Tension in a cable

You're looking for a catenary formula...

P245 Of Roark.

My web site (rmniall.com) has a spreadsheet that will calculate the forces and deflections in a cable (not necessarily horizontal) under the action of a point load (not necessarily vertical) acting anywhere along the cable.[ ] So if you have some specific cases you wish to analyse then it will probably help you.[ ] But if you actually want the formulae then it won't help you.

More than one?

The cable assumes a funicular shape similar to the shape of the bending moment diagram under the applied loading. With load P at midspan, span L and sag S, the horizontal component of the tension is H = PL/4S neglecting cable weight. To include cable weight, simply add W/2 to P where W is the total weight of the cable.

The maximum tension is the slope component of H which is approximately H/cosθ where θ is the slope of the cable which may be found using the Pythagoran theorem or alternatively, θ = arctan(2S/L). It is approximate because it neglects the slight difference in tension resulting from the additional slope due to the cable sag between the supports and the central load. If the cable is taut and the cable weight is small relative to the concentrated load, the result will be close enough for all practical purposes.

BA

Assume the cable is straight initially.
Assume a deflection at the center load, with cable straight between that point and the ends.
Calculate the change in length (hypotenuse of the triangle).
From change in length, calculate the strain.
From strain, calculate the stress. (Effective modulus of elasticity for a cable is different than a solid bar, available from manufacturers).
From stress, calculate the load in the cable, find horizontal and vertical components.
Compare vertical components to the applied load, adjust the assumed deflection as needed, and repeat.
Handily done with a spreadsheet.
I think this is the same as BARetired's version up there, except he didn't say how you find that deflection.

Usually, this is "wire rope" rather than "cable".

Note that if the cable has some initial sag, then its final sag will be greater than this derivation shows, which also means the forces in the cable will be lower, so it's conservative in that sense.

Hi
Click on the tenth title down on this list

Make sure that whatever you assume it is connected to can carry the horizontal loads.

If it's a cantilevered post, then the posts will lean in toward one another and increase your sag.

I've used this for cable analysis:
Just realized DesertFox already referred to it. I used it along with some other quick hand calcs that I found people suggested (from googling and looking for something reliable). They all matched up within a couple percent and it seemed very realistic to me. I had yet to find anything that was a major update from the handbook I linked to. All the suggestions that were agreed upon were essentially a variation of the equations the handbook has.