Table Of Contentm application op mm laplace tBANSPOEMTicm
TO TBS STUB! OP C33IIAIH PBOELEMS IB ®S
DYNAMICS OP THIN ELASTIC PLATES
A Thesis
Presented to the Paealty of the Graduate School
of Cornell University for the Degree of
Doctor of Philosophy
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CherLet !$yrroll Vest
^uae, 1951
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AlOTOB»S BXOGiUm
The author was horn in Greatline, Ohio on 1 July 1915, the only
son of hafey Allen and M&belle g ill son West, I© was educated in the
Orestlino Public Schools, the United States Naval Academy, the Ohio
State University and the University of Michigan.
go is employed as Assistant Professor of Mechanics in the Ohio
State University, m leave for the current year, Previous academic
appointments were as Hobinoon bellow (1939~19*&) and Instructor
(19^6-1947) in the same institution.
His ea^erienoe consists of field mxk for the Ohio Department
of Highways, structural engineering research for the Association of
American Hail roads and service In the United States havy (19^1-19^6).
He is an associate member of the American Society of Civil
Engineers, and a member of the American Mathematical Society, the
American Association of University Professors, Sigma XI, Tau Beta PI,
Pi Mu Epsilon and Sigma Pi Sigma.
the author1 a Special Committee was composed of Professor
0* p* Jtaher; Structural Engineering, Professor Harry Pollard*
Mathematics, and Professor H. D. Cosway; Mechanics of Engineering,
dhuirstea*... 2he proof of the mean convergence of the inverse trans
form glveu in Chapter I is due to p*e$es«b;e Pollard. The
assistance of all members of the committee is gratefully acknowledged.
TABLI OP CONTENTS
INTRODUCTION
Chapter 1
m m o p m OP THE TE101I
Chapter XX
PHQBLXMS ON THS RESPONSE OP SIMPLY SUPPORTED
MSTANSULAl PLATES TQ DYNAMIC LOADS
(hap ter IIX
AN APPLICATION TO THE CASE OP KINEMATIC
BOUNDARY CONDITIONS.
Chapter X?
A GENERAL PROBLEM ON TEN DYNAMICS OF SIMPLY
SUPPORTED RECTANGULAR PLATES
CONCLUSIONS
Appendix A. GLOSSARY OP SBSQLS
Appendix B. BIBLIOGRAPHY
INTRODUCTION
The full owing paper presents a method of set ring certain problems
In the dynamics of plates by use of the Laplace tr&nsforas&tlon.
Because erf the lim itations of thlstransform , the method is restricted
In its application to linear problems, vis. those dealing with the
small deflections of thin elastic plates with linear boundary
conditions. '
It lb shewn that the Laplace transform With respect to time of the
deflection of such a plate satisfies the equation of the middle surface
of a statically loaded plate of the same shape, resting on an elastic
foundation, and having essentially the same boundary conditions. If
the solution of this transformed problem is known, the solution of the
original problem may be obtained at once by an inverse Laplace trans
formation. A number of solutions of plates on elastic foundations are
available in the literature, and each corresponds by analogy to some
dynamic problem. In any ease, approximate methods are available (at
least In theory) for the solution of such static problems.
The theoretical basis of the problem is discussed, and a proof is
offered of the convergence of the inversion process in case only &
suitable approximate solution of the static analogue is available.
In this proof, the middle surface of the dynamic plat© is subjected
to conditions of continuity and differentiability which may appear
rather stringent from a mathematical viewpoint, but which are
physically quite reasonable. The derivations of the plate equations
and of the fundamental proper ties of the Laplace transformation are
not given, inasmuch as thee# matters are treated in the standard texts
and such knowledge is now a part of the equipment of all scientific
engineers.
In order to illustrate the use of the method, several problems
of engineering interest are solved. These deal with rectangular
plates subjected to dynamic loads of various distributions and time
characteri sties. Suggestions are made for the extension of the method
to more difficult types of problems.
Chapter X
Of THE THEORY
tor the development of the method presented in this paper,
ire shall confine our discussion to rectangular coordinates.
Special oberdlnate systems may be preferable in the solution ef
problems, but the technique set fo rth fo r the rectangular case
is applicable without modification, and the use of rectangular
coordinates in this chapter will simplify the discussion.
Consider the plate shown In Figure 1. Its thickness is h
and we take the y ^ - plnhe as the middle surface ef the plate
In itp undeflected position. The plate is supported in some
unspecified manner, and carries a distributed load per unit
area. The lead may be a function of position, and will be
regarded at positive when it acts downward.
i
z
Figure 1
If we consider the element hdxcfy , shown enlarged in Figure 2,
this element is noted upon toy moments and shears per unit of length
as shown.
‘/ nV 1 - / - Y -
i
Figure 2
3
la add! tie a to the so shear* and moments we haw the load ^ which
must include surface and body forces. fhe shears Q and the
moments M aw given per unit of length along the edge upon which
they act. forces are shown by single-headed arrows, while moments
are shown by double-headed arrows, the moment being in the sense of
a doctewis© rotation about the axis of the representative arrow.
It is shown in the treat!ses^ on plate theory that the
deflection vv of the middle surface of the plate from its unloaded
position satisfies the equation
VV|V = t|- (1.x)
whar.®
v * = v *(v .) s (Jg. + + | i )
= j. 7 -fa!— 4-
a;)1.
b
and . 3
T ) s E —
v ia(i-v * ) •
1, See any of the reference® in Appendix 1 on the theory of plates.
In particular, ef* Reference S, Chapters II and IP.
2. The symbols used are defined in Appendix A*
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