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This report summarizes work done as part of the Michell Trusses PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed around the study of a common problem. In this PFUG, we study cost minimizing trusses. We ask what is the structure, constructed of bars and cables, of least cost that will withstand a set of point forces. This work was studied in the Rice University VIGRE class MATH499 in the Spring and Fall of 2008.

Introduction

The Michell Truss PFUG studies a variational problem posed by mechanical engineer Anthony George Maldon Michell in the early part of the last century;what configuration of bars and cables is needed to withstand a system of balanced point forces is most economical?Suppose that some collection of forces with their points of application (point forces) is given and you are asked to build a structure which can withstand these forces.Your choice of materials are bars and cables. Bars can withstand compression (forces parallel and pointing inward) but will breakunder extension (forces parallel and pointing outward). Cables can withstand extension but will buckle under compression.You may also choose the strength of the bar or cable; the strength is the largestcompression or extension the bar or cable respectively can withstand. A structure built of such cables and bars will be called a truss.

The first question the PFUG has addressed is whether one can build such a truss at all. Notice that a bar or cable by itself canwithstand only one set of point forces; that of two opposite forces applied to the endpoints of the bar or cable. These point forces have the property that if the points of application were attached to a stationary body, the center ofmass would not accelerate and the body would not rotate. This property is called balanced. Since point forces can be linearly superimposed, a necessary condition that a set ofpoint forces be withstood by a truss is that it be balanced. Below we show that being balanced is also a sufficient condition; if the set of point forces is balanced then there is a truss withstanding it.

Next, the PFUG formulated necessary conditions for a truss to be economical. The cost of a bar or cable is its strength times its length. The cost of a truss is definedto be the sum of the cost of bars and cables constituted by the truss. In general, there are many trusses which withstand a given a set of point forces.The question is, which is most economical, i.e. which truss costs the least? In general this is a very difficult question to answer because a minimizing sequence,which we know to exist from the first part, may converge in a larger class of measures than described here and it is difficult to formulate any class of perturbations of a truss.Below we describe one class of perturbations on trusses with corners which yields a surprising necessary condition for economy. A set of perturbations to be studiedin the future are also described.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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