I was visiting San Francisco last week and took a boat across the bay. Looking at the Golden Gate Bridge made me wonder what shape the supporting cables should make.
Traditionally, two kinds of answers are given to this question. If a cable is suspended by two points and holds nothing but its own weight, then it assumes the shape of a hyperbolic cosine. If on the other hand it is holding a constant weight along its span and the weight of the cable is negligible, it is a parabola. But what happens when the cable is holding the weight of a bridge but we do not want to ignore the weight of the cable itself entirely? Wikipedia states that the shape is a parabola – but is it??? There is a great tool in a mathematician’s toolbox: asymptotic analysis.
Let’s begin by drawing a force diagram for a segment of bridge.
Balance of forces means that all the vector forces must add to 0. The key property of cables we exploit is the fact that they can transmit force only tangent to rope itself. Therefore we know that the sum of the vertical components must be equal to the weight supported. The weight supported has two components: and , the forces exerted by the mass of the rope and of the bridge respectively. The former is proportional to the line density of the rope multiplied by the arc length:
whereas the force exerted by the bridge is only proportional to and to the line density of the bridge :
Invoking balance of forces leads to:
Taking the limit as leads to a second order ordinary differential equation:
with two boundary conditions for y:
If the first term of the equation was the only one that mattered (the weight of the cable is ignored) the solution is a quadratic. If the second term is the only one that matters (the weight of the bridge is ignored) then the solution is a hyperbolic cosine. Let us non-dimensionalise the problem choosing and . Then dropping the bars and defining two new non dimensional constants we end up with the following problem:
I think that it may be possible to solve this (WolframAlpha gives this solution to the definite integral. However a far better approach is to realise that we are interested in the problem where is small (the bridge is “denser” than the cable) and (the reason for this will become apparent). Therefore expand the solution in the following power expansion:
Inserting this into the field equation allows us to write down the following problem:
Solving these equations with the correct boundary conditions leads to the following solution to order :
The first term is the quadratic corresponding to the weight of the bridge, the second term corrects the parabola for the weight of the cable and third corrects the parabola for the fact that the weight of the cable is exerted not over the length of the bridge, but over the arc-length of the cable. Now we can see why must be small. It corresponds to the square of the ratio of the droop of the bridge to half its span. Patently the bridge does not droop more than the half its width, therefore must be smaller than one.
Pheeew! So it looks like the solution is indeed a quadratic with some correction terms that are fourth order in space.