Matrix Methods of Structural Analysis

471 Pages·1964·25.225 MB·English

by  R.K. Livesley

Pergamon
Unified Engineering
Series
II
GENERAL EDITORS
Thomas F. Irvine, Jr.
State University of New York at Stony Brook
James P. Hartnett
University of Illinois at Chicago Circle
EDITORS
William F. Hughes
Carnegie-Mellon University
Arthur T. Murphy
Widener College
Daniel Rosenthal
University of California, Los Angeles
SECTIONS
Continuous Media Section
Engineering Design Section
Engineering Systems Section
Humanities and Social Sciences Section
Information Dynamics Section
Materials Engineering Section
Engineering Laboratory Section
The Stability of
Elastic Systems
S. J. Britvec
University of Stuttgart
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Preface
The place of stability theory in modern elastomechanics has gained prominence
in the last two or three decades. Demands in the analysis and the design of light
and highly stiff metal structures of many types capable of relatively high load-
carrying capacities have been indirectly responsible for considerable advances in
the theoretical knowledge in this area. Many contributions in this field have been
recorded in the last ten years throughout the scientific and technical literature, but
no unified approach has brought many of these results together. The purpose of
this work is to present to the student and the practicing engineer in one volume
some of the most important aspects of the stability and the non-linear behavior at
finite deformations of several types of structural elastic systems, which are
important for a more precise understanding of the statical performance of such
systems. The purpose is also to complete parts of classical (eigenvalue) theories
of buckling and to demonstrate the important role played by finite deformations
in the theoretical analyses of stability.
Some of the material in this work is derived from the author's original research
and some from his lectures delivered on this subject at several universities, prin­
cipally while he was teaching at Cornell, Harvard and the University of Pittsburgh.
The unstable and stable non-linear statical forms of symmetrical and non-sym­
metrical elastic systems, discussed in the subsequent chapters, were discovered
by the author at Cambridge University in England in the late 1950's. His work
was carried on by several other investigators, and the author wishes to use this
opportunity to acknowledge their contributions. These are referenced later in this
book. The writer has endeavored to present the material in a sequence helpful to
the reader unfamiliar with the more recent developments in this field, as well as to
include the topics that may be of interest to research workers in the future. The
emphasis in the book is on the basic principles which are illustrated in a variety of
applications. Experimental verification of the theory has been included wherever
this appeared to be desirable or even necessary.
The author is indebted to Professor W. Zerna for reading the whole manu­
script and for making valuable suggestions particularly in regard to its final
xi
xii Preface
format. He wishes to thank his graduate students Mr. Ming T. Yu and Mr. Donald
A. Hoecker for checking problems and exercises in the text. Further, he would
like to thank Professor A. H. Chilver for a discussion of a recent investigation
into the imperfection-sensitivity of some structural systems described in Sections
1-11 and 1-12 in Chapter 1. Finally, sincere thanks are due to Professor W. F.
Hughes for his continued interest in this work and for several helpful suggestions
during its preparation.
Pittsburgh, Pennsylvania S. J. BRITVEC
Introduction
Analysis and efficient design of many practical light, metal elastic structures are
sensitive to the precise relationship between the external loads and the states of
deformation, particularly when this relationship becomes non-linear. Then large
distortions may be accompanied by small changes in the external loads, while the
material deforms according to Hooke's law. Systems which most commonly
exhibit this property are two- and three-dimensional light structures, such as light
trusses, frameworks, reticulated shells, geodesic and triodetic domes, etc. This
type of behavior becomes apparent particularly when the members are slender and
when the axial loads in the members are high or critical. Non-linearities in the form
of paths of equilibrium states resulting from finite geometrical changes would not
be of material importance to the analyst and designer, in order that his design may
be safe, if a possibility of sudden motion from rest did not exist in these equilibrium
states. This concerns therefore, the change of a physical state in a conservative
static system (the structure and the loading it supports) when in a critical range of
static loading the system may be set into motion by a small disturbance.
To detect the unstable equilibrium states and to prevent this motion, or an
onset of large distortions, yïm te deformations of the whole structure must be con­
sidered in the analysis. The designer is therefore, faced with the problem: to
establish the actual critical state of the system and to ascertain if collapse by
motion is imminent. This requirement is essential in order that the system may be
safe and of practical use.
Unfortunately, it is not possible to give a simple or unique criterion for the
onset of unstable motion for all categories of practical structures, because the
criteria differ from system to system, depending on geometrical, material and other
properties.
It stands to reason to expect the systems which are so sensitive to minute
geometrical changes to be also sensitive to small initial geometrical imperfections,
and that these could radically alter the equilibrium configurations of a structure.
This, indeed, is confirmed later in this work. Such considerations become of
XIII
xiv Introduction
practical significance particularly in the design of slender space frameworks and
light but highly stiff reticulated shells.
A remarkable feature of many non-linear systems is that initially their
deformations depend linearly on the external loading. Sometimes the stiffness of
the structure is so high that initial deformations play a completely secondary part.
Most conventional analyses, in fact are based on an assumed linear relationship
and those which are, do not usually yield reliable solutions for statically unstable
structures.
Recently, elastic systems composed of discrete elements of one kind or
another such as geodesic domes, reticulated shells, etc. have gained in import­
ance. Their usefulness and economy in covering large areas, withstanding high
underwater pressures, or enclosing large volumes is well established. High
demands therefore, are placed on the accurate performance and reliability of
various shell-type structures composed of discrete elements. This requires a
correct understanding of the fundamental behavior of elastic systems at critical
deformations. Many systems of this kind have been developed and built by
empirical or semi-empirical methods, but so far, a synthesis of the fundamental
aspects in this behavior is lacking.
Independently, theoretical knowledge of the statical response in non-linear
elastic structures has progressed considerably in the last twenty years and
useful results have been obtained capable of imparting a more precise under­
standing of elastic instability. It is important to remember that instability results
from a change of physical states and that usually motion is the cause of failure
and not the material breakdown from a state of equilibrium. This breakdown may
occur subsequently in the motion or at large deformations which sometimes occur
in stable equilibrium states.
In this work some of the basic characteristics of the non-linear behavior of
elastic systems under static (or nearly static) loading are described, although
a detailed analysis of the more complex three-dimensional structures or reticulated
shells is outside the present scope. Also, only flexural post-buckling of single
members and whole systems under finite post-critical deformations is considered
at this stage. The author regrets that it was not possible to include a discussion
of other forms of post-buckling, such as the lateral, torsional, torsional-flexural
post-buckling of beams and columns of thin-walled cross-sections or buckling by
creep at finite post-critical deformations, as this would increase the size of the
book beyond the intended limits. It is his hope that at some later date these topics
can be put into book form to complement the present theories. Also, a statistical
treatment of the imperfection-distributions in various systems and the probabilis­
tic analysis of critical loads, depending on these distributions, had to be omitted
for the same reasons. Some relevant publications on this last topic are given at the
end of the book under "References and Related Bibliography," but these are far
from being exhaustive. The main purpose of this work is to present to the reader
the most important knowledge of statical stability necessary for the analysis and
the design of imperfection-sensitive structures composed of simple discrete
elements.
Introduction xv
All discrete elastic systems, or such systems that may be represented as
discrete, are subject to some common laws which govern their elastomechanical
behavior and usefulness. For example, the non-linear forms of the equilibrium
paths under conservative static loading which lead to dynamic failure are not
unlimited in number, but they are repeated in similar systems so that these can
be categorized accordingly. This in turn helps to distinguish the important
properties which unite or divide different categories.
We study in Chapter 1 the principal forms of equilibrium paths which are of
interest in practical structures composed of discrete elements. These paths occur
repeatedly in structural systems analyzed in subsequent chapters. Moreover,
we establish the relevant criteria which may be used to determine the stability
or instability of a certain equilibrium path and not merely its shape, in order that
similar paths may be readily recognized in different systems in practical situations.
This requires a systematic introduction to which Chapter 1 is devoted. Systems
are categorized according to their common properties on the broadest basis using
an established generalized analysis. Use is made of simple variational principles
in deriving some of the basic equations, since variational techniques based on
small perturbations lend themselves particularly well to the solution of this
problem. The non-linear forms of equilibrium paths studied in Chapter 1 do not
cover all known cases. However, the basic methods of analysis given here may be
used in a variety of other situations.
Basically two approaches are established. The energy and the equilibrium
approach of analysis and theorems and criteria are derived for their use. Many
examples of practical structures are solved theoretically using the established
theory and these are substantiated by experimental data obtained from tests on
model structures.
Solutions are given in the analytical or algebraic form. A general theory is
developed so that numerical techniques for the solution of more complex frame­
work problems may be devised. The emphasis in the theoretical treatment is
on the fundamental aspects to explain the physical behavior and to establish
the working tools for the analysis of practical structures for which the elastostatic
laws are non-linear.
Several features in the analysis should be emphasized for easier reading.
The exact theory in Chapter 2 leads in the end to very useful results which are
simple to apply in physical situations. It permits a reduction of all cases of plane
buckling of prismatic members to merely five basic relations to suit the more
accurate and First-ordert Theories which can be applied systematically to a
variety of otherwise involved problems using the coefficients tabulated in
Appendix I. This is illustrated by means of examples and by detailed solutions
of problems. The analysis of pin-jointed structural systems is mathematically
quite simple so that algebraic solutions of complex problems may be obtained.
The general law for pinned (or nearly pinned) members is established by Eq. (16.2)
in Section 2-16, Chapter 2, as a special case of the expressions derived in Section
tHere, the First-order Theory is not to be confused with the conventional analysis of buckling.
xvi Introduction
2-14. This law simplifies considerably the analysis of complex structures since it
is sufficiently general to give a correlation between the finite changes in the axial
forces and the deformations, regardless of the choice of buckling modes. Correla­
tion to a particular mode is automatic once the equilibrium equations are written
in the final form. Several cases of practical structures are solved and tested
experimentally which confirm this theory with reasonable accuracy. Two- and
three-dimensional systems are considered in Chapters 3, 4, 5 and 6 covering
the buckling analyses of pin-jointed, rigidly-jointed and portal plane frameworks
at finite deformations. Further classes of statically indeterminate two- and
three-dimensional systems and reticulated shells are analyzed separately in
Chapter 7. Non-conservative effects in plastic and dynamic buckling are studied
in Chapter 8.
Most chapters may be read independently after the reader has acquired
familiarity with the basic concepts which are presented in the first six or seven
sections of Chapter 1. Sections 1-8, a part of the Sections 1-10, 1-11 and 1-13 will
be found useful for a better understanding of the theory in later chapters.
Buckling of structural systems is shown to depend essentially on a branching
of equilibrium states. It is pointed out that the eigenvalue solutions of the classical
theories of buckling, which yield only the critical loads under ideal conditions,
are usually inadequate for the analysis and design of imperfection-sensitive
elastic structures and that more refined methods proposed in Chapters 1, 3,4, 5, 6
and 7 must be considered in an actual analysis. The branching paths in these
structural systems are usually unstable so that in the critical region of loading by
weights or mass the possibility of an unstable motion exists. Under static con­
ditions this is reflected in a marked reduction of the critical loads usually followed
by unstable motion. The actual behavior is in contrast to a widely ingrained myth
that elastic buckling of common frameworks takes place in essentially neutral
equilibrium. Most of the past studies of structural buckling are based on the
classical eigenvalue solutions of the equations of neutral equilibrium. This view­
point about practical structural systems has persisted for a long time even after
the initial discoveries of unstable motion in the buckling processes of thin-walled
shells were made. Only in the late 1950's it was discovered and confirmed
experimentally (7,9, 10, 13,21,93,95,107) that other common structural systems
of widely different geometries such as frameworks or shell-type structures
composed of discrete elements, which are not dominated by continuity and
non-conservative effects are, essentially, statically unstable, and, therefore
subject to an entirely different behavior from that previously accepted. It was
also established that equilibrium in the buckling range of all such systems may
be possible, but not feasible, because the non-linear equilibrium paths which then
exist are unstable. The basic non-linear forms of post-buckling equilibrium paths
of symmetrical and non-symmetrical elastic systems, discussed in Sections 1-4 to
1-9 and in Section 1-10 in Chapter 1 and their relevance to structural systems dis­
cussed in the subsequent chapters, were discovered by S. J. BRITVEC at
Cambridge University in England between 1957 and 1960 (7). Similar forms are
sometimes attributed to W. T. KOITER (58), but, initially, Koiter's work was