Another Catenary Question

[BenAustralia]

Getting myself all confused again!

Typically I find I have a cable I want to know the component reactions at ends of. Lets just assume a hanging cable, equal elevation on supports, gravity load only.

Information I have/can get:

L = Horizontal length of cable
w = weight of cable or UDL on cable, gravity for example
s = Sag (distance at centre from cable to horizontal)

The vertical reactions are fine, purely "w x L / 2". Simple.

The horizontal reaction is something I see conflicting information on.

1. Can I just use "w L ^2 / 2 s" to calculate my horizontal reactions?

2. What would I need to calculate/estimate cable TENSION at ends?

3. What would I need to calculate the Angle at ends? (if i can cal Tension and Angle I can break it back down into components).

Thanks!

 

[BenAustralia]

wL^2/8s, yes, typo.

So it comes down to calculation of the "Moment", which is dependent on cable shape.

Calculate this moment and divide back by the lever arm, Sag, and we have a component of Tension.

Assume Circular - Tension = P * R
Assume Parabolic - Component of Tension = wL^2/8s
Assume Hyperbolic/Catenary - Still not sure here. Uncertain on the 'a' scaling factor in the Catenary Formulas.

 

[BAretired]

So it comes down to calculation of the "Moment", which is dependent on cable shape.

I would have said the cable shape is dependent on the moment.

Calculate this moment and divide back by the lever arm, Sag, and we have a component of Tension...the horizontal component.

Assume Circular - Tension = P * R
Assume Parabolic - Component of Tension = wL^2/8s
Assume Hyperbolic/Catenary - Still not sure here. Uncertain on the 'a' scaling factor in the Catenary Formulas.

The scaling factor depends on the unloaded length of cable. Say the span is L and we have a point load at 'a' from the left support and 'b' from the right support (a + b = L). Neglecting the weight of the cable, the maximum moment occurs at the load and has a magnitude of Pab/L. If the sag is s at the load point, H=Pab/Ls. The moment diagram consists of two straight lines and that is also the configuration of the cable, two straight lines. The overall length of cable is (a²+s²)^1/2 + (b²+s²)^1/2 which is equal to the initial length of the cable plus the stretch due to tension. The tension will be the slope component of H which will be different on each side of the load except when it is at midspan.

Whatever the load is determines the moment diagram and the shape of the cable is similar. I hope that is clear now.


BA