Calculating the required strength of a cable?

[Sea Skimmer]

So for a story I’m writing I want a 30 ton submersible suspended underwater on a 40,000ft cable. I’m trying to get at least a rough idea of how thick a cable would have be needed for this record breaking task, and if it’s going to be possible at all? For simplicity I’m quite willing to ignore the added tension inflicted via ocean currents and the movements of the ship its hanging from, I just want to know what I need to dangle 30 tons at that depth under optimal conditions.

[wjs7744]

I'll probably be being corrected by one of the engineers soon, but I don't think the length is important. I think the numbers you need are the yield stress of the material the cable is made of, and the force applied by the submarine. The formula would then be P=F/A, where P is the yield stress, F is the force, and A is the cross-sectional area of the cable. Getting from that to diameter should be easy.

[Exonerate]

In a real life condition, the cable would have to support its own weight - since this is in water, the buoyancy helps out a bit, but 40k ft of cable probably still adds up to quite a bit of weight.

[wjs7744]

Will buoyancy make much difference to a vertical cable? I would have though that the buoyancy of the submarine itself would matter more. Naturally, the formula I gave will calculate the point at which the cable will give, so it should be treated as a hard lower limit.

[Kuroneko]

You might want to include the weight of the cable and a safety margin on top of that. Suppose that the cable length is L, cross-sectional area is A, the cable density is ρ, and the submarine weight is W. Then you're solving
(W + ρALg)/A ≤ σ, some maximum allowed stress [edit: typo here]
A ≥ W/(σ - ρLg), where g is acceleration of gravity
The buoyancy of the submarine and cable can be included in W and ρ, respectively: e.g., for the cable, replace ρ with the difference between the density of the cable and water instead of the actual density value.

[CJvR]

Well it depends on how your setup is intended to work.

Are you dangling a 30 ton load at the end of the cable?
In that case you need to scale the cable for 30 tons + it's own weight.

Are you dangling something massing 30 tons but perhaps neutrally bouyant?
In that case you just need to scale for the weight of the cable.

Add to that naturally the force you want to excert on the submersible.

[Kuroneko]

[Edit: I've just noticed that the length is 40,000ft in the OP instead of 30,000ft I've used. Mea culpa. Well, consider the below a sample calculation then; you'd have to recalculate for L = 4.8e5in. Incidentally, this makes the minimal yield strength to support a uniform cable sans submarine 120ksi.]

The density of steel is about 0.28lbm/in³, subtracting water gives ρ = 0.25lbm/in³, and likewise ρg = 0.25lbf/in³. It seems pretty clear that a uniform cable of length L = 3.6e5in will have quite a bit off difficulty. The minimal yield strength to support the cable alone is ρgL = 90 ksi.

For the moment, let's pick σ = 120ksi and submarine weight W = 6.0e4lbf. Then A ≥ W/(σ - ρLg) = (6.0e4lbf)/[1.2e5-(3.6e5)(0.25) lbf/in²] = 2.0in². I hope some engineers would comment on having a 120ksi cable, but it seems to me to be a bit too much.

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We could also have cable of variable cross-sectional area. Ignoring safety factors for a hard limit, the lowest area (at the bottom) is W/σ. For a distance λ from the bottom, the force differential is given by dF = ρgΑ.dλ, and so dA = dF/σ is the minimal required increase in area. Thus, letting B = ρg/σ, we have: dA/A = B dλ ---> Α = (W/σ) exp(λρg/σ).

So, for λ = L = 3.6e5 in and σ = 120 ksi, we decrease the maximal cross-sectional area to 1.1in². A marked improvement, although considering given the length, that's still excessive. Also, I suspect that σ ~ 75 ksi would be more reasonable, in which case the cable runs from 0.80in² at the bottom to 2.7in² at the top.

[Sea Skimmer]

Well it depends on how your setup is intended to work.

The submersible is a neutrally buoyant bathyscaphe (well slightly negative, to keep it on the bottom, but it also has lead ballest). However it’s being sent down to recover something of great importance under extreme conditions, so the cable needs to be able to physically pull up the sub even if the pressure happens to crush and flood its buoyancy tanks, or the crew sphere. The cable is also used to lift and reposition the sub during the course of its dive to increase the bottom area that can be searched, as any bathyscaphe is going to be slow as shit under its own power.

So, for λ = L = 3.6e5 in and σ = 120 ksi, we decrease the maximal cross-sectional area to 1.1in². A marked improvement, although considering given the length, that's still excessive. Also, I suspect that σ ~ 75 ksi would be more reasonable, in which case the cable runs from 0.80in² at the bottom to 2.7in² at the top.

Thanks for the help, I had a variable diameter cable in mind, but that was even more beyond my ability to figure out. 2.7-.8in for 30,000ft is smaller then I was expecting, so this should work out nicely

[Wyrm]

Kuroneko, are you basing your scientifically compliant pound system on force or mass? I say this because you're using both lbm and lbf in your calcs.

[Kuroneko]

Because it's easy to do in MATLAB, here's a table of the area (sq.in.) figures for W = 6.0e4lbf. The minimal area occurs at the bottom and is therefore independent of cable length.

Code: Select all

L (ft)  10,000    20,000    30,000    40,000  | Min
σ(ksi)+---------------------------------------+-------
50    | 2.1865    3.9841    7.2596   13.2278  | 1.2000
60    | 1.6487    2.7183    4.4817    7.3891  | 1.0000
70    | 1.3158    2.0198    3.1005    4.7595  | 0.8571
75    | 1.1935    1.7804    2.6561    3.9624  | 0.8000
80    | 1.0912    1.5878    2.3102    3.3613  | 0.7500
90    | 0.9304    1.2985    1.8122    2.5291  | 0.6667
100   | 0.8099    1.0933    1.4758    1.9921  | 0.6000
120   | 0.6420    0.8244    1.0585    1.3591  | 0.5000
150   | 0.4886    0.5967    0.7288    0.8902  | 0.4000
200   | 0.3486    0.4050    0.4705    0.5466  | 0.3000

Again, σ ~ 75ksi is my guess at being physically reasonable for cables--although there are steels with higher yield strength, I'm not so sure one could make steel wire at these quantities that performs that well (but here I should disclaim any competence at materials sciences).

As an aside, if the area of the cable until some given point is A, the total volume up to that point is V = (σA - W)/ρg = (4.0in³/lbf)[(σA - 6.0e4)lbf]. For σ = 75ksi and L = 30,000ft, A = 2.6561, and so V = 5.6e5in³. At the density of steel (0.28lbm/in³), that's 78 (short) tons in cable alone (not counting the mighty big winch you'd have to have). You better make sure your ship can handle that.

On a further note, if you want to aggressively pull up your submersible, make sure it's accounted for in the weight W. For example, if submersible is 28 tons, the maximum upward acceleration would be 30/28-1 = 0.071 gees, which would make the 28 ton submersible the requisite 30 tons of effective force on the cable.

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Kuroneko, are you basing your scientifically compliant pound system on force or mass? I say this because you're using both lbm and lbf in your calcs.

Eh, neither (I abhor that convention). I'm keeping them separate and simply using 1lbf = (1lbm)·(Earth gravity), so lbm is strictly mass and lbg is strictly force. All it does is allow a simple conversion from "30 tons" to "60,000lbm" to "60,000lbf under gravity" or ρ to a numerically equivalent ρg, etc. That's why I say ρ = 0.25lbm/in³ and ρg = 0.25lbf/in³ for the effective weight in water.

[Sea Skimmer]

So if I'm reading that chart correctly, the 75ksi cable that’s 40,000ft long will have a max thickness of 3.9624in and a minimum of 0.8000in right?

Again, σ ~ 75ksi is my guess at being physically reasonable for cables--although there are steels with higher yield strength, I'm not so sure one could make steel wire at these quantities that performs that well (but here I should disclaim any competence at materials sciences).

Well I certainly don’t want the cable to be the strongest thing ever made, it’s a 1924 industrial fantasy setting with only modest advances in certain technologies over what we had on real life earth.

that's 78 (short) tons in cable alone (not counting the mighty big winch you'd have to have). You better make sure your ship can handle that.

I’m figuring the cable wraps around a giant winch, and then is wound around a separate drum further forward, before the end is simply coiled up in a big compartment of the ship so that it can be inspected and cleaned. Winding 40,000ft of 4in thick cable all onto one big drum doesn’t seem too practical to me. Weight is no issue, the support ship displaces over 20,000 tons and mounts some very heavy lifting gear.

On a further note, if you want to aggressively pull up your submersible, make sure it's accounted for in the weight W. For example, if submersible is 28 tons, the maximum upward acceleration would be 30/28-1 = 0.071 gees, which would make the 28 ton submersible the requisite 30 tons of effective force on the cable.

I figure everything will move real slowly. Even in a free fall it took nearly five hours for the Bathyscaphe Trieste to reach the bottom of Challenger Deep at 35,800ft, with the craft sinking on the end of an unwinding cable it will take even longer. Start to finish a single dive with several search zones could last two days.

Thanks for the help again.

[Kuroneko]

So if I'm reading that chart correctly, the 75ksi cable that’s 40,000ft long will have a max thickness of 3.9624in and a minimum of 0.8000in right?

No, that's area in square inches. For diameter in inches assuming a circular cross-section, take 2sqrt(3.9624/π) = 2.2461. The chart would be as follows:

Code: Select all

L (ft)  10,000    20,000    30,000    40,000  | Min (in)
σ(ksi)+---------------------------------------+------- 
50    | 1.6685    2.2523    3.0403    4.1039  | 1.2361
60    | 1.4489    1.8604    2.3888    3.0673  | 1.1284
70    | 1.2943    1.6036    1.9869    2.4617  | 1.0446
75    | 1.2327    1.5056    1.8390    2.2461  | 1.0093
80    | 1.1787    1.4218    1.7151    2.0688  | 0.9772
90    | 1.0884    1.2858    1.5190    1.7945  | 0.9213
100   | 1.0155    1.1798    1.3708    1.5926  | 0.8740
120   | 0.9041    1.0245    1.1609    1.3155  | 0.7979
150   | 0.7887    0.8716    0.9633    1.0646  | 0.7136
200   | 0.6662    0.7181    0.7740    0.8342  | 0.6180

At 40,000ft and 75ksi, the cable would have to go from 1.01in to 2.25in in diameter. Please excuse the excessive precision.