Table Of ContentOTHER TITLES IN THE DIVISION ON
SOLID AND STRUCTURAL MECHANICS
Vol. 1. SAVIN- Stress Concentration around Holes
Vol. 2. GOL'DENVEIZER- Theory of Elastic Thin Shells
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Vol. 4. COX- The Bucklingof Plates and Shells
Vol. 5. MORLEY- Skew Plates and Structures
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THE BENDING A ND
STRETCHING
OF PLATES
by
Ε. H. MANSFIELD, Sc.D.
Senior Principal Scientific Officer
Structures Department
Royal Aircraft Establishment
Farnborough
PERGAMON PRESS
OXFORD · LONDON · NEW YORK PARIS
1964
PERGAMON PRESS LTD.
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Copyright © 1964
PERGAMON PRESS LTD.
Library of Congress Catalog Card Number 61-18328
MADE IN GREAT BRITAIN
PREFACE
I HAVE attempted in this monograph to present a concise, up-to-date
and unified introduction to elastic plate theory. Wherever possible the
approach has been to give a clear physical understanding of plate behav-
iour. The style is thus more appropriate to engineers than mathemati-
cians and although some of the topics have an aeronautical flavour, the
monograph will also be of value to structural engineers in civil, mechan-
ical and marine engineering, and to structural research workes and
students.
Small-deflexion theory is considered in Part I in six chapters entitled:
Derivation of the basic equations — Rectangular plates — Plates of various
shapes — Plates whose boundaries are amenable to conformai transfor-
mation — Plates with variable rigidity — Approximate methods. Each
chapter depends to some extent on the first, but is otherwise self-con-
tained. There are three chapters in Part II on large-deflexion theory,
entitled: General equations and some exact solutions — Approximate
methods in large-deflexion theory — Asymptotic large-deflexion theories
for very thin plates. These asymptotic theories are membrane theory,
tension field theory and inextensional theory.
To restrict the monograph to its present size it has been necessary
to omit virtually all numerical results and to focus attention on the methods
of solution. Numerous illustrative examples are presented but the steps
in the analyses are kept to a minimum. Attention is also drawn to the
classes of plate problem for which relatively simple solutions are available.
Certain topics are quite new while others have not previously appeared
in book form.
I am grateful to Professor W. S. Hemp for the invitation to write the
monograph, and to my wife Ola for her encouragement and for typing
the manuscript.
Ε. H. MANSFIELD
Farnborough
[ix]
PRINCIPAL NOTATION
x, y, ζ Cartesian coordinates, χ, y in plane of plate
r, θ, ζ cylindrical coordinates, r, 0 in plane of plate
U, V, w displacements in x, y, ζ directions'
€ By, €γ direct and shear strains in plane ζ = const.
χ χ
C, (Tyt Ty direct and shear stresses in plane ζ = const.
x X
Ν Ν Ν direct and shear forces per unit length in plane of plate
QX> Qy transverse shear forces per unit length
Qr> Qe
M, My, M
x xy bending and twisting moments per unit length
M, M, M
r e r9
curvatures, etc.
Ή>χ> Xy* Xy
X
E. G Young's modulus and shear modulus
ν Poisson's ratio
t plate thickness
μ Ht
D flexural rigidity, £/3/{12(l — v2)}
k foundation modulus
transverse edge support stiffness, or r/r
Q x
X rotational edge support stiffness, or complex potential
function
Ψ complex potential function, or — dw/dr
q transverse loading per unit area
coefficients in Fourier expansions for q
<ln> Imn
a, b typical plate dimensions
Φ force function
w w particular integrals
lf p
complementary functions
Ρ point load
t time, or tangent to boundary
η normal to boundary
s distance along boundary
Ψ angle between tangent to boundary and *-axis
[xi]
PRINCIPAL NOTATION XÜ
U strain energy
Π potential energy
Δ deflexion at reference point
α angle between generator and x-axis
M total moment about a generator
a
i) distance along a generator
dx2 dy2 dxdy dxdy dy2 dx2
CHAPTER I
DERIVATION OF THE BASIC EQUATIONS
ALL structures are three-dimensional, and the exact analysis of stresses in
them presents formidable difficulties. However, such precision is seldom needed,
nor indeed justified, for the magnitude and distribution of the applied loading,
and the strength and stiffness of the structural material, are not known
accurately. For this reason it is adequate to analyse certain structures as if
they are one- or two-dimensional. Thus the engineer's theory of beams is
one-dimensional: the distribution of direct and shearing stresses across any
section is assumed to depend only on the moment and shear at the section.
By the same token a plate, which is characterized by the fact that its thickness
is small compared with its other linear dimensions, may be analysed in a two-
dimensional manner. The simplest and most widely used plate theory is the
classical small-deflexion theory which we will now consider.
The classical small-deflexion theory of plates, developed by LAGRANGE
(1811), is based on the following assumptions:
(i) points which lie on a normal to the mid-plane of the undeflected plate
lie on a normal to the mid-plane of the deflected plate;
(ii) the stresses normal to the mid-plane of the plate, arising from the
applied loading, are negligible in comparison with the stresses in the plane of
the plate;
(iii) the slope of the deflected plate in any direction is small so that its
square may be neglected in comparison with unity;
(iv) the mid-plane of the plate is a 'neutral plane', i.e. any mid-plane
stresses arising from the deflexion of the plate into a non-developable surface
may be ignored.
These assumptions have their counterparts in the engineer's theory of
beams; assumption (i), for example, corresponds to the dual assumptions in
beam theory that 'plane sections remain plane' and 'deflexions due to shear
may be neglected'.
Possible sources of error arising from these assumptions are discussed
later.
[3]
4 THE BENDING AND STRETCHING OF PLATES
1.1 Stress-strain Relations
Let us consider now the state of stress in a plate with an arbitrary small
deflexion w(x, y) (See Fig. 1.1). The mid-plane is a neutral plane and accordingly
ο
FIG. 1.1
we shall focus attention on the state of strain, and hence the state of stress,
in a plane at a distance ζ from the mid-plane. The slopes of the mid-plane
dw dw
are — and — so that the displacements u and ν in the x^-plane at a
dx dy
distance ζ from the mid-plane are given by
dw \
and 1.1
dw
V = —ζ—·
dy J
The strains in this *,>>-plane are therefore given by
du
dx
d*w
— z
dv
1.2
du ^ dv
ε
dy dx
DERIVATION OF THE BASIC EQUATIONS 5
Now by virtue of assumption (ii) of para. 1 a state of plane stress exists
in the χ,^-plane so that the strains ε e, e are related to the stresses a %
χ9 y xy x
a, r by the relations
y xy
ε = — (σ -va)
χ Ε χ y
e = — {a—va) 1.3
y Ε y x
2(1 +v)
Equations (1.2) and (1.3) may be combined to give
Ez lldd2*ww dV
σ„ = —
Ez ld*w , d*w\
y + v 1.4
y = ~\ t \— ' dx*
1 — v2 \c
Ez d*w
1 +v dxdy
These stresses vary linearly through the thickness of the plate and are
equivalent to moments per unit length acting on an element of the plate
as shown in Fig. 1.2. Thus,
ζσάζ
χ
λ , d*w\
n
= — D\ \-v
\dx2 dy*f
M. ZOyUZ
1.5
id2w
\dy* dx*J
M = zTdz
xy xy
2*
2
dW
= —D(l —v)
dxdy
where the flexural rigidity D of the plate is defined by
Efi
D = 1.6
12(1-v2)
