Sag of elastic / non-elastic catenary

In summary, the conversation discusses the difference between sag elastic and non-elastic catenaries, with the question of which would have a larger sag when they have the same horizontal tension. The conversation also clarifies the equations used for each type of catenary and addresses the potential energy involved. It concludes that the "elastic catenary" has a smaller tension and therefore a smaller sag than the non-elastic catenary.
  • #1
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
 
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  • #2
A catenary, by definition, has no bending stiffness, so your question really does not seem to make much sense.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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 = Th1/(w1*g)
c2 = Th2/(w2*g)
Because w2<w1, also Th2<Th1. 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.
 

1. What is a "sag" in relation to an elastic/non-elastic catenary?

A "sag" is the amount of vertical drop or curvature that occurs in the shape of a suspended rope or cable, such as a catenary, due to the weight of the object it is supporting. In an elastic catenary, the sag increases as the weight increases, while in a non-elastic catenary, the sag remains constant regardless of weight.

2. How does the elasticity of a catenary affect its sag?

The elasticity of a catenary refers to its ability to stretch and return to its original shape. In an elastic catenary, the sag increases as the weight increases due to the stretching of the rope. In a non-elastic catenary, the sag remains constant regardless of weight as the rope does not stretch.

3. What factors affect the sag of a catenary?

The sag of a catenary is affected by the weight of the object being supported, the length and material of the rope or cable, and the distance between the two supporting points. Additionally, the elasticity of the rope or cable can also impact the sag.

4. How is the sag of a catenary calculated?

The sag of a catenary can be calculated using the catenary equation, which takes into account the weight, length, and distance between the two supporting points. The elasticity of the rope or cable can also be factored into the equation if it is known. Alternatively, the sag can be measured directly using specialized tools.

5. What are some real-world applications of elastic/non-elastic catenaries?

Elastic catenaries are commonly used in suspension bridges, where the weight of the bridge and its traffic causes the cables to sag. Non-elastic catenaries are used in power lines, as they need to maintain a constant distance from the ground despite changes in weight and weather conditions. They are also used in construction and in the design of arches and domes.

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