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transcript
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
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Non-Classical Problems in Stability
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?
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Non-Classical Problems in Stability
What is buckling? Leonhard Euler (1707-1783)
2)( L
EI P c
π
=
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Non-Classical Problems in Stability
What is buckling?
New structural shape
Buckled mode
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Non-Classical Problems in Stability
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.
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Non-Classical Problems in Stability
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.
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Non-Classical Problems in Stability
Many books on Classical Buckling/Stability Problems
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Non-Classical Problems in Stability
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).
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Non-Classical Problems in 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
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Non-Classical Problems in Stability
Energy formulation for a discrete structural system
Another school, butStill the same classical
theory
No general proof available
Eigenvalue
problem
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Non-Classical Problems in Stability
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
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Non-Classical Problems in Stability
Things you would like to know
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Non-Classical Problems in Stability
Classification of Critical States, according to Koiter
i
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Non-Classical Problems in Stability
Simple experiments on axially loaded cylindrical shells, onset of buckling
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Non-Classical Problems in Stability
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).
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Non-Classical Problems in Stability
Critical loads from 20 tests on axially loaded cylinders
(same students, same testing machine, same day)
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Non-Classical Problems in Stability
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)
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Non-Classical Problems in Stability
More things you would like to know
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Non-Classical Problems in Stability
Two classical forms of buckling, and their post-critical paths
Limit Point Buckling Bifurcation buckling
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Non-Classical Problems in Stability
2)( L
EI P c
π
=
Even more things you would like to know
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Non-Classical Problems in Stability
Examples of Buckling, leading to collapse or part of normal operation
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Non-Classical Problems in Stability
Influence of small and large imperfections on buckling strength
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Non-Classical Problems in Stability
Simulation of buckling process of a tank due to wind pressures
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Non-Classical Problems in Stability
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
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Non-Classical Problems in Stability
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).
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Non-Classical Problems in Stability
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)
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Non-Classical Problems in Stability
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
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Non-Classical Problems in Stability
Reformulation of the energy functional for non-holonomic systems
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Non-Classical Problems in Stability
Perturbation analysis of the Lagrangian
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Non-Classical Problems in Stability
Equilibrium paths
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Non-Classical Problems in Stability
Non-holonomic Systems: Changes in the boundary conditions
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Non-Classical Problems in Stability
Summary
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Non-Classical Problems in Stability
And that’s the end