Why do suspension bridges need to be parabolic? Why can't they connect the towers directly?
Would it not use less material and cost less?
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6$\begingroup$ Just so you know, the curve isn’t paraboli, it’s a catenary. $\endgroup$ – Eric S Jan 2 at 14:22
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4$\begingroup$ @EricS no it's not a catenary. That would only be the shape for a free-hanging cable, or a bridge cable that is much heavier than the roadway. In a real bridge it's the other way around, the roadway is much heavier than the cable, and then the shape is indeed a (near-) parabola. $\endgroup$ – leftaroundabout Jan 2 at 15:11
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$\begingroup$ I'll take your word for it, but I'm not sure why a parabola would be the shape. $\endgroup$ – Eric S Jan 2 at 17:34
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$\begingroup$ @Eric, because the suspenders are vertical and have a load that is uniformly distributed across the span. In a catenary, the load is uniformly distributed along the length of the catenary curve, so it has a denser load where the curve is steeper. $\endgroup$ – Phil Sweet Jan 2 at 19:19
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$\begingroup$ @EricS I derive how the catenary vs. parabola arises here. $\endgroup$ – Chemomechanics Jan 2 at 23:31
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The next time you see some kids playing jump rope, go out and ask them to hold the rope perfectly straight. If two kids hold the two ends of the rope with it dangling until it is almost on the ground, they have to hold the ends of the rope up but they don't have to pull much, end to end. The straighter the rope is, the more they will have to pull. If you want the rope to be close to perfectly straight, they will be pulling with all their might. Even then, if you sight along the rope, you will see a curve.
Similarly, for a suspension bridge, the longer the dangle, the less the tension required at the ends. The bad news is, we can't have infinitely tall towers so there has to be some tension which is why the cables are tied off on each end of the bridge.
Interestingly, the curve of the cable changes as the bridge is being built. If you just have the cable, with no roadway, the curve is a cosh curve. As the roadway is added, it moves to a parabolic curve. There is a story about the math department that had a good view of the Golden Gate Bridge construction (which is shown in your picture). They plotted the curve of the cable on their window during construction to see the transition.
If we had adamantium cables with appropriate anchors at the ends, it would be neat to have a bridge with mostly straight cables but adamantium is kind of hard to come by.
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$\begingroup$ Think you mean a catenary curve if all it supports is its own mass... $\endgroup$ – Solar Mike Jan 1 at 19:50
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3$\begingroup$ @SolarMike, Yup, A cosh curve is a catenary curve, see Catenary $\endgroup$ – user1683793 Jan 1 at 20:07
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$\begingroup$ @user1683793: While you're right that a catenary curve is a cosh curve, in my experience the term of art for the shape is a catenary, no? $\endgroup$ – Wasabi♦ Jan 2 at 14:36
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$\begingroup$ Am I right in assuming that the mass of the cable is substantial compared with that of the deck? Not necessarily 50:50 but still an appreciable percentage of the overall structure, which goes some way towards explaining why it's best to let the cable adopt its natural catenary. $\endgroup$ – Mark Morgan Lloyd Jan 2 at 14:45
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$\begingroup$ @user1683793 "There is a story about the math department that had a good view of the Golden Gate Bridge construction..." Are you referring to this story about the George Washington Bridge connecting Manhattan and New Jersey? If not, can you provide a reference? I'm open to the possibility that multiple scholars have done this, although if you read the note, it seems more likely that the change in shape depends predominantly on cable stretching. $\endgroup$ – Chemomechanics Jan 2 at 23:37
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A suspension bridge cable is the most effective use of material. Because it uses high tensile cables in a position where they are the most effective material: tension. The parabolic shape is the closest natural geometry a cable assumes under uniformly distributed horizontal load ( not the self-weight, that is a catenary curve).
It has reduced the structure in a way that all other stresses are basically eliminated. Hence the parts and components that were needed to support them are eliminated saving material and weight.
Any other shape like a truss or concrete hollow structures wastes a lot of efficiency by resisting the forces of gravity by the beam action and resisting the bending stresses by the inefficient balance of compressive and tensile forces.
The parts that are tasked to support the compressive stresses have to be thick and wide to resist buckling, so they are very heavy and unnecessarily add to the weight of the structure.
It comes to a point where for some long spans any other type of structure other than cable can not even support their own weight.
The beauty of cable is it automatically assumes the best shape to resist the gravity forces with the least amount of unpredicted stress concentration or excessive loads on a member, such as if you have a suspended bridge that say, due to a traffic accident has all the traffic-jammed in a certain part of the bridge the suspended cables will squirm a little to accommodate the imbalance.
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$\begingroup$ "uniformly distributed horizontal load" Do you mean "load with a horizontally uniform distribution"? Your wording makes it sound like the load is horizontal, not the distribution. $\endgroup$ – Acccumulation Jan 2 at 4:10
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$\begingroup$ I mean a load that is either the weight of the actual horizontal deck or equal wights that could be suspended from the cable but are distributed not evenly along the arc of the cable but are hung from the cable with equal horizontal spacing say Dx=1m but Ds along the cable is concenterated vloser in the center of the arc. $\endgroup$ – kamran Jan 2 at 5:14
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Consider this simplified analogue to a suspension bridge: we model the roadway as a single point mass $m$, which is suspended from a rope attached to the cables. Now think about what's the best angle for the cable to make at the attachment point: completely straight, or hanging through a bit?
The answer is, it must hang down at least a bit. Recall that cables transmit (approximately) only tension forces, i.e. the force that the cable transmits must always point along the cable itself. So with the cable completely horizontal, the forces must also be purely horizontal and there's no vertical component that could hold the mass up!
In the angled version meanwhile, the force decomposes into a horizontal component (which is cancelled by the opposite end of the cable) and a vertical one (which is added by both sides of the cable):
Note that with a completely horizontal cable, the force-addition parallelogram would be perfectly flat, i.e. the vertical component would be zero. You could make the angles shallower, but then the magnitudes of $F_\text{L}$ and $F_\text{R}$ would need to be very big to still get enough vertical component.
In a real suspension bridge, you don't have only one vertical rope but multiple, but the principle is the same: at each attachment point, the main cable need to make a bit of an angle in order to give you a vertical force component. And in the limit of infinitely many attachment points, this means you want a smooth curve with constant positive second derivative (the infinitesimal version of these vector-addition parallelograms). That's a parabola.
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It's not possible for the cable to be perfectly horizontal. Cable can only pull; that is, the force they apply at a connection point is along the direction of the cable at that point. So a horizontal cable would be able to exert only a horizontal force. But the cable needs to transmit the weight of the bridge, and to do that it must apply a force with a vertical component.
The cables here make an angle of about 45 degrees with the towers, which means that about half the force they are applying to the towers is vertical, which means that the total force they are applying is twice what they would need just to support the weight of the bridge. The other half of the force from each cable is horizontal force cancelled out by the cable on the other side of the tower.
If the angle between the cable and the tower were larger, then an even smaller fraction of the total force would be vertical, and so we would need even more horizontal force to get enough vertical force. The cables would be shorter, but the total force they would exerting would be greater, so they would have to be stronger, which means making them thicker.
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A suspension bridge is an efficient use of material, because all the forces in the structure are carried by tension, and not by the parts of the structure resisting bending. Beams which resist bending are often useful components of structures, but they are an inefficient way to use material because the stress in the beam is not constant through its depth, and is small near the axis of the beam.
You can imagine the road deck of a suspension bridge to be made of short segments which are hinged together, so they can not carry any load at all without some additional support. That support comes from the vertical cable attached to each "hinge point" which pulls upwards to balance the weight of the deck plus whatever is crossing the bridge.
You can also imagine that the top cable is a set of short segments hinged together. Thinking about the vertical cable in the middle of the bridge, the two segments of the top cable on either side of it must be sloping in a symmetrical V shape to balance the force in the vertical cable. Moving from the center to the next vertical cable, the next segment of top cable must slope at a steeper angle to support the downwards pull of both the vertical cable and the top cable segment on the other side of it. Working along from the center to the end, the top cable must have a curved shape, which turns out to be a parabola. The shape of the cables at each end tower form an inverted symmetrical V, so the force in the tower is directed vertically downwards, with no forces that would bend the towers.
If you tried to replace the top cable with a horizontal beam, in fact there would be no point in having the top cable at all, since you could just make that beam part of the deck of the bridge and get rid of all the vertical cables. But doing that would make a structure that was much heavier than the cable suspension system.
(Of course a real suspension bridge structure does have to withstand some bending loads caused by wind loading, etc, but that can be ignored when understanding the basic principle of how the design works.)
