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Why is the shape of the cable a parabola and not a catenary? I've always been taught that the shape a cable makes hanging between two supports is a catenary. Is

by nsxwolf 11y ago

Why is the shape of the cable a parabola and not a catenary? I've always been taught that the shape a cable makes hanging between two supports is a catenary. Isn't a suspension bridge the same thing?

deleted 11y ago
[deleted]
TeMPOraL 11y ago
Wikipedia says [0] it's "due to the weight of the roadway being much greater than that of the cable". [0] - https://en.wikipedia.org/wiki/Catenary#Suspension_bridge_curve https://en.wikipedia.org/wiki/Catenary#Suspension_bridge_cur...
- msandford 11y ago
  Yeah, catenary is the shape a cable makes when under its own weight, and with only its own tension holding it up. If you hang a single 1lb weight in the middle of a clothes line, it forms very nearly a perfect V. That's because the 1lb totally dominates the 1/2 oz that the rest of the line weighs.
  - function_seven 11y ago
    Right, but isn't the roadway's weight uniformly distributed along the length of the main cable? Maybe a linear weight distribution favors the parabola, while a curved weight distribution (i.e. the cable's own weight when hanging free) causes the catenary? That makes sense to me, there's more weight per linear meter at the ends than in the center when considering just the cable's weight.
    - abduhl 11y ago
      The reason for this is how the force distributions are applied. A catenary forms when self weight is the driving force. In other words, when the weight per unit length of the cable is the same. For a bridge, the weight of the deck is much alrger than the cable. The weight of the deck is the same per unit horizontal length which is not the same as the unit length of the cable due to the curvature.
    - tesseract 11y ago
      It's uniformly distributed along the length of the roadway. The arc length of the main cable between the suspension cable hanging points is greater near the towers where the slope of the main cable is steep, than closer to the center of the span.
cmsmith 11y ago
The parabola is the equilibrium shape of a cable which is supporting an even load per unit horizontal distance. The catenary is the same for an even load per unit cable length. As for why, it's been a few years since I had to derive this, but it goes something like this: * The horizontal component of the force in the cable doesn't change along its length * This leads to the fact that the extra vertical load picked up in the cable, per unit horizontal length as you approach the support, is proportional to the change in slope of the cable per unit length * The change in slope is the second derivative of the cable height, and since it is a constant the cable shape is parabolic Fun extra fact - the parabola is the equilibrium shape because the cable can resist tension but not bending. If you reverse the sign on all the equations then they still work, but your parabola is pointing away from the direction of gravity and you have only compression, no bending, in the material. That's why arch bridges are shaped like a parabola too.

by nsxwolf 11y ago

Why is the shape of the cable a parabola and not a catenary? I've always been taught that the shape a cable makes hanging between two supports is a catenary. Isn't a suspension bridge the same thing?

deleted 11y ago

[deleted]

TeMPOraL 11y ago

Wikipedia says [0] it's "due to the weight of the roadway being much greater than that of the cable". [0] - https://en.wikipedia.org/wiki/Catenary#Suspension_bridge_curve https://en.wikipedia.org/wiki/Catenary#Suspension_bridge_cur...

msandford 11y ago
Yeah, catenary is the shape a cable makes when under its own weight, and with only its own tension holding it up. If you hang a single 1lb weight in the middle of a clothes line, it forms very nearly a perfect V. That's because the 1lb totally dominates the 1/2 oz that the rest of the line weighs.
- function_seven 11y ago
  Right, but isn't the roadway's weight uniformly distributed along the length of the main cable? Maybe a linear weight distribution favors the parabola, while a curved weight distribution (i.e. the cable's own weight when hanging free) causes the catenary? That makes sense to me, there's more weight per linear meter at the ends than in the center when considering just the cable's weight.
  - abduhl 11y ago
    The reason for this is how the force distributions are applied. A catenary forms when self weight is the driving force. In other words, when the weight per unit length of the cable is the same. For a bridge, the weight of the deck is much alrger than the cable. The weight of the deck is the same per unit horizontal length which is not the same as the unit length of the cable due to the curvature.
  - tesseract 11y ago
    It's uniformly distributed along the length of the roadway. The arc length of the main cable between the suspension cable hanging points is greater near the towers where the slope of the main cable is steep, than closer to the center of the span.

cmsmith 11y ago

The parabola is the equilibrium shape of a cable which is supporting an even load per unit horizontal distance. The catenary is the same for an even load per unit cable length. As for why, it's been a few years since I had to derive this, but it goes something like this: * The horizontal component of the force in the cable doesn't change along its length * This leads to the fact that the extra vertical load picked up in the cable, per unit horizontal length as you approach the support, is proportional to the change in slope of the cable per unit length * The change in slope is the second derivative of the cable height, and since it is a constant the cable shape is parabolic Fun extra fact - the parabola is the equilibrium shape because the cable can resist tension but not bending. If you reverse the sign on all the equations then they still work, but your parabola is pointing away from the direction of gravity and you have only compression, no bending, in the material. That's why arch bridges are shaped like a parabola too.