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# Calculating Tension and Sag in a Cable

## Calculating Tension and Sag in a Cable

(OP)
I am designing a double angle fence post member that has a tension cable attached to its top. See the attached document for a graphical representation. The tension cable spans the width of a pedestrian pathway and is attached to double angle fence posts on either side of the pathway. The cable is only for aesthetic purposes, and thusly will be loaded only by its self-weight and a design ice thickness of approximately 10 lb/ft.

In order to arrive at the moments at the base of the post, I need to determine the amount of tension in the cable at the post's top. From what I can tell, there are a few complicating items:

1) The basic catenary tension/sag equation is T = wl^2/(8*d)
where

T=horizontal tension
l=undeflected cable span
d=mid-span cable sag

So from this equation, I need T to solve for d, and vice versa. Which variable is known? I can imagine 2 scenarios: the cable being installed to a limited maximum sag (in this case d is known and T can be derived) or the cable being pre-tensioned to a minimum tension value (in this case T is known and d is derived).

2) When the cable undergoes the design ice load, the post/cable system will deflect to reach a point of equilibrium. The cable will sag and the posts will bend inward (creating additional sag in the cable). In order to determine these forces, I have modeled the system in RISA.

Using the RISA manual's recommendations for modeling cable action, I've broken the cable up into 5 equal members and  given the member joints an initial displacement downward (essentially a guess at the catenary shape). See second attached image. After running the analysis and recording the new displacements, I move the cable joints halfway between their initial location and the new displacements. After a few iterations, I have converged to an equilibrium state. This seems like a sensible way to arrive at a solution. Does anyone else have experience or suggestions on an analysis model?

I've used an E=12,000 ksi and an effective area Ae=0.5Ag which approximates the midrange of values listed in the Wire Rope Engineering Handbook. I've used a 1" diameter cable with Ae=0.393 in2.

Thanks guys; any and all help is appreciated.

### RE: Calculating Tension and Sag in a Cable

A couple of thoughts...

If this is a hand rail you may have to design it for more than just ice...I think there are minimum forces to design for (200lbs pt load I think, or a min. lb/ft)

Also, I think modelling the cable in RISA might be a bit overkill...I'm looking at my structures book from school and the calc's aren't that bad

I would think you would want to pick an acceptable deflection, then calc your tensions, then size your cable.

### RE: Calculating Tension and Sag in a Cable

the cable's 14' high, i think that's a bit much for a hand-rail ?

i agree with the modelling comments ... surely a cable design guide (such as you've referred) is sufficient ?  else "simple" hand calcs; the biggest concern (IMHO) is the impact on the columns.

### RE: Calculating Tension and Sag in a Cable

This is probably not a great application for RISA.  The problem is that a transversely loaded cable is a highly geometrically non-linear problem. And, the RISA features for geometric non-linearity are more geared towards building structures (i.e. P-Delta analysis).  Therefore, the analysis (at least as of early 2012) is not really sophisticated enough to handle that type of transversely loaded cables.

The technique used in the Modeling Tips section can be used. However, it has a number of limitations:

1) It is really most effective when you ALREADY know the final deflected shape of the cable. In that way, it can be used to VERIFY the accuracy of another calcualtion.  But, it is a difficult and iterative process if you are trying to use trial and error to guess at the correct geometry to input into the program.

2) It is only valid for a single load combination.  If you have multiple load combinations, then each one would require a different geometry.

### RE: Calculating Tension and Sag in a Cable

#### Quote:

1) The basic catenary tension/sag equation is T = wl^2/(8*d)

Actually, that is not the equation of a catenary.  It assumes a parabolic drape, but it is probably close enough for your purposes.

Start with the desired sag, d and calculate T(cable) for the non-iced condition.

Estimate d for the added weight of ice.  Calculate T(ice), deflection of each post and cable stretch due to ice.  Using the expression for the length of a catenary (or parabola) check the value of d.

Repeat until your estimated value of d is acceptable.

BA

### RE: Calculating Tension and Sag in a Cable

Horizontal lines are a B*&*tch.  Your horizontal loads will sky rocket the tighter you pull them and try to reduce the sag!!

But as BA says - its a very simple calc.

### RE: Calculating Tension and Sag in a Cable

If this were a pole line, both poles would be "dead ends" and have guys bracing them. If this is in the US, the NESC has icing and wind combinations.

I looked for a site to explain instead of me writing, I'm a hunt and peck typist, and I found this site that should be of some help.

http://www.shermanlab.com/science/physics/mechanics/Catenary.php

The wire is an elastic catenary and it will stretch further when carrying ice so you will have to consider that as well as the deflection of the poles, it helps to reduce the tension so it shouldn't be ignored.

If I were doing this, I would do some quick and dirty approximations to get a feel for the problem. I might try ther iced up wire at a deflection limit to rough out what size pole, sorry, angles would be needed. When I was assigned to the transmission and distribution group, I found that the wire calculations always seemed to be backward, or perhaps I had to think backwards, start with a deflection and find the wire to match. I don't know what headroom you must maintain, but the sag increases when the temperature of the wire increases but this usually doesn't control if there is an icing case.

Michael.
Timing has a lot to do with the outcome of a rain dance.

### RE: Calculating Tension and Sag in a Cable

Josh, are you hinting that cable members are going to arrive soon????

### RE: Calculating Tension and Sag in a Cable

1" is a big cable for just sw and ice

### RE: Calculating Tension and Sag in a Cable

I didn't mean to hint at any arrival times for cable elements in RISA. I just wanted to clarify this in case someone Googles, "RISA Cable Sag" a few years from now and sees my reply.

Better treatment of cables is a frequent request. For that reason, I wouldn't be surprised if gets into the program within a couple of years.  But, I seriously doubt whether it could happen any time in 2012.

### RE: Calculating Tension and Sag in a Cable

I certainly don't know all the reasons people make the choices they do, but I must really be losing it on this one.  Why double angle posts, when a round or square tube would be so much cleaner and a more appropriate shape for the purpose of a cantilevered post?  And, a cable attached to the top, only for aesthetic purposes?  What aesthetic, if it's not an electric conductor, and I do understand that my taste doesn't rule your design here.  As MiketheEngineer and Paddington suggest trying to tighten that cable to get rid of any sag causes a very large cable tension, a high cantilevered end loading on those posts, whatever they are made of.  And, adding any loads vertical or lateral only makes this problem worse.  BA's or Paddington's design approach should be adequate for this design problem, if other things are properly taken into account.  You can't make the cable straight and taut, because you can't build a strong enough post, even with guy wires.

Just my aesthetic, but...,   why not get a 20' long 2x12, or have someone saw a, 20' long, 2 or 3" plank out of a log.  Carve "MakeYourself's Walking Path" in both faces, and put  6 or 8 eye bolts in the top edge of this plank; run the cable through these eye bolts, btwn. the posts.  Now when you tension the cable, which you should do, you will put the plank in compression btwn. the post, but cause almost no cantilever bending or foundation problems in the posts.  A small angle bracket at the bottom of the plank to both posts, fixes the plank against swinging.  I grant that you will have some cantilever action from wind loads, but no where near the problem the cable tension would cause.  And, you will have something to memorialize your great design effort.  After all, it is so difficult to stencil "MakeYourself's Walking Path" on a cable and have it be legible from the ground.  In fact, with a 21' long plank bolted btwn. the double angle legs you could save the cable to tie up your horsey, but not from the top of the posts.

### RE: Calculating Tension and Sag in a Cable

I'm with dhengr here.  I read the first post & thought "you're kidding", but then everyone weighed in with all sorts of esoteric analysis.  I thought someone was nuts, either me or most of the responders, but the answers kept coming.  Tell the architect, or landscape designer, or whomever decided on the aesthetic, that the post has to be "X" size(pick something you can live with), the cable sag will be "Y", and call it a day.  Resolving cable forces is a never-ending exercise.

### RE: Calculating Tension and Sag in a Cable

(OP)
Thanks for the input everyone. Sounds like there are a lot of questions concerning the use of double angles and an "aesthetic tension cable"; however, this is the look and feel the architect likes and is happy with.

On the analysis side of things, I am heeding the advice of many of you (BA's and Paddington's in particular) and scrapping the use of RISA and using simple hand calcs taken from the Wire Rope Engineering Handbook I initially referenced.

My next question for those who have dealt with cables:

What shows up on the drawings/specifications regarding cable erection? The erection cable tension or erection cable center deflection (sag)?

### RE: Calculating Tension and Sag in a Cable

i'd get them to check the deflection of the columns as well (but maybe the arch wants a nice gracefull curve to these as well ?)

### RE: Calculating Tension and Sag in a Cable

Make -

You are the engineer - you specify!!

### RE: Calculating Tension and Sag in a Cable

@MakeYourself, I don't know how far you are getting, into means and methods. For transmission lines they use sag, they are given a table of sags to match variations in the ambient temperature. I think it would be easy for the Contractor to mark the sag on each post and sight the sag in, but that is "means and methods", his problem.

Another question occurs to me, you should perhaps, CYA in case somebody hangs a significantly large sign from the wire, perhaps after a couple of years have passed under the bridge.

P.S. I do like mixing metaphors, it seems to make them more powerful.

Michael.
Timing has a lot to do with the outcome of a rain dance.

### RE: Calculating Tension and Sag in a Cable

maybe hang a big sign on the wire ... "hang no signs here" ?

### RE: Calculating Tension and Sag in a Cable

LOL.
I was thinking of either; a letter to the architect saying that MakeYourself was going to add the effect of a large sign in the hope that he would send back a letter guaranteeing it wouldn't happen, or just add an allowance into the calculation.

Michael.
Timing has a lot to do with the outcome of a rain dance.

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