Suspending Cable under own weight

[Kronustor]

Homework Statement

In this project, the methodology of Finite Elements and convergence of numerical solution is explored.

A cable has the following properties:

weight per unit length of wo = 5 N /m
length l= 24.2m

The cable is suspended with its two ends a distance of L = 20 m apart, as shown below.


The analytical solution for this type of problem is presented in an example in Chapter 7 of your textbook. The middle of the cable would drop a distance of h’ = 6 m vertically from its two ends, and the cable tension force has a maximum of Tmax = 75.9 N which occur at its two ends.

Procedure:
Part 1. You are now using the finite element approach to solve this problem. The approach is to divide the cable into equal sections. The number of sections N will be 2, 3, 4, 5 respectively. Consider each section as a rigid rod which is then pined to the next section. Lump the weight of each section to concentrated forces on the section ends. Then solve for the forces in each section and the positions of the section ends. From your solution, determine the drop of the middle point of the cable from the ends in the vertical direction hN, and the cable tension force at the ends TN, where the subscript N denotes the number of sections considered. Make sure that you take fall advantage of the symmetry in the problem. After your solutions for 2, 3, 4, 5 sections are achieved, group hN and TN into a table, and plot them versus N . Compare your plot with the follow plot of numerical solution convergence patterns. Answer what type of convergence it is for the solution of h, and what type of convergence it is for the solution of T.

Homework Equations

ƩFx=0
ƩFy=0
ƩMo=0

The Attempt at a Solution

I have done the first permutations from the problem. Where there 2 rods instead of the cable and where there are 3 rods intead of the cable. What i am having problems with is in the third and fourth permutations. I can't figure out how to solve the angles. I am sure if i could find one angle i for both of the last permutations i could finish the rest on my own.
The teacher has stressed that we use the symmetry in the problem but i can't figure out how to do it.
In the part where we use 4 rods i know each rod has a length of 6.05 and there is 3 suspended weights of 40.333. i have tried projecting the 4 rods over the length of 20m to form an octagon but that didnt work. I also thought of using isosceles triangles but didnt know where to go from there. Can someone point me in the right direction?

 

[SteamKing]

Your work is upside down.

 

[pongo38]

If the suspended cable were uniform, the shape would be a cosh function (catenary) for which a parabola is a good approximation when the span is (say) greater than 8 times the dip. The nodes of the rigid bar solutions should also lie on a catenary (or parabola).