations Check Please

[Andy873]

I've been working on this morning calculations of a 9 mile branch line (most of you will know which it is), however that's not important.

I'd like a math's check on this please.

1. The route is exactly nine miles.
2. It's a double line.
3. Each rail is 21 foot in length.
4. sleepers are 9 feet wide.
5. 7 sleepers to each 21 foot section of permanent way (track).

Calculation:

1 mile = 5,280 feet
5,280 feet x 9 miles = 47,520 feet
Sleepers are 9 feet long
Fishplates are 1.5 feet long.

Calculations:

9 miles = 47,520 feet requiring 21 foot lengths of rail = 2,263 single rail sections
2,263 single rail sections x 2 rails (pair of) for one, 21 foot section of track = 4,526 rails.
Double track = 4,526 rails x 2 = 9,052 rails
2,263, (21 foot sections) x 7 sleepers each = 15,841 sleepers (single line).
Double the sleepers (for double line) = 15,841 x 2 = 31,682 sleepers.
For 31,682 sleepers, x 2 chairs, one at each end of a sleeper = 63,364 chairs.
63,364 chairs x 3 bolts (2 screws, 1 Trenail) = 190,092 bolts, (126,728 screws, 63,364 nails).
63,364 Keys for chairs.
Double track sections = 4,526 x 2 = 9,052 track sections = x 2 joints = 18,104 fishplates
18,104 fishplates x 4 nuts and bolts = 72,416 nuts and bolts (same number of, for each).
Each joint needs 2 fishplates = 18,104 fishplates x 2 = 36,208 fishplates in total.

Am I right, or have got these calculations wrong please.

 

[DelW]

When I was buying used timber sleepers in bulk for falsework foundations, they were virtually all 8' 6" long not 9', though I don't think that affects your calculations at all.

I don't know what standard sleeper spacing is so I can't comment on exact number of sleepers etc. but your arithmetic looks correct for 3' spacing.

However I think your joint calc's are astray. 9052 rails = 9052 joints each with 2no fishplates and 4no bolts c/w nuts = total 18,104no fishplates and 36,208no Nuts & bolts.

As a check: 9 miles x 5280 ft/mile x 4 rails ÷ 21ft per joint x 4 bolt sets/ joint = 9x5280x4÷21x4= 36206 bolt sets (rounds up to 36,208)

I hope that's right!

 

[Gloster]

8’ 6” long sleepers are now the standard type, but until the 1930s they were 9’. The differences is so that complete track panels will fit within the loading-gauge when loaded on wagons.

 

[bishdunster]

Where uses 21ft rails? normally 60ft or 45 ft, at 24 sleepers/60ft
=2112 /mile.

 

[DelW]

8’ 6” long sleepers are now the standard type, but until the 1930s they were 9’. The differences is so that complete track panels will fit within the loading-gauge when loaded on wagons.

Well, you learn something new every day on here! The ones I was buying can't have been as old as I sometimes suspected they were (this was late '70s and early '80s).

 

[midland1]

Where uses 21ft rails? normally 60ft or 45 ft, at 24 sleepers/60ft
=2112 /mile.

If is used by a pacer it will be a rough ride.

 

[Rescars]

Where uses 21ft rails? normally 60ft or 45 ft, at 24 sleepers/60ft
=2112 /mile.

Although a different part of the country, I have read that the Port Road to Stranraer was originally laid with 21ft rails using 8ft 9in sleepers laid 3ft apart. As an aside, these rails were laid "unfished" (i.e. without fishplates). Whilst this would make Andy873's calculations easier, this style of construction certainly failed to impress the BoT!

 

[eastdyke]

When I was buying used timber sleepers in bulk for falsework foundations, they were virtually all 8' 6" long not 9', though I don't think that affects your calculations at all.

I don't know what standard sleeper spacing is so I can't comment on exact number of sleepers etc. but your arithmetic looks correct for 3' spacing.

However I think your joint calc's are astray. 9052 rails = 9052 joints each with 2no fishplates and 4no bolts c/w nuts = total 18,104no fishplates and 36,208no Nuts & bolts.

As a check: 9 miles x 5280 ft/mile x 4 rails ÷ 21ft per joint x 4 bolt sets/ joint = 9x5280x4÷21x4= 36206 bolt sets (rounds up to 36,208)

I hope that's right!

I think that it is!

This seems to be where it went astray:

Double track sections = 4,526 x 2 = 9,052 track sections = x 2 joints = 18,104 fishplates

 

[Andy873]

Thanks everyone for the double check, I see what you're all saying!

Regarding my calculations and why I did them:

I'm trying to picture the volume of rails, sleepers etc that the contractors had to deal with and be responsible for. In the case of the Hoddlesden branch (not my branch of interest), was only just under 2 miles long and single track. My branch, as said double, 9 miles in total. For this one, the line was (for various reasons) split into two sections, one was 6 miles, 60 chains, the other 2 miles, 20 chains. So, my calculations need to be split accordingly.

Now, even the short Hoddlesden line would have needed around 1,000 rails, and that's still a lot to store yourself as the contractor.

8’ 6” long sleepers are now the standard type, but until the 1930s they were 9’. The differences is so that complete track panels will fit within the loading-gauge when loaded on wagons.

Regarding specific measurement of P/W bits and pieces:

Sleepers - In this case, they are indeed 9 feet long, 10 inches wide, and 5 inches deep. The spacing between two sleeper's centre points in this case is 3 feet, 2 inches.
When two 21 foot rails join, the spacing between two sleeper's centre points is 2 feet.

Fishplates - they are 1 foot, 6 inches wide and have 4 holes.

Rails - As said, they are 21 feet long.

Regarding the rails themselves, I wanted to make this kind of rough calculation some time ago, but I couldn't make it work because the only information I had about them was from the only book dedicated to the branch line from 1993. The author gives all the dimensions correct for the P/W except, would you believe the length of the rails! He says they are 24 foot (as originally laid). He gave the correct sleeper distance, and the number of them, but it quite literally didn't add up.

Now I can see the P/W drawings, and you left in no doubt that the rails were indeed 21 feet and not 24.

Some general notes you might find interesting:

In the case of the Hoddlesden branch, the contract says that the P/W rails, sleepers etc are to be delivered to s specific station, and that the contractor has to take them away a.s.a.p. or the L&Y will charge demurrage.

An off topic question if I may ask, It's not the taking possession of the P/W parts, but the insistence of the railway company that you not only keep all of it safe (guarded), but also dry? doesn't dry imply a roof, and in turn some sort of temporary building?

Thanks once more for double checking my calculations.

 

[eastdyke]

An off topic question if I may ask, It's not the taking possession of the P/W parts, but the insistence of the railway company that you not only keep all of it safe (guarded), but also dry? doesn't dry imply a roof, and in turn some sort of temporary building?

Or perhaps the use of tarpaulins?

 

[Rescars]

Thanks once more for double checking my calculations.

One aspect your calculations illustrate are the huge number of rail joints, fishplates, etc needed with shorter length rails which would have to be tightened and otherwise maintained for every mile of route. Thus the armies of lengthmen employed in past times, some living in remote railway owned lineside cottages, to keep the P/W in good order.

 

[Andy873]

Or perhaps the use of tarpaulins?

I hadn't considered tarpaulin, would that be sufficient to keep the P/W items dry enough? Also, that would be less secure than some sort of building?

One aspect your calculations illustrate are the huge number of rail joints, fishplates, etc needed with shorter length rails which would have to be tightened and otherwise maintained for every mile of route. Thus the armies of lengthmen employed in past times, some living in remote railway owned lineside cottages, to keep the P/W in good order.

@Rescars, as always, you're one step ahead of me! Now you say it, it's obvious the shorter the rails, the more fishplates and bolts you need to keep an eye on. Something I hadn't considered.

More calculation facts:

I decided using the correction made, to calculate the 6 miles 60 chains part of the branch line. Rounded slightly I get 6,788 rails, 23,758 sleepers, and 47,516 chairs etc. In total (including everything else), 336,006 individual P/W items.

If we just look at the numbers regarding rails and sleepers, surely these are going to be delivered in batches?

Wouldn't the resident engineer of the L&Y give the nod to the company that more rails etc are need?

If we take the Hoddlesden line contract and say you can only lay the rails when you have 1,500 foot of trackbed ready for the rails, that works out at just under 19 chains, perhaps batches of 20 or 40 chains worth of P/W were delivered at a time?

The above contract states "the rails will be delivered when the works are ready to receive them".

Rounded up / down slightly for even numbers (double track):

20 chains worth of P/W
252 rails, 1,764 sleepers.

40 chains
502 rails, 3,514 sleepers.

For me, 20 chains looks good, as that's a quarter of a mile.

If you're a railway company and you have to deliver the 20 chains option, then that's more manageable, this could be done over a few days. It's also more manageable for the contractor to cart away, and certainly easier to guard than larger amounts of P/W.

What does anyone think about deliveries in batches of say 20 or 40 chains? or perhaps even 5 or 10 chains at a time?

One more thing I need to consider is (as this is a double line), would initially only one track be laid down? this would give more space for general movement along it especially for those horses. As far as I can see, because most of this section is on a downward gradient, you'd roll the wagons down to the embankments, unload earth, and then get the horses to pull the empty wagons back up the hill.