The prompt is to find the parabola equation for the main span of Golden Gate bridge. Based on the $3$ points on the parabola: $(0,227)$; vertex $(640,75)$; and $(1280,227)$ I found the equation:

$$y = 0.00037109375x^2-0.475x+227$$

It is required to calculate the heights of vertical ropes under the parabolic main cable, which spans across a length of $1280$ m.

enter image description here Vertical ropes are distanced $15.24$ m apart with a diameter of $0.09736667$ m. The $x$-range is $[0,1280]$.

If I assume that the heights of vertical ropes follow the geometrical progression I get that $u = 219.8471693*0.968^{n-1}$. Where $u=$ height of vertical rope, $n=$ the number of the vertical rope in sequence.

I need a way to to prove if this equation is correct or not.

I tried by writing that \begin{align}219.8471693\cdot 0.968^{n-1} = & 0.00037109375\cdot(15.24n+0.09736667\cdot (n-1))^2 \\ & -0.475 \cdot (15.24n+0.09736667\cdot(n-1))+227\end{align} where L.H.S. is the function of height of vertical rope calculated by geometrical induction and R.H.S. is the height of vertical ropes calculated by using parabola function. $x$ at this point is the position of vertical rope on $x$-axis which is calculated by using the distance between ropes and diameter of one rope, and $n$ is the number of rope in the sequence. I tried to prove this equality by mathematical induction, but I couldn't.

How could it be proven that the heights of vertical ropes follow or not the geometrical progression?

  • $\begingroup$ You may substitute 3 sets of numbers found from the quadratic equation at the top, the height of vertical ropes and calculate the common ratio. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Feb 24 '17 at 21:35
  • $\begingroup$ Do you know that considering the curve is a parabola is only a raw approximation ? The true curve has equation $y=a \cosh(bx)$. $\endgroup$ – Jean Marie Feb 24 '17 at 21:46
  • $\begingroup$ Common ratio to was 0.968 on 3 points, I assume it is a GP. $\endgroup$ – Archetype2142 Feb 24 '17 at 21:54

Ignoring (x,y) shift

$$ y= k x^2, \quad k= y/x^2 $$

$$ k = (227. - 75)/640^2 = 0.000371094 $$

Considering $(x,y)$ shift $ = (h,k)$

$$ (y - k) = k (x - h)^2 $$

$$ (y - 75) = k (x - 640)^2 $$

The above is a parabolic progression. Please check your calculation. If a table of values (x,y) is given a parabola can be verified by numerically taking first differences twice ( like differentiating twice) from a table of (x,y) values.

Geometric progression is for an exponential curve!

We have a deep catenary suspension bridge cable of shape $ y =c \cosh (x/c)$ if rate of loading $ dQ/ds = $ constant with respect to sloping arc/arch/cable direction and is of shallow parabolic cable shape $ y= c x^2 $ if $ dQ/dx =$ constant for constant loading with respect to span , the x direction. For the Golden Gate bridge the deck is horizontal and bears a constant load per meter, the error is not so high for its shallow arch and the latter is more accurately described as a parabolic bridge.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.