Tasked to find the equation of a bridge, I chose the Tyne Bridge.
What's a "Tyne Bridge"?
The Tyne Bridge links Newcastle upon Tyne with Gateshead. It was designed by the firm “Mott, Hay and Anderson”, and built by “Dorman Long and Co.” (same people who made Harbour Bridge). The construction of the Tyne Bridge started in August 1925, and was completed in February 1928. It was built over the Tyne River, in Northeast England. There have been many bridges going over this river, all the way back to "Pons Aelius", the Roman Bridge. When it was completed, it was opened by King George V. It was the largest single-span steel arch bridge in England, and at the time, in the world.
To start this task, I got the dimensions of our bridge.
Height: 59m
Span: 389m
Width: 17m
FUN FACTS ABOUT THE TYNE BRIDGE:
- It was originally painted green
- It was made simultaneously from both ends, and connected in the middle.
- Constructed to include lifts, but they were never completed
- Some speculate that it may have been a “big brother” bridge to the Sydney Harbour Bridge
Keep scrolling to see more photos of the Tyne Bridge!
THE STEPS TO FINDING OUR PARABOLA
Putting our bridge onto a cartesian plane, I put it to scale (in m).
Then I found some key points.
Now, the next part is to actually find the equation of this bridge.
FINDING THE EQUATION OF THE TYNE BRIDGE
To do this, I took two points, the vertex and a random one.
So, I had points (0, 5.9) (the vertex) and (10, 4.46154)
Using this, I could substitute into vertex form, which looks like this
So substituting this, it ends up being
- 4.46154 = a(10-0)² + 5.9
- 4.46154 = 100a + 5.9
- -1.43846 = 100a
- a = - 0.00143846
Our final equation is
y=-0.0143846x^2+5.9
Or, for an answer a bit more in scale...
KEY FEATURES
- Axis of Symmetry: X = 0
- Vertex: Maximum, found at (0,59)
- Intercepts: X = 197.4842, -197.4842. These are physical - they're were either side of the bridge meets the ground.
- Width/steepness: A = 0.00143846. The parabola is very wide, and not very steep.
THE FINAL RESULT
A REFLECTION (and not the Parabolic kind)
While our parabola lines up almost exactly with our bridge, there's a slight discrepancy on one side. This would come up to construction and any image distortions.
The parabolic shape is used in bridges for the purposes of weight distribution. It is a necessity. But, a parabola can't possibly have exactly even weight distribution - you can't say within reason that all of the weight across all of the bridge will be exactly even at every single point, and furthermore that every single point of the bridge requires the same level of support (ie, that the middle is as supported as the floor-connected sides). Therefore, the parabolic shape is warped slightly so that the unsupported areas get support, and that it can hold any random combination of weight that the bridge may experience.
Credits:
Created with images by Peppygraphics - "Night view of Harbour bridge in Sydney Australia" • moomusician - "Tyne Bridge, an arch bridge over the River Tyne in North East England, linking Newcastle upon Tyne and Gateshead. With Gateshead Millennium Bridge on background"