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This module attempts to present the paper "Hencky-Prandtl Nets and Constrained Michell Trusses" by Gilbert Strang and Robert Kohn in a way that is accessible to undergraduate mathematics students.

Hencky-Prandtl Nets and Constrained Michell Trusses

Introduction

The paper “Hencky-Prandtl Nets and Constrained Michell Trusses” by Gilbert Strang and Robert Kohn [link] provides many interesting insights into the problem posed by Michell in his 1904 paper “The limits of economy of material in framed-structures” [link] . This module attempts to present the former paper in a manner that can be understood by undergraduate students in mathematics.

Michell's problem assumes a material with given tensile and compressive yield limits, say ± σ 0 , and allows concentrated forces. A force F can only be carried by beams with cross-sectional area F / σ 0 , so if the points of application of the forces are too close together, there will be difficulty in finding enough space. Because there is no restriction on the magnitude or the spacing of the external loads, which are surface tractions f distributed along the boundary Γ of the given simply connected domain Ω , one needs to be careful to ensure Michell's problem is self-consistent. To do this, we remove the limitation imposed by σ 0 and allow increasingly strong beams whose cost is proportional to their strength. This way, the cross-sectional area doesn't get too large, so there isn't a problem with finding enough space.

Variational forms of the stress problems

The following is a brief derivation of the Michell problem, where the stress constrains come from the equilibrium equations:

div σ = 0 in Ω , σ : n = f on Γ

where

σ = σ x τ x y τ x y σ y
div σ = d d x σ x d d y τ x y d d x τ x y d d y σ y

and its eigenvalues, or principal stresses, are

λ 1 , 2 = 1 2 ( σ x + σ y ± ( ( σ x - σ y ) 2 + 4 τ 2 x y ) 1 / 2 )

The formula for [link] comes from the quadratic formula for the 2-D Lagrangian strain tensor. Bars are placed in the directions of principle stress, which, by definition, come from the orthogonal eigenvectors of σ . The required cross-sectional areas at any point are proportional (scaled by σ 0 ) to | λ 1 | and | λ 2 | and the total volume of the truss is proportional to

Φ ( σ ) = Ω ( | λ 1 | + | λ 2 | ) d x d y

Michell's optimal design can be determined from the solution to

Minimize Φ ( σ ) subject to div σ = 0 , σ : n = f

To express the problem in a way that suggests a computational algorithm, we introduce a stress function ψ ( x , y ) . For any statically admissible stress tensor σ there is a function ψ such that

σ = ψ y y - ψ x y - ψ x y ψ x x

where the subscripts indicate partial derivatives. It is easy to see that div σ = 0 . In the first column, for example,

x σ x + y τ x y = x ψ y y + y ( - ψ x y ) = ψ y y x - ψ x y y = 0 .

ψ is determined up to a linear function, because all second derivatives of a linear function are zero. Given the unit normal n = ( n x , n y ) , the unit tangent t = ( - n y , n x ) , and the condition

σ x τ x y τ x y σ y n x n y = f 1 f 2 ,

ψ y y n x - ψ x y n y = f 1 or ( grad ψ y ) · t = f 1 and similarly for f 2 . Therefore the tangential derivative of ψ y is, by definition, ψ y t = f 1 or ψ y ( P ) = P 0 P f 1 d s by the fundamental theorem of calculus, and similarly for f 2 . In order for the constraints to be compatible, the external loads f 1 and f 2 must be self-equilibrating, and therefore the boundary values ψ x and ψ y are well defined: the integrals around the closed curve Γ are zero. Provided that there is no net torque from the external forces, ψ itself can be determined. Write ψ = g , ψ n = h , where ψ n is the normal derivative of ψ .

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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Source:  OpenStax, Michell trusses study, rice u. nsf vigre group, summer 2013. OpenStax CNX. Sep 02, 2013 Download for free at http://cnx.org/content/col11567/1.2
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