Non Classical Problems

8/3/2019 Non Classical Problems

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Non-Classical Problems in Stability

Non ClassicalProblems

in the Theory of Elastic Stability

Luis A. Godoy

Department of Civil Engineering andSurveying, UPRM




Topics covered in this presentation

What is buckling?

What is a classical problem (or a field, or a concept, or…)?

What are the present classical theories of Elastic Stability?What is a non-classical problem?

What are non-holonomic problems?




What is buckling? Leonhard Euler (1707-1783)

2)( L

EI P c

π

=




What is buckling?

New structural shape

Buckled mode




Lessons learned

Euler did not study what we are told he studied in 1742

Euler needed the input of previous research on elastica by Bernoulli

Euler was not the first to find the phenomenon: experimental work of van-Musschenbroek of

1729 showed the inverse relation between critical load and square of the column length. He

tested wood columns.

This was not an important contribution for Euler, only published one memoir on the topic. What we now call critical load is the value of the control parameter at which the behavior

changes.

What we call the critical mode is a deflected shape associated with reaching the critical load.

There may be several different frames to explain a phenomenon, and frames change withtime.




Knowledge may be considered “Classical” if…

Sociology of Science

It has permeated into formal levels of instruction:

Included in graduate education (INCI 6057).

Included in undergraduate education.

Included in K-12 as research projects.

It may be found in textbooks or monographs:

Graduate books (Thompson and Hunt, Godoy).

Undergraduate textbooks and manuals

(Timoshenko)

Popular science books.

It was formulated long time ago.

There are conferences on the subject (i.e.

Dynamics and Stability of Structures) It has been around for so long, that most

researchers in related fields know about it.




Many books on Classical Buckling/Stability Problems




Knowledge may be considered “Classical” if…

Epistemology

It falls within the structure of a theory:

“Normal science”, according to Thomas Kuhn.

“Protective belt”, according to Imre Lakatos.

It is an illustration of an established theory and providesconfirmation through evidence.

It is a new technique to solve problems defined within thetheory (finite element solution of stability problems).

It is an application of an established theory to a new case,under special conditions (Buckling of threedimensional solids).

It is an application of an established theory to a new field.

It provides the theoretical basis to reformulate another field(i.e. stability approach to fracture mechanics: it becomes non-classical for fracture mechanics, but stillit is classical stability).




Knowledge is “non-classical” only with respect to something which is considered

to be “classical”.

Classical theories

1744, Leonhard Euler (1707-1783)

1890, G. Bryan, Theory of Elastic Stability

1945, Warner T. Koiter, (1914–1997)

General Theory of Elastic Stability




Energy formulation for a discrete structural system

Another school, butStill the same classical

theory

No general proof available

Eigenvalue

problem




Lessons learned

You can formulate the theory first, then

discretize (the path followed by Koiter). Thereare schools of mechanics that accept this as the

only formally smart way to work. This is a

dominant school in France, Italy, Brazil,…

Your can discretize first, then formulate the

theory. This is a dominant school in England,

Canada, parts of the US,…

Energy

Continuous

Form-

ulation

Discrete

formulation

Discrete

Energy

Discretization

FE

model

FE

model

Discretization




Things you would like to know




Classification of Critical States, according to Koiter

i




Simple experiments on axially loaded cylindrical shells, onset of buckling




Lessons learned

What really matters is the onset of instability.

The initial postcritical behavior is determined by the stability of the

critical state.

To visualize the initiation of instability you need to perform

“displacement controlled” experiments, not “load controlled”.

Students prefer to do their experiments using Heineken, rather than Coca-Cola (limited evidence, but conclusive).




Critical loads from 20 tests on axially loaded cylinders

(same students, same testing machine, same day)




Lessons learned

Small geometric imperfections are responsible for a drastic reduction in

the buckling load with respect to the classical (Timoshenko) critical load.

The theory was applied to this problem by Koiter in 1963.

Koiter considered a deterministic imperfection, with a fixed shape and

variable amplitude, which was included as a new control parameter.

The theory was extended to account for small imperfections, of the order of the thickness of the shell.

The worst imperfection shape is that provided by the classical critical

load.

Imperfection-sensitivity may be

– High (Cylinder/axial load, sphere/uniform pressure)

– Moderate (cylinder/lateral pressure, cylinder/wind pressure)

– Low (column/axial load, plate/in-plane load)




More things you would like to know




Two classical forms of buckling, and their post-critical paths

Limit Point Buckling Bifurcation buckling




2)( L

EI P c

π

=

Even more things you would like to know




Examples of Buckling, leading to collapse or part of normal operation




Influence of small and large imperfections on buckling strength




Simulation of buckling process of a tank due to wind pressures




Examples of Non-Classical Problems in …

Non-Classical Problems in the Theory of Elastic Stability

Non-Classical Vibrations of Arches and Beams

Non-Classical Shell Problems

Non-Classical Elastic Solids

Non-Classical Continuum Mechanics

Non-Classical Physics




Knowledge may be considered “Non-Classical” if…

Sociology of Science

It is very new (nonsense). Most recent knowledge is

classical.

It may not be found in textbooks. “… none of the

subjects, touched upon in this monograph, have

been discussed exclusively in the existing books

on buckling analysis” (Elishakoff et al.).

Epistemology

It falls outside the structure of a theory.

It is not an application of an established theory to a

new case.

It goes beyond the basic assumptions of the theory, sothat it needs to be extended (stability of non-

holonomic problems). This pushes the frontiers

of a theory.

It merges a theory with another one to combine twofields (stability and probability analyses, stability

and sensitivity/optimization).

It has not yet found a proper place within a theory.

It challenges the theory to which it is associated

(Catastrophe Theory).




Basic assumptions in the General Theory of Elastic Stability

Discrete or continuous systems

Elastic material behavior

Static conditions

Conservative systems (an energy

functional exists)

Deterministic analysis (no

uncertainties)

One load parameter (even if there are

many loads)

Holonomic system (there are no

constraints on the values of

displacements, and the boundary

conditions do not change)

Instability of Inelastic solids

Dynamic Instability

Dynamic Buckling (Budiansky)

Non conservative systems (Leipholz)

Probabilistic systems (some random

features, Bolotin, Elishakoff)

Multiple parameter theory (Huseyin,catastrophe theory)

Non-Holonomic systems (Burgess,

our work)




An example of non-classical problem: Stability of Non Holonomic Systems

There are constraints on the values of

displacements, and the boundary conditions may

change

One-Way Systems




Reformulation of the energy functional for non-holonomic systems




Perturbation analysis of the Lagrangian




Equilibrium paths




Non-holonomic Systems: Changes in the boundary conditions




Summary




And that’s the end

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Non Classical Problems