Table Of ContentLecture Notes in
Engineering
Edited by C. A. Brebbia and S. A. Orszag
65
J. Holnicki-Szulc
Virtual Distortion Method
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J. H. Seinfeld . P. Silvester· P. Spanos' W. Wunderlich' S. Yip
Author
Jan Holnicki-Szulc
Institute of Fundamental
Technological Research
Swietokrzyska 21
00-049 Warsaw
Poland
ISBN-13: 978-3-540-53779-3 e-ISBN-13: 978-3-642-84453-9
001 10.1007/978-3-642-84453-9
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INTRODUCTION
Virtual distortions are incompatible deIormations imposed on
structures (e. g. , by nonhomogeneous heating or by local
imperIections) causing a compatible state OI deIormation (initial
deformations) and a selI-equilibrated state OI stress (initial
stresses). The theory OI virtual distortions provides an eIIicient
tool which can be used to treat many problems that diIIer Irom the
physical point OI view. Thermal stresses, material deIects,
residual stresses in plasticity are examples OI problems that can
be analysed emploing the virtual distortions approach, where the
temperature Iield, dislocations and plastic distortions (permanent
plastic deIormations) are described by virtual distortions. The
presented approach allows the development OI eIIicient computational
methods Ior numerical analysis OI such problems.
States OI compatible deIormations and selI-equi1 ibrated
stresses caused by virtual distortions in homogeneous and compound
(biphase) bodies are analysed in the Iirst part OI the book. The
results OI these considerations are explored in the next two
chapters. The Iormulation OI the approach discussed in Chapter 1 and
based on continuum mechanics was presented previously in the papers
[5,17] while its application to the analysis OI skeletal structures
was discussed in [47].
The second part OI the book is devoted to the simulation
(modeling through virtual distortions) of structural modifications,
where modiIications are understood as any changes OI structural
geometry (e.g. ,material redistribution) or physical properties
(e. g. ,generation OI Iractures or plastic zones) in the design or
loading process. In part icular, the Virtual Distortion Method (VDM)
was applied to various problems:
IV
a) successive modifications of material distribution [the
procedure SIMAT - SImulation of MATerial modifications],
b) optimal material redistribution [the procedure SIREM
SImulation of optimal REModeling],
c) material nonli near it ies due to stress limits [the procedure
SISCO - SImulation of Stress COnstraints],
-progressive collapse by the successive simulation of
fractured elements
-progressive collapse of structures with yield conditions
-simulation of rod-cable structures with non-negative stresses
in the cable elements.
d) simulation of structures with locking properties [the procedure
SINCO - SImulation of strain COnstraints],
e) simulation of slackened structures (with initial clearances)
[the procedure SISGA - SImulation of Structures with GAps],
All the above procedures have been developed together with Dr
Jacek T. Gierlinski on the computer system FADSYS [Frame Analysis
and Design SYStem] [64].
In the case of design changes a comprehensive review of
relevant re-analysis methods based on the concept of .so-called
compensative forces was presented by Kassim and Topping [13].
However, the modeling of structural modifications based on the VDM
concept (following presentation in the papers [15,16]) provides a
simpler formulation of the simulation rules and gives an easy
physical interpretation. This is particularly important in the case
of combined problems (e.g., optimal design of elasto-plastic
structures) where one component of the distortion field can describe
permanent plastic deformations and the other can describe material
redistribution.
The problems of optimal structural control by means of
passively or actively imposed virtual distortions are described in
the third part of the book. The optimal structural prestressing in
different formulations for homogeneous and compound bodies as well
as for large rod structures are analysed in the section devoted to
passive control. The effect of significant reduction of stress or
v
strain concentrations can be achieved by prestressing (due to
distortions imposed in a purposeful way). Structures with
distortions controlled in real time are discussed in the sections
dealing with active control. Problems which are very important from
the engineering point of view like active shape control of large
parabolic mirrors or active damping of vibration in large, flexible
space structures can be efficiently solved by making active
corrections of virtual distortions.
Passive control problems such as optimal prestress of
structures are well known and explored in many structural
applications. However, such active control problems as quasi-static
shape control (cf. [36-39]) or problems of active damping of
vibration in flexible structures (cf.[41,44,46,48-50.53,60]) began
to be widely discussed in the eighties. In all these formulations
the force actuators applied in the control systems are modeled as
external loads. On the other hand, the VDM concept applied to
structural control results in a closed, self-controlled system with
control parameters describing the virtual distortions introduced
directly into the elements of the structure, either by thermal
heating, the piezoelectric effect or by hydraulic or pneumatic
servo-systems.
The objective of this book is to present a general concept of
the VDM method with the necessary theoretical background (Chapter 1)
and a variety of its applications to problems of structural analysis
and design. The book is focussed more on theoretical aspects of the
problems than on the practical design of structures. Nevertheless, a
number of numerical algorithms discussed below has already been
developed as a computational system capable to solve various
problems of structural analysis.
The results presented in this book have already been published
in part. In particular, Chapter 2.1 contains some excerpts from the
paper [15], Chapter 2.2 from the paper [27] and Chapter 3.3 from the
papers [26,66].
CONTENTS
Introduction
1. Analysis of initial stresses and deformations caused by
virtual distortions 1
1.1. Distortions in elastic body 1
1.2. Analysis of stresses and deformations caused by
distortions 6
1.3. Examples of structures with distortions 12
1.4. Distortions in biphase body 17
1.5. Surface distortions 25
2. Simulation by virtual distortions 30
2.1. Simulation of geometrical and physical modifications
in structures 30
2.1.1. General description of simulation concept 32
2.1.2. Material distribution modifications 35
2.1.3. Simulation of unilateral constraints on
stresses 40
2.1.4. Simulation of unilateral constraints on
deformations 47
2.1.5. Simulation of slackened structures 52
2.1.6. Discussion 56
2.2. Simulation of shape modifications in continuous bodies 59
2.2.1. Shape modification in elastic bodies 59
2.2.2. Optimal remodeling in 2D continuum 63
2.2.3. Generalized formulation - the Michell
structure 69
2.2.4. Conclusions 72
3. Optimal control by means of imposed virtual distortions 74
3.1. Passive control of stresses (prestress) 74
3.1.1. Formulation of prestress problems 74
3.1.2. Minimization of global measure of stress and
deformation states 76
3.1.3. Maximization of load capacity 81
3.1.4. Examples of prestress applications 84
3.1.5. Optimal surface prestress 96
VII
3.2. Quasi-statical active control problems 106
3.2.1. Active control of structure stresses 107
3.2.2. Examples of active control of stresses 114
3.2.3. Active control of structure deflections 123
3.3. Active damping of vibration 128
3.3.1. The concept of damping through actively
controlled virtual distortior$ 129
3.3.2. Simulation of viscous damping 136
3.3.3. Modal strategy of damping 141
3.3.4. Four degrees of freedom example 145
3.3.5. Conclusions 146
3.4. Active strategy of avoiding resonance 150
3.4.1. Optimal strategy of avoiding resonance 153
3.4.2. Example of beam with controllable support
stiffness 158
3.4.3. Free vibration of beam with elastic support 165
3.4.4. Conclusions 169
References 171
LIST OF THE MAIN SYMBOLS USED IN THE BOOK
CO state of virtual distortions
compatible component of the virtual distortion state
£0
-c
eO self-equilibrated component of the virtual distortion state
-r
L L
~ ,~ states of strain and stress due to the external load
~ R ,~R ,~ R initial strains, stresses and displacements due to
the virtual distortions
L R
c=e +c
] final states of strains and stresses
L R
0=0- +0-
A tensor of elastic properties
p external load
u externally forced displacements
U specific energy of deformations per unit volume
D infl uence matrix describing de:format ions caused by uni t
distortions
A set o:f modi:fied elements o:f the structure
E Young's moduli
v Poisson's ratio
o Kronecker's symbol
9 temperature
M mass matrix
K sti:f:fness matrix
C damping matrix
A eigenvalues
Qi eigenvectors
j
1. ANALYSIS OF INITIAL STRESSES AND DEFORMATIONS CAUSED
BY VIRTUAL DISTORTIONS
1.1 DISTORTIONS IN ELASTIC BODY
Problems of initial states caused in an elastic body by
distortions have been analysed from three points of view:
as thermal distortion problems (e.g., [11,56])
as plastic distortion problems (e.g., [30,32])
as problems of distortions model ing local material defects
(e.g., [19,21,33,51,57]).
Let us consider a generalized description of virtual
distortions introduced to the body. This chapter discusses the
analysis of stresses and deformations caused by a distortion state
and also the problem of identification of distortions causing
specified initial states of stresses and deformations.
Consider a homogeneous elastic body which occupies in R3 a
simply connected region V limited by a regular boundary A (Fig.1.1)
The external load is composed of the field X of body forces in
the region V, the field p of boundary load over the section A of
p
the edge as well as the field u of forced displacements over the
section A of the edge (A = A v A ). Assume that the sect ion
u p u
A and A are simply connected areas on the surface A.
p u
The solution is confined to ?tatic problems, small
deformations and regular tensor fields, the components of which (in
the Cartesian system xi' i=1,2,3) are functions of the class C2.
Along the boundary surface A of the body an orthogonal
coordinate system tK (K=1,2), n is defined where n is the unit
vector externally orthogonal to this surface (cf.Fig.1. 1).
2
E
O..J
ow ';'---a:::---f-O
« o
Ap
I
01o1
Fig.l.l Notation used in Fig.l.2 Strain-stress
description of loaded body. -distortion relation.
The initial state of deformation can be presented as a sum of
introduced distortions and deformations related by the Hooke law
to induced initial stresses [5]:
£ R (1.1)
where A denotes the tensor of elastic properties,
whence, defining the virtual stresses I!.0 = A ~o (cf.Fig.1.2), we
obtain constitutive relations for the initial states:
(1. 2)
Considering the final states which are a superposition of
the initial states and the states due to the external load
R L R L
~ = I!. + I!. ' ~ = ~ + ~ , we obtain modified constitutive relations
in which the effect of virtual states on the redistribution of the
final states is taken into account:
~ = A ( £ _ £0 ),
(1.3)
Here the restrictions imposed on the final state of stresses I!.
