Sag of elastic / non-elastic catenary

[cornis]

Hi,

I would like to ask about the difference of sag elastic and non-elastic catenary.
When these catenaries (their mathematic model) have a same horizontal tension, it is bigger sag elastic or non-elastic catenary?

I have my own calculations and I want to verify them.

Thanks
Joseph

 

[Dr.D]

A catenary, by definition, has no bending stiffness, so your question really does not seem to make much sense.

 

[cornis]

I will clarify my query. I have wire with horizontal tension Th, unit weight w, modulus of elasticity E, cross section A and span length a.
Parameter of catenary is c = Th/(w*g).

1) When is consider only non-elastic catenary, the equation is y = c*cosh(x/c)
2) When is consider elastic catenary, the equation is more complicated because of the wire elasticity.

I calculate the maximum sag using 1) and 2). Which sag is bigger?

I hope the question is clear now.

 

[Dr.D]

So, as I understand you, in case 1 you have a true catenary, while in case 2 you have a tension beam bending under its own weight.

To address your question as to which as the larger sag, consider a similar question. Assume that both are deformed by gravity to exactly the same shape. In that case, which has the higher stored potential energy?

I think you make a mistake in using your term "elastic catenary." This contradicts the definition of a catenary.

 

[cornis]

Yes, now you understand it well. I apologize for inaccuracies.

I'm a bit confused. In case 2) is added potential energy of elasticity - ½*k*(Δx)². On the other hand, it is smaller potencial energy of gravity, because stretched wire has a smaller unit weight. From this assumption and assumption that the catenaries have the same shape (same parametr of catenary) c₁=c₂
c₁ = T_h1/(w₁*g)
c₂ = T_h2/(w₂*g)
Because w₂<w₁, also T_h2<T_h1. So the "elastic catenary" has a smaller tension then non-elastic catenary with the same shape.
So, if the tensions are equal, the sag in case 2) is smaller than sag in case 1).

I do not know if that's right.