Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

## #1 2006-03-31 21:22:57

George,Y
Super Member

Offline

### The Shape of a Suspension Bridge's Main Cable

Aesthetic, light, and strong, suspension bridges can span distances from 2,000 to 7,000 feet -- far longer than any other kind of bridge. They also tend to be the most expensive to build. True to its name, a suspension bridge suspends the roadway from huge main cables, which extend from one end of the bridge to the other. These cables rest on top of high towers and are secured at each end by anchorages.

The towers enable the main cables to be draped over long distances. Most of the weight of the bridge is carried by the cables to the anchorages, which are imbedded in either solid rock or massive concrete blocks. Inside the anchorages, the cables are spread over a large area to evenly distribute the load and to prevent the cables from breaking free.

Suspension bridge
Golden Gate Bridge, San Francisco, CA

Now my question is, what's the shape of a suspension bridge's main cable in between the two towers?  I got an answer from my rough model. And one of the two alternatives is my real answer.
A a PARABOLA  B an ARC

Of course you may have your own answer. Anyway, say your answer and demonstrate it in this topic

Last edited by George,Y (2006-04-04 22:57:25)

X'(y-Xβ)=0

## #2 2006-04-01 13:30:17

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

Actually, my friend disagrees with my answer, thus A and B are our answers.

And the shape should be like a parabola or an arc, not exactly for sure.

Different people, different models, so

X'(y-Xβ)=0

## #3 2006-04-03 03:58:24

jchristophm
Novice

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

I would have assumed the shape would be a catenary. But it appears I am wrong:

See here

Last edited by jchristophm (2006-04-03 03:58:57)

## #4 2006-04-03 16:39:37

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

Thank you for your enthusiasm, jchristophm. It's really natural to think it as a catenary. (actually it was also my instinct)

Myorigenal model (edited):
My model is to cut the whole by the same horizontal distance, Δx,  and assume that cablepieces are straight, and that the concret road pieces are don't support each other. It finally derives the tangents  of the cable pieces develop by 0, k, 3k, 5k, 7k...to each side, where k is an unknown constant related to cable flexibilty. And that's similar to the motion that an arrow does without air resistance.

Still, to be exact, allow me to say that it's only near parabola for in-continousity.

Any way, next time we see a SB, we will enjoy its beauty and know its cables rest almost parabolas.

Last edited by George,Y (2006-04-04 23:00:30)

X'(y-Xβ)=0

## #5 2006-04-03 16:44:02

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

Why is my uploaded image so small ??? Anyone help me ...

X'(y-Xβ)=0

## #6 2006-04-03 23:11:39

jchristophm
Novice

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

#### George,Y wrote:

Still, to be exact, allow me to say that it's only near parabola for in-continousity.

I don't quite understand why you would say "near parabola". With the assumption that each "piece" of cable supports the weight of the bridge directly beneath it you get a function x^2. And this assumption is born out of engineering reality.

## #7 2006-04-04 10:21:34

John E. Franklin
Star Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

George said "Why is my uploaded image so small ??? Anyone help me ...".
Answer: Click on the tiny picture and it gets bigger.  This confused me once too.

I think George said near-parabola because the weight of the cables throws it off slightly, but probably only millimeters.

Imagine for a moment that even an earthworm may possess a love of self and a love of others.

## #8 2006-04-04 14:51:22

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

The discontinuity:

n,m,g,W,+
if we name the mid verticle cable the 0th verticle, we can also name all five verticle cables -2nd verticle, -1st verticle, 0th verticle, 1st verticle and 2nd verticle. and then we can define on each Δx is distributed with same mass m. g represents acceleration due to earth gravity.

1/2mg=def=W hence mg=2W

at the upper end of nth verticle(the tri-knit), forces are always 3, one downward(gravitional force), one leftward, and one rightward. define horizontally rightward and verticly upward +,and use cartesian expression of vectors. Using Newton's 1st and 2nd Law,we will easily get the table below.

nth                     0            1             2               3 (not displayed in graph)
leftwardforce (-h*,W)  (-h,-W)#  (-h,-3W) (-h,-5W)  *h is unkown
rightwardforce(h,W)#  (h,3W)     (h,5W)    (h,7W)
LF tangent         --          W/h         3W/h         5W/h
height-increament--     (W/h)Δx   3(W/h)Δx  5(W/h)Δx
height              0(set)  (W/h)Δx   4(W/h)Δx  9(W/h)Δx
distance           0(set)      Δx           2Δx           3Δx

# (h,W) and (-h,-W) are anti each other (Newton's 2nd Law)
at nth knit, LF+RF=2W, always(from 1st Law)
hence height= C distance², ignoring measure

but if the main cable behave more like a soft rope rather than a solid dome, it will bend more at the knits, thus not perfect parabola.

Last edited by George,Y (2006-04-05 23:07:43)

X'(y-Xβ)=0

## #9 2006-04-04 14:59:50

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

cannot enlarge them in posts don't know how others could do it

X'(y-Xβ)=0

## #10 2006-04-04 21:54:31

mathsyperson
Moderator

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

Alternatively, you can upload an image to imageshack or another free hosting site, and then put it here with the [img] tag.

Why did the vector cross the road?
It wanted to be normal.

## #11 2006-04-04 23:01:18

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

Many thanks, mathsyperson

X'(y-Xβ)=0

## #12 2006-04-04 23:41:35

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

an alternative model has been developed

A lot of films have such a horrible scene:
heros dashing on a broken rope bridge, and luckily reaching the end.

The rope , similar to the cable of a suspension bridge, should break near one end, rather than in the mid point-- you may verify this.

Last edited by George,Y (2006-04-05 22:30:13)

X'(y-Xβ)=0

## #13 2006-04-06 21:31:39

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

NOTICE the KNOTS (corrected), they are streched downward.

Franklin's guess is right, too. Even the knits doesn't form a parabola but a parabola-catenary mix.

For Post 12, you cannot get a parabola knits trajectory.
the tangent (slope) grows by 0, 2, 4, 6
height grows by 0, 2, 6, 10 =n(n+1)

length l= (n-1/2)Δx set Δx=1 n=l+1/2
n(n+1)= (l+1/2)(l+3/2) the function of right knits isn't symmetric about the axis in the graph, thus  left knits have another function (l-1/2)(l-3/2)
Two parabolas

Last edited by George,Y (2006-04-07 11:22:53)

X'(y-Xβ)=0

## #14 2006-04-07 01:55:18

jchristophm
Novice

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

Ok.

But if we only plotted the points were hanger meets main cable, then we have a parabola.

But if we had a bridge that had only one hanger that was planar across the entire main cable, then it would be parabolic.

## #15 2006-04-07 01:57:08

jchristophm
Novice

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

#### jchristophm wrote:

But if we had a bridge that had only one hanger that was planar across the entire main cable, then it would be parabolic.

Of course, then the massive weight of this planar hanger is going to have to be considered.

## #16 2006-04-07 03:46:41

krassi_holmz
Real Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

There's an interesting math function, called chain fnction. It's exactly the hyperbolic cosinus.

IPBLE:  Increasing Performance By Lowering Expectations.

## #17 2006-04-07 03:50:33

krassi_holmz
Real Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

A plot:

Last edited by krassi_holmz (2006-04-07 04:02:24)

IPBLE:  Increasing Performance By Lowering Expectations.

## #18 2006-04-07 11:29:25

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

But if we had a bridge that had only one hanger that was planar across the entire main cable, then it would be parabolic.

Yes, you could say so. but to be exact, near. because  matter is made of sufficient smalls instead of continuous infinite smalls.
My point is, most of caculus's success  is built on descrete approximation.

Last edited by George,Y (2006-04-07 11:30:13)

X'(y-Xβ)=0

## #19 2006-04-07 12:47:46

mikau
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

This is an awsome problem! :-D

A logarithm is just a misspelled algorithm.

## #20 2006-04-08 01:04:24

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

Thank you for your appreciation. Thank you krassi_holmz for your reference.

Let's take a break and appreciate some history.

The first modern suspension bridge ever built is Brooklyn Bridge, which connects Manhatten and Brooklyn, New York. The idea came from a German immgrant engineer, John Augustus Roebling . Avant as he was, he prepared 2 years from 1867 for every detail before start, and worked for 14 more years before he died of an infection.

His son Washington, also an engineer and bridge builder, took over his father's dream. He and his wife co-si[ervised the construction and completed the bridge.

The Brooklyne Bridge

Total Span: (Measures the distance between the two anchorages.)  3,455 feet
Main Span: (Measures the distance between the two towers.)         1,595 feet
Height of the Towers:                                                                        276 feet
Engineer(s):                John Roebling,   Washington A. Roebling
Cost:                                                                                             \$15 million

Washington and his wife

More details in suspension bridge development

Last edited by George,Y (2006-04-10 11:21:10)

X'(y-Xβ)=0

## #21 2006-04-10 11:11:30

George,Y
Super Member

Offline

### Re: The Shape of a Suspension Bridge's Main Cable

Liquid-mirror telescope

When liquid spins, it makes a parabola surface. and that's a good stuff for astronomors

Liquid-mirror telescope set to give stargazing a new spin

http://www.europa.com/~telscope/LMT.txt

it's really amazing to see these inventions are made  little by little rather than spontanously.

Last edited by George,Y (2006-04-10 11:20:35)

X'(y-Xβ)=0