Cable Support Structure 3

[heislert]

If a cable is used to support uniform loads in a catenary with fixed supports, the cable tension can be determined. What happens to the cable tension if the length and total load is doubled but a vertical support only is added in the middle? I think it stays the same but it does not seem correct.

 

[rb1957]

i think the boundary condition at the middle support is not the same as the RH anchor point ... i think at the middle support there is no slope in the cable.

if you allow zero bending stiffness in the cable, then the two cables should have the same tension. why not ?

 

[BAretired]

What is the height of the middle support?

BA

 

[BAretired]

If all three supports are at the same elevation, the tension is the same in each cable and is the same as the single cable described in the OP.

BA

 

[rb1957]

good catch BA ! ... another way to destroy the similiarity between the single span and the double span cable. i'd automatically assumed the same height.

 

[paddingtongreen]

Not to be picky, but, a catenary is the shape of a cable or chain supporting itself or a load uniform along the cable, not uniform along the span. If the cable is supporting a uniform load along the span, and the weight of the cable is small, the shape is of a parabola.

Otherwise, I agree with BAretired as long as everything is balanced. You could place the support somewhere off the center and still balance the tensions with different sags.

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

 

[BAretired]

The tension is not constant in a catenary as noted by paddingtongreen. The horizontal component of the tension is constant, so maximum tension occurs at the supports where slope is maximum. Minimum tension occurs at the low point where the slope is 0.

BA

 

[heislert]

Thanks everyone.

The reason I am having difficulty understanding is that on a two span cable (same elvation), if you load the left span, you get a sag on the left span and a tension carried through to the right support. If you load the right span, you get the same tension to the left support. So would you not add the two values by superposition to get double the tension?

Regards!

 

[rb1957]

ok, the middle support is the same height as the anchor points, and has a roller on it so the cable can run over it.

you apply a load to the left span, it sags, draws cable from the right span ... intuitively the right span becomes horizontal (assuming zero weight).

now if you apply a load to the left span and the right span, the cable won't run through (as both spans are equally loaded), so the displacement of the left span will be less than the previous case (assuming an inelastic cable) and i suspect the cable tension will be different ... the vertical components at the anchors will be the same (but they'll sum at the middle support) but the horizontal components will be different (due to the different slope of the cable)

 

[paddingtongreen]

heislert, I'm guessing you have a pulley in the middle. If so you don't transmit horizontal force to the middle support. The cable has to carry it to a fixed point. If you check the "self weight" sag of the unloaded span you will find the same tension as on the loaded side. This will be less than when both sides are loaded but the sag on the loaded side will be more.

However you do it, with a pulley in the middle, the tensions will equal out because the sags are different.

Try taking moments about the center of each span, you will find that the deeper sag of the heavily loaded cable produces the same horizontal reaction as the lightly loaded smaller sagged span.

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

 

[BAretired]

heislert,

Superposition cannot be used when you change the geometry of the structure. By loading the left side only, you increase the length of cable and the sag on the left side while decreasing the length of cable and sag on the right side.

rb1957 and paddingtongreen have given you a couple of ways of looking at the problem. The following is another way.

First, load the left span while temporarily holding the midpoint against translation. The cable exerts a force to the left.

Second, load the right span, still holding the midpoint against translation. The right hand exerts a force to the right.

Next, release the middle support. It has force H pulling left and force H pulling right, so it is in equilibrium and doesn't move. H = M/s throughout both spans where M is the simple span moment in each span and s is the sag.

The force at each end of each segment of cable is M/(s*cos[θ]) where [θ] is the angle the cable makes with a horizontal line. The vertical reaction on the middle support is one half the total load, assuming uniform load per lineal foot of cable. The left and right supports each take one quarter of the total load.

BA

 

[heislert]

So in the end, I was asking about the cable tension.

It seems then that the tension does not change when another span and load is added. So the cable could be ten spans long (one anchor each end, pulleys at each interior) and ten times the load and the tension is the same as one span (assuming no cable weight).

I just have difficulty comprehending this. Does a ski chair lift tension change when fully loaded as compared to one span only loaded between towers? Perhaps it does because the elevations are changing significantly.

Thanks again everyone!

 

[BAretired]

Does a ski chair lift tension change when fully loaded as compared to one span only loaded between towers?

In the case of a ski lift, the supports are usually not at the same elevation. If ten spans are loaded equally, the sag in every span is equal assuming equal spacing of supports and constant slope between adjacent supports.

If load varies from one span to another and only gravity loads are considered, sag varies as required to maintain a constant horizontal component of cable tension throughout all spans.

BA