Some Problems Relating to the Deflections of Rectangular Membranes and Plates

66 Pages·4.768 MB·English

by  Chang, Chi-Chuan

m m pmui&m mtATim to the deflect sons
of rectangular m m R m & s am plates
A Thesis
Presented to the Faculty of the Graduate School of
Cornell University for the Degree of
Doctor of Philosophy
By
Chl-Chuan Cheng
June, 1951
ProQuest Number: 10834581
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u ift th# Central University Aft
fte aftfttfitaat in tli# H* t * daisatt&ieni until
when ft# cam* to th# M M States. After
year in H M M $til# ClM«Pt M t
M raaeivin# theft.S*4mgmm in in
he can* to Cornell university* H» f$ now a eandidat# for
tfc*M*la§lit *f'-Mftt#r ftf ^Aiioftoj^iy with ssajor subject in
Haehanift# endminors t« £l%*id Mechanics and Mathematics,
The work os has bmmn carried
OiH th« of Profoisor
f§# .0, t Cftalrstan of the Special Otwiittee* The
author wishes to take this
his high indebtedness «sd dt«p to hist for his
valuable suggestions sod helpful criticisms during the
of this thesis
In addition, the author wishes to express his sincere
appreciation to the other member* of his Special Committee,
Frofsssor 0, f, Curate? on© Professor -fc. X talker, for
their kind guidance throughout the progtom of his graduate
study at Cornell University*
Chi^Chuan Chang
Comoll diversity, Ithaca, Um V
June,19£l
ill
TABUS OF OOifTEOTS
Peg©
The Largo Deflections mi fleetangola* Fist## * * * 1
dotations « * * . » « • • * » . • < « * * * » * * 2
Introduction , . . « * . < * . « « » « « • • • « 3
III# lA&XfoxssXy Loaded Membrane . . . * • « * * • * -6
The Uniformly Loaded Flat## « . . * .......... W
Tables and Graphs 22
Siblingraphy . . . . . . . . . . . . . . . . . . 29
PART XI
The Banding ©f Rectangular Plato# with All Edges
Clamped Subjected Simultaneously to thilomiy
Distributed Lateral Load and to Tonsil# or
Coiapresslv© Foroao In the Flan# of the Flat#. * * 31
dotation# * . * . * * « . * « * . . * » . . . * * 32
Introduction 33
General Solution ........... 36
Results 43
Tab!## and Graph# . . . . . . . . .............. 46
Bibliography . . . . . . . . . . . . . . . . . . 59
Appendix . . . . . . . . . . . ........ . . . . 60
itr
P A R T I
rm or , mczAmmm
fiates
i
m m r w m
a In x*»directien
b--:- in
h
ixtt il^ origin
M t
m at
m&ximm deflection
w©
y t v displacements in x~ and y~
q
normal forces par unit length of sections of
plate perpendicular to x~ and y«disections,
respectively
shearing force in direction of y~axl$
length of section of a plate
x^-axis
unit elongations in x- and y-directions
16*
shearing strain
^xy
p the Airy stress function
Young*s modulus
Polsson9 $ ratio
flexural rigidity of plate
* n9 pt q» rf st represent Integers
2
tt CSVfflsQ mutt be thin and their deflections may be
la rg e *
^ Hie metal covering is generally divided Into a number
heads or other structural members. Each area can he con«
aids rod as a rectangular elastic plats subjected to
various losds ufidor cartair* edge conditions, The analysis
of plates loaded in this manner is complicated by the
fact that the deflections of the piste may ho comparable
in magnitude to the plats thickness* In such cases
KXrchhoff *£> linear plats theory may yield results that
are considerably in error and a more rigorous theory
that takas account of deformations in the middle surface
should he applied.
The fondkmentsl norw linear large deflection equations
4
for the more ewitt by Van kanfrn
if* !fI0^1^ These equations have fe##b solved by three
is«#ly# the energy method,^) the finite-
difference method, t3) and th# Fourier series method. (4)
Among these methods levy*® solution by Fourier soria*
is th© only m m mi a theoreticsily ©scact nature but the
numerical result® can be obtained only af^er gnat labour.
FeppiH method consisting of a combination of th© known
solution given by th© theory of small cleflection and the
result obtained from th© membrano theory by energy method
so©##-'to bo the simplest and mot applicable.
in the linear small deflection theory we simply ne­
glect the membrane effect and consider bending only*
Stretching the middle surface, however, Is a necessary
consequence of the transverse deflection. As the deflec­
tion Increases, the membrane effect becomes more prominent,
until for deflections mny times larger than the plate
thickness, the membrane effect Is predominant whereas
the bending stiffness is comparatively negligible* This
is the basic reasoning of Fdppl • a method which proceeds
as follows; In a uniformly loaded, initially flat,
rectangular thin plate, consider the uniform lateral
pressure q as consisting of two separate pressures
* Numbers in the parentheses refer to the Bibliography*
5
and <|2# the form** earning only bending and the latter
only pur# stretching ©f thm plat#. The pressures q^ an^
q$ 3H M la terms ef th# m t i m deflection
Wo and ax# added t© five expressions fox uniformly loaded
thin plat## in tslileb both bending and stretching take
In th# following investigation, Levy*# Fourier
series method Is first adopted to solve th# membrane
problem. Th# sqya re plat# 1# then used 3s an example
to illustrate Foppl*# method for obtaining th# solution
of th# large deflection of plat##. Th# results ar# com*
pared in graphical for® with those obtained by other
methods.