2

When flexible metal transmission lines are hung between two poles, the shape they assume to minimize the action of gravitational and tensile forces is a called a catenary. While assuming that shape, the line undergoes stretching and internal shear which will cause it to degrade. Further, the shape keeps changing due to the extensibility of the line.

Consider a transmission line design consisting of flexible transmission wires wrapped around a rigid supporting metal beam (pole).

Is there a benefit (in terms of material degradation) in manufacturing the supporting rigid beam (made of metal) in the shape of the catenary to begin with?

kbakshi314
  • 2,382
  • The lines hang in catenaries precisely because they're nearly perfectly flexible; i.e., they can't support a bending moment and therefore can't be manufactured to maintain a certain shape. Unless you mean supporting them by a rigid external frame, in which case, why keep the catenary shape? Another option is to decouple the conductivity and strength requirements by linking/hanging/wrapping/intertwining a strong support cable with the (weaker) conductive transmission line to reduce the load on the latter to nearly zero. Is this what you're describing? – Chemomechanics Mar 07 '21 at 01:51
  • Yes, I'd like to comment that the purpose of the question is better served by sticking to the idea of rigid support cables wrapped with flexible conducting wires. – kbakshi314 Mar 07 '21 at 01:55
  • This then seems like an engineering economics question rather than a physics question. – Chemomechanics Mar 07 '21 at 01:58
  • Although the question is inspired by engineering application, I'm trying to understand if the basic principle makes sense. The question is whether shaping the rigid supporting cables as a catenary will reduce the stress on it, as opposed to shaping it as a straight line. – kbakshi314 Mar 07 '21 at 02:01
  • That would add large bending stresses in the case of any system perturbation (support tower movement, thermal expansion, wind, ice accumulation) that were essentially absent before. The ideal cable has no bending stresses, only axial stresses—which would still be present in the new scheme. Put another way, just become something sags doesn't mean that it's not strong; it only means it's compliant, which isn't necessarily a shortcoming. It would be useful if you clarified your question with a reference describing the details of the problem (e.g., "internal shear") you're trying to solve. – Chemomechanics Mar 07 '21 at 02:11

1 Answers1

1

The shape of the catenary is the result of the requirement that internal forces are only acting tangential to the curve shape, and there is no shear or bending moment along the wire. If $H$ is the horizontal tension, and $V$ the vertical (shear) tension then you have $$ \frac{V}{H} =\frac{{\rm d}y}{{\rm d}x} $$

The segment's total weight is balanced by the difference in vertical forces (and hence zero internal shear forces).

$$ {\rm d}V = \lambda \sqrt{1+ \left(\tfrac{{\rm d}y}{{\rm d}x}\right)^2} {\rm d}x $$

Here is a post of mine about the development of the equations.

So now what can happen if you design a cable with built-in curvature to match the shape? well, essentially a thin wire has zero resistance to bending (an assumption for the catenary shape development since internal moments are zero). So the initial shape of the cable does not matter. In fact, it comes in a spool that is much more curved than when installed.

Or maybe you are thinking of replacing the stranded wire with a solid one of the correct shape, which has other problems. For one, the internal tensions are still going to be tangential only, but due to the extra weight, natural frequencies will be much lower and hence more prone to galloping vibrations. And the friction between the strands provides a level of vibration damping that protects the cable.

Also, the current capacity will be diminished as electrons like to travel on the surface of the metal and the stranded shape has a much higher surface to volume ratio.

Finally, mechanically speaking it would be harder for a solid wire to respond to changing environmental conditions (like temperature, ice and wind) which change the loading and the force balance shape.

John Alexiou
  • 38,341