Table Of ContentTHEORY OF BEAMS
The Application of the
Laplace Transformation Method
to Engineering Problems
2nd enlarged edition
by
Dr. T. IWINSKI
Institute of Mathematics
Polish Academy of Sciences Warsaw
Translated from the Polish by
E. P. BERNAT
Professor of Mechanical Engineering
at the
Ahmadu Bello University, Zaria, N. Nigeria
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First edition 1958
Second edition 1967
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PREFACE
TO THE SECOND EDITION
THE amount of material contained in the second edition is
roughly doubled compared with the first edition. It was a
great encouragement for the author to continue his work
when the first edition of the book met with the approval of
readers and reviewers and was then translated into Japanese
[32].
The introductory sections of Chapter I and sections 1-9
of Chapter II of the first edition are retained unchanged,
but section 10 of Chapter II, which deals with beams of vari
able cross-section, has been substantially expounded and
appears now as Chapter III. It contains the fundamental
equations of the theory and also some examples on boundary
conditions, but not as numerous as those in the Chapter II.
This is because the theory of Chapter II may, without much
difficulty, be applied also to the corresponding cases of beams
with variable rigidity.
Throughout the whole book the stress is put rather on the
method used than on the comprehensive coverage of all the
important cases which may be met in engineering practice.
This approach has influenced the contents of Chapter IV.
Although the problems selected for discussion should be of
interest to all who specialize in the field of structural engineer
ing, the choice was decided upon primarily with a view to
illustrate the applications of the method (or more precisely
to illustrate some important subtle variations of the same
method).
vii
viii Preface to the second edition
The application of the Laplace transformation and the step
functions necessitates a regular use of some transformation
formulae. These formulae are tabulated, for convenience, in
Chapter V.
PREFACE
TO THE FIRST EDITION
THIS monograph is addressed mainly to technically minded
readers. Its principal aim is to discuss the solutions of the ordi
nary differential equations with assigned boundary conditions
which occur in the theory of beams by means of integral trans
forms, now called the "Laplace integrals" or "Laplace trans
formation".
This method of solving linear differential equations was
originally published by Laplace nearly one hundred and fifty
years ago (1812), and has been used by mathematicians along
side the other methods as a regular training in higher mathe
matics for a good many years. Engineers, on the other hand,
whose mathematical training, of necessity, covers a more re
stricted range are still not very familiar with the method. But
following the natural evolution of events, the engineer finds now
adays that he is continuously requiring more and more math
ematical knowledge, i.e. more powerful mathematical tools to
be able to solve the more complicated problems in ever ex
panding fields of engineering science. And so the engineer came
in recent years to realize the usefulness of the Laplace trans
formation method.
The method provides a form of shorthand by means of which
very quick and elegant solutions of a great many engineering
problems may be obtained, often saving much of the laborious
calculations involved in the classical approach. It may be ob
jected that these solutions are in a rather symbolic form and in
fact require conversion into the more practical form in order to
X Preface to the first edition
be directly amenable to physical interpretation and this, in
turn, may be either a difficult or laborious process. However,
the solutions can be given in standardized forms, e.g. arranged
as tables similar, say, to those of logarithms or other specific
functions, thus greatly simplifying the actual procedure of
analysis. And once the technique has been mastered, the meth
od of solution becomes more direct and straightforward than
the "classical" methods.
E.P.B.
Sheffield,
September 1957
Chapter I
INTRODUCTORY INFORMATION
1. General Introduction
The topics discussed in this book are taken from the field of
the theory of structures but from the point of view of the meth
od employed for solving these problems the book may be
considered as the study of some linear differential equation.
The solution of several simple equations proves to be not so
simple a matter. Much attention has been given to this problem
both by engineers and mathematicians, but it does not appear
that the subject is fully exhausted. The reason for this is the
nature of the practical engineering problem: complex boundary
conditions and complex loading.
The mathematical representation of these complications is
the appearance of discontinuities in the functions describing
the physical quantities or the geometry of the problems. Usu
ally, this is an ordinary discontinuity (see the note in section 2
of Chapter II), which leads to the appearance of the so-called
step functions.
The step functions are so characteristic for the structural
problems that without them it would almost be impossible to
formulate any general problem. Hence the concept of the step
functions (in one form or the other) had been introduced at
the very beginning of formulation of the theory of structures.
Interesting comments on this are given by J. E. Goldberg
[19]. Even such classical writers as A. Clebsch [10] and A.
Föppl [17] are connected with this subject. The interest in the
1
2 Theory of beams
matter has been recently revived afresh. In addition to the
names listed in ref. 19 the following may be added: W. H.
Macauley 125], J. Case [24], R. V. Southwell [23], A. J. S.
Pippard, J. F. Baker [22], C. L. Brown [21], E. Kosko [30],
S. Timoshenko [20], J. E. Goldberg [19], T. Iwinski [18],
J. Nowinski [9], T. van Langendonck [7] and S. Drobot
[12].
There are several good textbooks on the subject of the theory
of beams. A special mention deserves the book by S. E. Mike-
ladze [8] in which some very interesting methods of the mathe
matical analysis and of the differential equations are employed.
The lively interest in the subject in the present state of know
ledge is aroused not so much by the need of solving the prob
lems as for the sake of introducing new methods with the possi
bility of simplification of the solutions. The author feels that
a real simplification results from the use of the Laplace trans
formation method or more accurately from the use of Laplace
transformation together with some rationalized properties of
the step functions. Such a treatment not only leads to simplifi
cation of the theory of beams, but also makes this theory more
general.
Some interesting remarks on the application of Laplace
transformation to the technical problems are given by the
reviewer of the first edition of this book in the journal Physics
To-day, July 1959, pp. 43-44. The method of Laplace trans
formation as well as other methods of integral transformation
or of differential operators such as Carson-Laplace method
[31], Ditkin-Kuznetzov method [3] or Mikusinski method
[27], have been widely used for sometime in several branches
of engineering (e.g. in the theory of electric circuits, in the
theory of servomechanisms or in the theory of vibrations).
But in the theory of beams, or more generally in the theory of
deformation of solids, these methods have been introduced rela
tively late. Some examples on the theory of beams are included
Introductory information 3
in the excellent textbooks on the application of Laplace trans
formation by R. V. Churchill [2] and W. T. Thomson [26].f
In the last decade this lag has been well compensated for.
A substantial contribution is due to the Polish Scientists.
A reference to the Polish Scientific Publications (Archives of
Applied Mechanics, Engineering Dissertations, Proceedings of
Vibration Problems, etc.) will disclose that the operational
methods are a useful tool in solving problems in the fields of
strength of materials, theory of structures, elasticity, thermo-
elasticity, plasticity and theory of heat. A special mention here
deserve publications by W. Nowacki [28], [29].
The acceptance of the operational methods is understandable.
They provide easy mathematical algorithms similar to those
of the classical algebraic transformations; even in the more
difficult, non-typical cases, the solution often reduces to the
mere use of the tables of transforms.
As far as the engineers are concerned a difficulty may arise
in mastering of the theory, which, for example, in the case of
Laplace transformation will require familiarity with the ana
lytical functions. However, it may be pointed out that the
content of the mathematical knowledge of engineers nowadays
is constantly increasing and also that it is not imperative to
acquire a deeper knowledge of the theory in order to be able
to use the numerous available tables of the transforms (just
as the use of the mathematical or physical tables does not re
quire the knowledge of how these tables have been compiled).
In Chapter II beams of constant rigidity or composed of
sections of constant rigidity are discussed. Starting with the
differential equation of the deflection curve and with a uni
formly distributed loading, the solution of the equation is
obtained by means of the Laplace transformation. Then, by
means of a suitable definition of concentrated forces and cou-
t Compare the observations on the subject by J. E. Goldberg in Applied
Mechanics Reviews, May 1959, rev. no. 2280.
4 Theory of beams
pies, the author generalizes the solution to include the case of
simultaneous loading by concentrated forces and couples as
well as distributed loads (uniform and non-uniform).
It is shown that the elastic curve of a beam loaded in such
a general manner and resting on two or more supports is given
by the relation y = W(x) + S(x), where W{x) and S(x) are step
functions defined and briefly discussed in the introduction. The
function W{x) is a polynomial whose coefficients depend en
tirely upon the nature of the supports and can be evaluated from
the boundary conditions, while the second function S(x), called
by the author the load function, depends solely upon the load
ing of the beam. If the load distribution is known, the load
function can be determined beforehand independently of the
step polynomial W{x) either by direct application of the Laplace
transformation or by means of tables of the Laplace transforms.
In this way the problem of finding the deflection curve is
transformed from the problem of solving a differential equation
to that of solving algebraic equations. In short, it reduces to
finding the coefficients of the polynomial W(x) by solving a
system of simultaneous algebraic equations. Moreover, the
use of this method does away with the necessity of different
treatment of the iso- and hyper-static beams since the same
general relation holds in both cases.
The chapter ends with solutions for beams on two supports
under various conditions of end fixing (freely supported, built-
in, etc.) as well as for continuous beams both with rigid and
elastic supports.
Chapter III deals with beams of variable flexural rigidity
under the action of transverse loading. The loading is most
general. The elastic curve is deduced by the use of transfor
mation. It is shown that also in this case the deflection of the
beam y(x) contains two terms, i.e. y = 0(x) + S(x). The second
term S(x) is, as before, a function of the load (however, much
more complex because of the variable rigidity B(x)). The first
