UK Centre for Materials Education
Working with you to enhance the student experience
Working with you to enhance the student experience
Author: Dr Nigel Mills
Institution: University of Birmingham
Abstract: Group work on the materials in rope-based sports, with web searches for specific product details, and presentations of the findings. A vehicle to consider polymeric fibre and rope production, and mechanics-of-materials related to sports activities.
The case study involves four 3-hour sessions. In the initial session the students are organised into groups of approximate size 5, and the rules of the case study are explained. The problem is how to make rock climbing a safe activity. Ropes are used to protect climbers who fall off a rock face. 50 metres of rope has to be carried to the rock face, so its mass must be moderate. The rope must be affordable and it must survive several years of use.
provides the sport, and some mechanics background
a) Introduction to climbing techniques - show
climbing video with falls for 10 minutes. A student debate follows to
establish the rope performance areas - expect to get:- tensile strength,
low weight, flexibility so can be clipped through carabiners, durability,
doesn't absorb water...
b) Rope flexibility in bending - If the polymer
were in the form of a solid rod of diameter D,
this would have 2nd moment of area I
If the solid rod is replaced by n fibres, the total cross sectional area A of polymer must be kept constant, so the rope has constant tensile failure load. Each fibre has diameter d = D / n1/2, so has 2nd moment of area
In a bent rope, as there is no bonding between the fibres, these act a separate 'beams', so the total 2nd moment of area is the sum of that of individual fibres (consider bending a stack of plastic rulers which are free to slide over each other)
Substituting typical values of D = 12 mm and d = 0.05 mm, the reduction in I is by a factor of 240. The bending stiffness of the rope is E IR and the Young's modulus E of polyamide fibre is only about 2 GPa, compared with 210 GPa for steel. Consequently the bending stiffness of the rope is low.
c) Fall factor - The mechanics analysis of the forces on the body, taken from the paper by Smith, assumes that the rope has negligible bending stiffness and constant tensile stiffness. The fall factor is defined as
The maximum value of the FF is 2, when a climber has climbed to the full
extent of the rope above the belay before he falls. The analysis uses
the energy transfer between
a) the potential energy of the climber mgh, before he/she falls
b) the kinetic energy of the falling climber, which increases with the
distance fallen, before the rope becomes taut
c) the strain energy of the extending rope, once it becomes taut. When
this reaches a maximum, the climber's downward motion is arrested.
It is assumed that the rope has a linear force F
vs extension x relationship,
with slope, for a metre length of rope, k
(N/m)
A rope of length L has stiffness kL
A longer rope is less stiff, because it extends more for the same force. However the kinetic energy of the climber increases with the rope length, so in the final equation the rope length does not appear. The maximum body acceleration, relative to the acceleration of gravity, is given by
If the rope has an effective Young's modulus E, the tensile stiffness of a metre length is given by
where A is the total cross section of polymer.
a) An introduction to rope manufacture (OU video)
followed by discussion of the difference between a twisted and a spliced
rope. Note how the polypropylene fibres are collected into yarn, then
into strands, which are twisted and heat set.
b) Information on the polymers used (nylon, polypropylene,
Kevlar, rubber) and how they are produced as high tensile strength
fibres. The table shows some data
|
polymer |
weight per unit of strength for 100 ft of 12.7 mm rope |
% elongation at failure |
cost |
|
Nylon(polyamide) |
2.9 kg |
20 | 4 |
|
polyester |
3.4 |
14 | 4 |
| polypropylene | 2.3 |
30-40 |
2 |
|
Kevlar 29 |
3.9 |
4 |
45 |
For use in a dynamic rope, the material must have
1) a high tensile strength. The materials selection
criterion for a light-weight rope of high tensile strength is 
where
is the tensile strength and the 
the density. The tensile strength of nylon and polyester fibres can reach
600 MPa, as a result of warm stretching of the solid fibres. In these
semi-crystalline polymers, the stretching aligns the c
axes of the polymer crystals almost perfectly with the length axis of
the fibres. In the polymer crystals, the c axis
is the direction of the covalently bonded polymer chains (Mills, 1997)
consequently this is the direction in which the crystals have the highest
modulus and tensile strength.
A double braided nylon rope of diameter 12.7 mm has a tensile breaking
load of 3700 kg compared with 2700 kg for a three-strand rope. This shows
that braiding can allow effective load transfer when some of the fibres
have failed. The braided mantel of the 'kern-mantel' (core/sheath) ropes
protects the core against abrasion damage from rocks etc.
2) a moderate tensile stiffness. Heat treatment of the nylon fibres at 120ºC is used to reduce the Young's modulus, without much of a reduction in tensile strength. Twisting the cords to provide a cord-straightening mechanism, and reduces the effective Young's modulus by a factor of about 2. In bungee cord, natural rubber filaments of Young's modulus only a few MPa is used. This allows the jumper to be arrested by low forces, consequently experiencing at most 2g acceleration. However bungee cord would cause the climber to be scraped up and down the rock face. For dynamic climbing rope, it is hoped that the falls are rare, and a g level of 15 is allowed.
Using the analysis of equation (6), if the peak acceleration of a 80 kg climber is less than 15 g for a fall factor of 2, the tensile stiffness of 1 m of rope must be less than 39 kN/m. For an 11 mm diameter nylon rope, since the density of nylon fibre is 1150 kg m-3, the average cross-sectional area of the nylon is 63 mm2. This means by equation (7) that the effective Young's modulus of the rope must be less than 0.6 GPa. However the Young's modulus of straight high strength nylon fibre is 2.0 GPa. Twisting or braiding the fibre reduces the effective Young's modulus by a factor of about.
In sessions 2 and 3 the students groups work on one of the following topics. For each topic, a useful web address and/or some starting article is provided. Four topics are
1) Dynamic climbing ropes
Useful web site: www.Bealropes.com
2) Durability of climbing ropes
See class.et.byu.edu/mfg340/quality/report/nylon.htm
for a report on factors that reduce the strength of nylon ropes
3) Bungee ropes
See www.fettke.com/bungee
and www.sci.wsu.edu/idea/Bungee
for a video clip of the jump and the partial differential equations related
to it
4) New sports using bungee ropes
See www.delphion.com to search
for patents. US 5421783 patent Human Slingshot machine is a description
of one such sport.
Further areas could be: static ropes for caving, or the webbing harnesses
used by climbers. During this period, students need access to the internet,
and if they are stuck, access to the staff member.
The group members may wish to allocate roles and tasks.
In the 4th session one or two members of each group make a 15 minute presentation, emphasising the materials properties and processing necessary to produce the particular product. There is 5 minutes for questions. The presentations are assessed in terms of a) structure, b) quality of visual aids, c) interaction with audience, d) whether the message got across, e) dealing with questions. The assessment, together with suggestions for improvement, are returned to each group later.
They write a 4 page illustrated report, linking their particular product with the materials used and the manufacturing method.
The learning outcomes are that students can explain
how
a) high strength is achieved in polymer fibres
b) rope bending flexibility is achieved
c) the tensile stiffness of a rope and the fall factor affects the maximum
force experienced
d) the rope properties relate to some sports applications
The students should, by attending the presentations, get a broad picture of the area, including environmental effects on rope durability (does absorbed water affect the strength of nylon?) and wear.
Feedback from the students
A feedback form asks the students for the best parts of the case-study,
and areas where they feel it can be improved.
Future plans - to develop rope testing related to this case study. Students could carry out dynamics drop test on a small scale, using a 10 kg mass falling 2 m, and measuring with a quartz load cell + computer interface, the dynamic force versus time. This could be converted to force vs rope extension, if the numerical integration of F/m is carried out.
