A general non-linear theory of large elastic deformation of layered plates

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A general non-linear theory of large elastic deformation of layered plates
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Thesis (Ph. D.)--University of Florida, 1990.
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Includes bibliographical references (leaves 80-83).
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by Aref Altawam.
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A GENERAL NON-LINEAR THEORY OF LARGE ELASTIC
DEFORMATION OF LAYERED PLATES













By

AREF ALTAWAM


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1990













In the name of God, Most Gracious, Most Merciful


"God there is no deity save Him, the Ever-Living, the Self-Subsistent Fount of All
Being. Neither slumber overtakes Him, nor sleep. His is all that is in the heavens and all
that is on earth. Whlo is there that could intercede with Him, unless it be by His leave?
He knows all that lies open before men and all that is hidden from them, whereas they
cannot attain to aught of His knowledge save that which He wills [them to attain]. His
eternal power overspreads the heavens and the earth, and their upholding wearies Him
not. And he alone is truly exalted, tremendous."
(Qur'an 2:255)











dedicated to my parents

Nizar and Lamia Altawam


with all my love and gratitude












ACKNOWLEDGMENTS


The author would like to express deep appreciation to Dr. Ibrahim Ebcioglu

for his assistance and support which made this dissertation possible. Thanks are

extended to Dr. Bhavani Sankar for his continued assistance in both academic and

professional concerns. Thanks are also extended to professor Lawrence Malvern

whose insights were very much needed. And to Dr. Clifford Hays and Dr. David

Zimmerman who served as members of the supervisory committee, the author is

thankful.












TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ......................................... iii

ABSTRA CT .................................................. vi

CHAPTERS

I. INTRODUCTION ................................... 1

1.1 Introduction to the Theory of Plates ................... 1
1.2 A Prospective of the Theories of Multi-Layered Plates ..... 3
1.3 Principles of the Present Theory ...................... 14

II. GENERAL THEORY ................................ 16

2.1 Mathematical Preliminaries. ......................... 16
2.2 Strain-Displacement Relations ................ ...... 18
2.3 Stress-Strain Relations ............................. 19
2.4 Displacement Field ................... ........... 21
2.5 Hamilton's Principle The Variational Integral........... 25
2.6 The Constitutive Equations ............. .......... 28

III. THEORY OF THREE-LAYERED PLATES AND
SPECIAL CASES .............................. 32

3.1 Variational Integral for the Three-Layered Plate .......... 32
3.2 Kirchhoffs Hypothesis ............................. 35
3.3 Von KIrmin's Equations of Plate Large Deflection ....... 39

IV. THE SOLUTION TO THE SANDWICH PLATE
BENDING AND BUCKLING EQUATIONS ......... 43

4.1 The Assumptions of Eringen's Equations ............... 43
4.2 The Equations of Equilibrium and Boundary Conditions .... 43
4.3 Applications: Numerical Results and Discussion .......... 49
4.4 Stability Analysis of the Sandwich Plate ................ 56
4.5 Reduction to the Classical Plate Bending and
Buckling Equations ........ ..................... 61







V. CONCLUDING REMARKS ........................... 63

APPENDICES

A. NOTATIONS OF SOME EARLIER WRITERS ............ 65

B. SOLUTION TO PDE AND LISTINGS
OF COMPUTER PROGRAMS ................... 66

REFERENCES ............................................... 80

BIOGRAPHICAL SKETCH ..................................... 84












Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

A GENERAL NON-LINEAR THEORY OF LARGE ELASTIC
DEFORMATION OF LAYERED PLATES

By

Aref Altawam

December 1990

Chairman: Professor Ibrahim K. Ebcioglu
Major Department: Aerospace Engineering, Mechanics, and Engineering Science

Layered plates fall into several categories where the preliminary assumptions

as well as the methods of analysis are different, thus leading to multitudinous of

specialized theories. Starting from the three-dimensional theory of non-linear

elasticity, and using a displacement approach, much like in the Reissner-Mindlin type

theories, a general layered plate theory is developed. It has the advantage of

producing a complete set of fundamental equations consistent with various stages of

linearization in the general strain-displacement relations. The notion of stresses in

the reference state is employed in the fundamental equations, and a variational

procedure is extended to derive the equations of motion and the boundary conditions

of layered plates subjected to large displacements and large angles of rotation.

The present theory incorporates the effects of transverse shear and transverse

normal strains as well as rotatory inertia, with different thickness, material densities

and material constants in each layer of the plate. All stress components in each layer







are considered, and the continuity conditions between the contiguous layers are

maintained for all tractions. Each layer is assumed to be anisotropic, having elastic

symmetry with respect to its middle plane. The assumed displacement field is

piecewise linear in the thickness coordinate in all displacement components and

fulfills the geometric continuity conditions between the contiguous layers.

Theoretical development of the mathematical form of the nonlinear equations

of motion and the boundary conditions is presented for the case of three-layered

plate. In demonstrating the generality of the theory, several theories available in the

literatures are derived from the present as special cases. A numerical investigation

is conducted to correlate the results of the linear theory with a simplified version of

the present theory; and the effects of compressible layers on a multi-layered plate

instability are examined.












CHAPTER I
INTRODUCTION


1.1 Introduction to The Theory of Plates

The classical plate theory came as a result of the significant treatment of

plates in the 1800s principally by Navier, Kirchhoff, Levy, Lagrange and Cauchy.

Based on the fundamental assumptions known as the Kirchhoff hypotheses, the

theory is limited to the small deflection and bending of isotropic, homogeneous,

elastic, thin plates [1]. In the case of thin plates with large deflection, Kirchhoffs

second assumption (as defined by Love [1]) is satisfied only if the plate is bent into

a developable surface [2]. Otherwise, bending of a plate is accompanied by strain

in the middle plane. The corresponding stresses must be taken into consideration

in deriving the differential equation of plates. In this way we obtain non-linear

equations, and the solution of the problem becomes much more complicated [2].

Two coupled non-linear, partial differential equations governing the large deflection

problem were first introduced by von Kirmin in 1910 [3]. The approximate theories

of thin plates become unreliable in the case of plates of considerable thickness. In

such a case, the thick-plate theory which considers the problem of plates as a three-

dimensional problem of elasticity, should be applied. The stress analysis becomes,

consequently, more involved. Thus Kirchhoffs first assumption (also known as

Kirchhoffs hypothesis) becomes no longer valid. However, it was first recognized

that transverse shear strains in the plate can not be neglected when plate vibration





2

was studied [4]. In 1951, Mindlin [5] developed his plate theory to include transverse

shear in a similar manner to Timoshenko's beam theory of 1921 [6].

A trend toward a theory of plates which do not possess an isotropic behavior

later gained larger interest. Multi-layered plates is one example of such behavior.

The simplest approach, in the beginning, to these types of plate structures was to

assume a single layer orthotropic plate. In general, flexural and extensional motions

are coupled in a layered plate. Composites were also analyzed using similar theories

[7]. However, sandwich plates, which were very popular thirty some years ago,

attracted more attention; hence, several sophisticated approaches led to a level of

sandwich plate theories in a class by itself.

Laminated composites have been for a long time and are still being analyzed

as anisotropic plates of the classical type, anisotropy being the major additional

feature that was accounted for [7]. For symmetrical laminated plates flexure and

extension are uncoupled. If the bending-twisting stiffness coupling terms in the plate

equation further vanish, the symmetric laminates are referred to as specially

orthotropic and were usually treated as orthotropic plates, also of the classical type.

But since, recently, the transverse shear effect has been shown to have a great

influence on the overall behavior, whether dynamics and vibration or bending and

buckling analysis are concerned, the analysis of laminated composite plates has taken

new directions from the classical theory of laminates [7]. In contrast, the shear effect

was shown to be important in the analysis of sandwich plates a long time ago;

sandwich plate theories have developed very well and several higher order theories

have been introduced. Only recently, upon this development, the connection of





3
laminated composites to layered plate analysis in general has been examined and

exploited [7]. Thus, the much advanced theories in sandwich plates became

attractive tools to the analysis of layered plates in general and laminated composites

in particular. In the following section we will examine this trend from its early

stages.


1.2 A Prospective of the Theories of Multi-Layered Plates

The expression "sandwich plate" designates a composite plate consisting of two

very thin layers of high-strength "face" material, between which a thick layer of ultra-

lightweight "core" is sandwiched. The early sandwich plate structures had identical

thickness for the faces which preserved symmetry about the midplane. By the late

1940s Hoff and Mautner [8] had derived differential equations and boundary

conditions for the bending and buckling of sandwich plates. They have reflected in

their derivation the general understanding of the behavior of sandwich plates at that

time. The moment of inertia of the cross section of the sandwich plate is large

because of the comparatively great distance between the two faces, and thus the

buckling formulas derived from classical plate theory give buckling stresses that, as

a rule, far exceed the yield stress of the face material. Since the modulus of elasticity

of the core in the plane of the plate is of the order of magnitude of one-thousandth

of that of the faces, the normal stresses in the core are of little importance in

resisting bending moments even though the usual ratio of face thickness to core

thickness is between one-tenth and one-hundredth. The lightweight core permits

unusually large shearing deformations, and performs a task of transmitting shear

forces. Hence shearing deformations must not be disregarded in the analysis of





4

sandwich plates. Moreover, relative displacements of the two faces are possible

because of the small extensional rigidity of some of the core materials used.

The theory of Hoff and Mautner, with the aid of the principle of virtual

displacements, is developed from a consideration of these strain energy portions

stored during deformations, the strain energy caused in the faces by extension and

bending and that caused in the core by shear and by extension perpendicular to the

plane of the faces. Later in 1950, Hoff [9] modified the theory and added the strain

energy of in-plane shear in the faces, but continued to disregard that of extension

parallel to the faces in the core. He justified the assumption that the in-plane

stresses in the core contribute only negligible amounts to the total strain energy, by

taking the elastic and shear moduli of the core to be small as compared with those

of the faces. Also, normal strains in the core are disregarded. Furthermore, the

strain energy stored in the faces because of shear perpendicular to the faces is

neglected this is permissible when the ratio of the length or width of the plate to

the thickness of a face is always large. The displacement field was described by three

functions corresponding to the coordinate system directions. The in-plane deflections

are of the faces and they are equal and opposite for each face, while the vertical

deflection for the entire plate takes place through shearing of the core. This vertical

deflection does not cause force resultants (corresponding to membrane stresses) in

the individual faces but it gives rise to bending and twisting moments in them

because of the non-vanishing bending and torsional rigidity of the faces. Small

deflection elasticity is assumed, and the material of each layer is taken to be

isotropic. One last note to Hoffs work is that he acknowledged that the contribution

of the core to resisting bending cannot, as a rule, be neglected.





5

In 1948, E. Reissner introduced a large deflection theory which extends F6ppl-

von Kirmin theory of large deflection of ordinary plates to sandwich plates [10]. His

equations permit the analysis of the effect of transverse shear stress deformation and

transverse normal stress deformation in the core on the overall behavior of the plate.

Reissner's results led him to an important conclusion at the time. It states that the

range of deflections for which the linear "small deflection" theory is adequate

decreases in accordance with a simple explicit formula as the core is made softer

relative to the faces. This was also encountered by Hoff's experimental results.

Reissner extended the two assumptions that the thickness of each face tf is

small compared with that of the core t, and that the value of the elastic constants E,,

Gf for the face layers are large compared with the values of the elastic constants E,,

Gc for the core layer, to further assume that the products tfEf, tfGf are large

compared with the values of tE,, and tcG,. On the basis of the first assumption, he

assumes that the stresses in the faces parallel to their planes are distributed

uniformly over the thickness of the face layer. On the basis of the third assumption,

he neglects the face-parallel stresses in the core layer and their effect on the

deformation of the composite plate. As a result, the theory treats the sandwich plate

as a combination of two plates without bending stiffness (the face layers), and of a

third plate (the core layer) offering resistance only to transverse shear stresses and

transverse normal stresses. The displacement field is described by six displacement

functions, three components in each face layer corresponding with the coordinate

system. An interesting note to be made is that Reissner had opted to follow an

approach parallel to the method of elasticity in deriving his system of differential

equations. That is he started by integrating the equations of equilibrium and





6
consequently he had to define several other terms in his theory. Then, through an

elaborate algebraic manipulation the stress-strain and the strain-displacement

relations are included, and the result is a set of several algebraic relations along with

a system of non-linear partial differential equations for three displacement functions

and an additional three functions defining the changes of slopes.

In the meantime, various others had been working on the problem of

sandwich plates. However, various basic assumptions were made, such as (1) the

core undergoes shear deformation only; (2) the bending rigidity of the core is

neglected; (3) some of the shear and normal stresses which occur in the core are also

neglected; and (4) the faces are assumed to be membranes.

In 1952, Eringen presented a theory of bending and buckling of sandwich

plates which became one of the important works in the area and one which would

last for years [11]. To the present day it is still being used as an important reference.

The interest in this work came mainly from the fact that the above mentioned

restrictions are removed. Consequently, the theory is more general in that it

considers all six components of the stress tensor in the core, face layers having

bending rigidity, and it encompasses two types of bending and buckling, namely:

overall bending and buckling and bending and buckling with flattening

compressibilityy) of the core.

The simplifying assumptions are within the scope of the usual small

deformation theory, i.e. (1) faces are thin as compared to the core, but are not

membranes-thus, for the bending of the faces the Bernoulli-Navier hypothesis is valid

(plane sections perpendicular to the median plane of the plate remain plane during

deformation); (2) the displacements are linear functions of the distance from the





7
median plane of the plate; and (3) the linear theory of elasticity is assumed. Since

the faces are taken to be thin and symmetrical, he uses one set of displacement

components for the entire plate while assuming linearity in distribution up to the

median plane of the faces. In other words, assumption (2) is the result of expanding

the displacement components into a power series of the transverse coordinates and

taking into account only the terms which are linear in the transverse coordinate.

In the analysis, the total potential of the sandwich plate is expressed in terms

of the deflection components, and thus with the aid of the principle of the minimum

of the total potential energy, four partial differential equations are obtained. The

usual methods of the variational calculus are used in order to obtain the extremals

of the total potential. Eringen next solved an example problem using Navier method

and Fourier series type solution.

It is clear that the simplicity of Eringen's theory, while not compromising some

of the generality of the problem, is due mainly to its linearity. One aspect of the

linear theory is assuming linear distribution for the deflection components through

the entire plate, whence, limiting the application of the theory to sandwich plates

with very thin face layers.

Y. Y. Yu generalized Eringen's concept by assuming thick faces and using

different displacement functions for each layer, though this was done indirectly. Yu

had started from three-dimensional elasticity and followed the procedures of small

deformation theory as defined by Novozhilov [12]. Yu's contribution to the problem

of sandwich plates in particular and that of layered plates in general spans over thirty

years and includes a long list of published literatures. It started in 1959 when he

introduced a new theory of sandwich plates [13]. By assuming the transverse





8

displacement to be uniform across the plate thickness, and the displacements in the

plane of the plate to be of linear variations, with the slope in the faces taken to be

different from that in the core, it was possible to include in the theory the effects of

transverse shear deformations in the core and faces, the rotatory and translator

inertias of the core and faces, the flexural rigidity of the core, and the flexural and

extensional rigidities of the faces. Hence, although the sandwich as a whole is under

flexure, each of the two face layers is under combined flexure and extension. Since

no restrictions were imposed on the magnitudes of the ratios of the thicknesses and

elastic constants between the core and face layers, the results were therefore in

general applicable to any symmetrically arranged three-layered plate. Continuity of

tractions as well as displacements is maintained at the interfaces between adjacent

layers. In Yu's early work, the non-linear equations of motion of sandwich plates

together with the appropriate boundary conditions were derived from the variational

equation of motion of the non-linear theory of elasticity. In his later work [14,15],

Hamilton's principle in dynamics, which is the counterpart to the minimum total

potential energy principle, was applied in the derivations.

In all of the foregoing investigations the core and facings of the sandwich

plates were assumed to be isotropic. Alwan [16] in 1964 introduced an analysis that

adopted the same assumptions as Reissner [10] but the core was taken as an

orthotropic honeycomb-type structure.

While Yu may be considered one of the first to adopt a continuous piecewise

linear displacement distribution, the concept of piecewise displacement distribution

has become an obvious representation in the theory of layered plates. Another

breakthrough came in the mid- to late 1960s when Ebcioglu and Habip introduced





9

the non-linear equations of motion of plates and shells in the reference state [17,18].

The theory of the reference state involves the notion of stress measured per unit area

of the undeformed body, taken as a reference, in contradistinction from the

conventional representation of stress measured per unit area of the deformed body.

The distinction is of special significance for the correct interpretation of the

"geometrically non-linear," "finite-" or "large deflection" theories of plates and shells,

where this point is often overlooked. The non-linear equations of motion in their

theory had been obtained by integrating the corresponding three-dimensional stress

equations of motion through the thickness of the undeformed body. The concept of

a reference state makes it possible to properly identify the origin of various

additional terms that appear in the field equations of non-linear theories of plates

and shells in comparison with those of linear theories. Alternatively, Habip gave a

new derivation of these equations of a non-linear theory of elastic, anisotropic and

heterogeneous plates and shells by means of the modified Hellinger-Reissner

variational theorem of three-dimensional continuum dynamics [19,20].

However, the further extension of these considerations to the case of elastic

sandwich plates is due to Ebcioglu [21,22]. He considers a sandwich plate taking into

account all the stress components in each layer. As in other works by Yu [13,14], this

theory incorporates the effects of transverse shear and transverse normal strains as

well as rotatory inertia, with different material constants in each layer of the

sandwich panel. It is assumed that the facing and the core are anisotropic, each

having elastic symmetry with respect to its middle plane. An added consideration to

the theory was including the effect of steady thermal gradients in the stress-strain

relations. In a system of convected general curvilinear coordinates, the stress





10

equations of motion are incorporated into the variational integrals which stem from

the application of Hamilton's principle. In the displacement field, it is no longer

assumed that the transverse displacement component is uniform across the plate

thickness, and the angle of rotations are assumed to be different in each layer. The

continuity of tractions and displacements at interfaces are preserved.

Ebcioglu extended his geometrically non-linear theory of sandwich plates to

the range of material non-linearity [23] (physical non-linearity, in the sense of

Novozhilov [12]). Simplified non-linear strain-displacement relations are used. For

the elastic case, each layer is of a different thickness and of a different anisotropic

material having one plane of elastic symmetry. Transverse shear, rotatory inertia and

thermal effects are included in each layer. However, the influence of transverse

normal strain on the deflection is neglected. For the plastic case, the stress-strain

relations of the Henky-von Mises deformation theory of plasticity are used including

the effect of temperature and compressibility. For the latter case, only isotropic

materials are assumed.

Vasek and Schmidt [24] introduced a theory for the non-linear bending and

buckling of multi-sandwich plates. Their plate consisted of N stiff layers and N-1

weak layers. The assumptions made were as follows: (1) the linearly elastic,

homogeneous, isotropic stiff layers possess bending stiffness and deform in

accordance with Kirchhoffs hypothesis for thin plates; (2) the linearly elastic,

homogeneous, orthotropic weak layers can transmit the transverse normal and

shearing stresses but not the in-plane stresses; (3) the layers behave non-linearly in

a sense of F6ppl-von KArmin; (4) the thicknesses and the materials may be different





11
from layer to layer; and (5) the layer-to-layer bonds are strong enough so that under

all loadings no bond failure will occur.

In a series of studies, including one on laminated and sandwich plates in 1970

[25], Pagano constructed three-dimensional elasticity solutions and compared them

to the analogous results in classical laminated plate theory. He concluded that the

laminated plate theory (LPT) leads to a very poor description of laminate response

at low span-to-depth ratios, but converges to the exact solution as this ratio increases.

This convergence, he observed, is more rapid for the stress components than plate

deflection.

Based on the same conclusion, laminated plate theory based on the Kirchhoff

hypothesis is inaccurate for determining gross plate response and internal stresses of

thick composites and sandwich type laminates. Whitney in 1972 [26] introduced a

procedure which is an extension of the LPT to include the effect of transverse shear

deformation. In 1979, Bert [27] stated that thickness-shear deformation is important

even in single-layer panels of composite material due to their very low ratio of shear

modulus to Young's modulus as compared to isotropic materials.

A departure from the approach of Pagano and Whitney of considering all

layers as one equivalent single anisotropic layer was presented by Di Sciuva in 1987

[28]. He assumed the displacement field to be piecewise linear in the in-plane

components and uniform in the transverse component.

Before we depart from our review, we must mention one last important

contribution to the laminated plates analysis. It is the 1984 Reddy's higher order

theory for laminated composite plates [29]. As was stated earlier, laminated

composite plates were until recently analyzed as anisotropic plates. First-order shear





12

deformation plate theories emerged, which consider linear distribution of the

displacement of the laminate and the linear strain-displacement relations of the small

deformation elasticity. The stress-strain relations are then incorporated in the layers,

and hence the laminate is defferentiated from an anisotropic plate.

Reddy attempted to improve the accuracy of the prediction of stresses and

displacements in the laminate while not increasing the number of dependant

variables and satisfying the boundary conditions at the surfaces. To maintain the

satisfaction of the condition that the transverse shear stresses vanish on the surfaces

and are non-zero elsewhere, a parabolic distribution of the transverse shear strain is

required. Reddy then required the use of a displacement field in which the in-plane

displacements are expanded as cubic functions of the thickness coordinate and the

transverse deflection is constant through the plate thickness. He justified the uniform

distribution of the transverse deflection by comparing the in-plane and transverse

normal stresses. The number of generalized coordinates is reduced by setting to zero

the transverse shearing stresses at the top and bottom surfaces. The linear strain-

displacement relations are also used, while the plate is assumed incompressible, i.e.,

transverse normal strain is ignored. The stress-strain relations in each layer possess

a plane of elastic symmetry. The stress continuity across each layer interface was not

imposed. The theory is then characterized, according to Reddy, as a simple two-

dimensional theory of plates that accurately describes the global behavior of the

laminated plates and seems to be a compromise between accuracy and ease of

analysis.

Several scholars attempted to improve on Reddy's theory; most notable are

Pandaya and Kant [30], and Librescu, Kudeir and Frederick [31]. Several refined





13

anisotropic composite laminated theories substantiated on the basis of different initial

assumptions have been considered and compared by Librescu and Reddy; and it was

shown analytically [32] and numerically [33] that they represent but different

formulations of a single theory, designated as the moderately thick plate theory.

The higher-order shear deformation theories, as well as the first-order theory

will not fulfill the continuity conditions for the transverse shearing stresses at the

interfaces [28].

On the numerical analysis application to the theories of layered plates, the

most notable work is the finite element method implementation into the solution of

the problem of layered plates. The earliest attempt was introducing the triangular

element for multi-layer sandwich plates by Khatua and Cheung [34]. Recently,

Rajagopal, Singh and Rao [35] introduced a quadratic isoparametric element having

five degrees of freedom at each node for the non-linear analysis of sandwich plates.

Limited to linear conditions, a composite finite element analysis requiring only

continuity of the field variables, thus permitting the use of simple Co elements as

opposed to the requirements for the more complicated finite element analysis, was

presented by El-Hawary and Herrmann [36].

The preceding review of numerical analyses of layered plates by no means is

representative of the computational work in this area. Clearly, our main concern is

the theoretical and analytical advancements in the field; and the review of these

advancements is unequivocally complete as of the date of this study. To summarize,

layered plates fall into three categories, sandwiches, multi-sandwiches and laminated,

as illustrated in Figure 1.1. The preliminary assumptions as well as the methods of

analysis in each of these categories are different, thus leading to multitudinous of



























PRESENT
JI.) THICKNESS OR MATERIAL LIMITATIONS
THEORY /


Figure 1.1: Categories of Layered Plates.

specialized theories. The present theory is a general theory in the sense that it is not

limited to any one of these categories, rather it is applicable to layered plates in

general.


1.3 Principles of the Present Theory

Starting from the three-dimensional theory of non-linear elasticity, and using

a displacement approach, much like in the Reissner-Mindlin type theories, we will

develop our three-dimensional layered plate theory. The present work can be

considered as an extension of the theory of sandwich plates formulated by Ebcioglu

[21,22], and as a refinement of the contribution to the theory of sandwich plates by

Eringen [11] and by Yu [13,14]. The notion of stresses in the reference state, used





15
by Habip [19] and Ebcioglu [21,22], is employed in the fundamental equations. Also,

a variational procedure used by Yu [15] is extended to derive the equations of

motion and the boundary conditions of layered plates subjected to large

displacements and large angles of rotation.

The present theory incorporates the effects of transverse shear and transverse

normal strains as well as rotatory inertia, with different material densities and

material constants in each layer of the plate. All stress components in each layer will

be considered, and no assumption underlying the significance of a stress component

is implemented anywhere in the derivation. The continuity conditions between the

contiguous layers is maintained for the stress vectors (tractions). In addition, each

layer may be of a different thickness, and no a prior limitations are imposed upon

the displacement functions which define collectively the displacement field, which,

otherwise, would produce appreciable constraints during the deformation of the plate

as discussed by Ebcioglu [21], especially for the case of symmetrical flattening

discussed by Eringen [11]. Each layer is assumed to be anisotropic, having elastic

symmetry with respect to its middle plane. The assumed displacement field is

piecewise linear in the thickness coordinate in the in-plane as well as the transverse

components and fulfills the geometric continuity conditions between the contiguous

layers; furthermore, it takes into account the distortion of the deformed normal.












CHAPTER II
GENERAL THEORY


2.1 Mathematical Preliminaries

Let every point of the continuous three-dimensional body, called briefly the

body Bo, be at rest, at time t=to relative to a fixed rectangular Cartesian system of

axes x, (see Appendix A). The position vector of a typical point Po of the body Bo

referred to the origin is given by

r = k ik (2.1)

where ik are unit vectors along the fixed axes.

We suppose the body Bo is deformed so that at time t a typical point Po has

moved to P. The position vector of P referred to the same origin is

R = yk ik (2.2)

The position vector of P relative to Po is denoted by V and is called the displacement

vector. Thus

V = R r = (yk -k) ik (2.3)

We assume that each point P, at time t, is related to its original position Po at time

t=to by the equations

Yi = Yi (x1 x t) (2.4a)

i = xi (yI y2 ,y3 t) (2.4b)

where, and xi are single-valued and continuously differentiable with respect to each

of their variables as many times as may be required. If this deformation is to be







possible in a real material then

ayi >0 (2.5)
I 5x, I

The general theory of the present work is based on the derivation in terms of

the reference state of the plate. Thus, the rectangular Cartesian system of axes

defined in the undeformed body at time t=to are used as the base system for all

vectors and tensorial quantities.

The base vector gi and metric tensor go" may be defined for the coordinate

system xi in the body Bo, so that

gi = r,i = ii (2.6a)

gij = gi g = 6, (2.6b)
Similarly, base vector Gi and metric tensor Gy may be defined for the coordinate

system xi in the body B at time t. Thus

Gi = R,, (2.7a)

Gi = G, G, Ox (2.7b)
& xi ax.
where the metric tensor Gy is called in some literature the Green's Deformation

tensor when it is referenced to the coordinate system in the undeformed body, as is

the case here, instead of a general curvilinear system Oi as an example.

The symmetric strain tensor, Yi, is defined by the equation

,ij = 1/2( Gij-gi g ) (2.8)

From (2.3) we see that Gi = R,i = r,i + V,i

Hence, using (2.6), (2.7), and (2.8),

Yij = 2 ( gi. V,j + g. V,i + V,i. V,) (2.9)

The displacement vector V may be expressed in terms of the base vectors of B. Thus





18
V = v, g V,i = V,,i g~ (2.10)

By introducing (2.10) into (2.9) and using (2.6), the strain tensor Yj is expressed in

terms of the displacement components and becomes as

Yij = (ViK+ Vii+Vri V j) (2.11)
where VJ are the components of displacement referred to the axes x, in the

undeformed body.


2.2 Strain-Displacement Relations

The general non-linear strain-displacement relations for large elastic

deformations are given in tensor form in (2.11) which are identical to those given in

most literature on elasticity and non-linear continuum mechanics, e.g.[37]. The terms

of (2.11) denote three-dimensional space functions. However, in the study of plates

and shells, it is more convenient to refer to surface functions (without losing any of

the transverse and out of plane terms.) Thus we write

YO= *(V, a+Vv, a V13 +V3,1 V310)

Ya3 =( 3+V3, +V y,3 +V3~ V3'3) (2.12)
Y33= 1(2 V3,3 + V,3 V 3 + V V3,3)
Expressions (2.12) are identical to (2.11).

In the theory of large deflections of thin plates, a simplified non-linear strain-

displacement relations have been used extensively, i.e. [3],

Ya = ( V + V3,a V,) (2.13)
These are based primarily on the following assumptions: (1) plane sections before

deformation which are perpendicular to the middle surface of the plate, remain plane

and perpendicular to the middle surface after deformation; and (2) the distance of





19
every point of the plate from the middle surface remains unchanged by the

deformation [3]. However, in developing a general theory of large deflections of

multi-layered plates, the transverse shear and normal strains, (yx, y,, and Yz,) can

not be ignored. Consequently, the above assumptions become invalid.

A simplified non-linear form of (2.12) has been used in the theories of

sandwich plates [21],

vp' = 1(V +V.,a +V3,a V3',)
YV ( 3+V3 (2.14)
'a3 1' a' 3 3'a/
Y33 = 33
where the assumption that the maximum deflection is assumed to be of the order of

magnitude of the thickness is removed. The relations (2.14) will be used in the

present theory. The in-plane relations are non-linear, while the transverse shear and

normal strain-displacement relations are linear, and this is by no means unjustifiable

since these components are defined in each layer rather than throughout the entire

plate thickness.

A more general form than (2.14) has been used in [18] when considering a

large deflection theory of anisotropic plates,

S= ( Vij + Vi + V3,i V13j) (2.15)



2.3 Stress-Strain Relations

The stress vector ot, per unit area of the undeformed body, associated with a

surface in the deformed body, whose unit normal in its undeformed position is on, is

ot = Sij i Gj (2.16a)

on = oni g (2.16b)





20
The stress tensor Si (second Piola-Kirchhoff stress tensor) is measured per unit area

of the undeformed body while defining the state of stress in the deformed body.

The equations of motion in terms of Si are

in v : [Sir (6jr + Vj,r)],i + P (oF) = Po ()), (2.17)

where, F = Fi gi, and f = ofi g

are, respectively, the body force vector and acceleration vector, p, is the density of

the undeformed body, and vo is the volume of the undeformed body. From the

moment equations of motion, it follows

Si = Sji (2.18)
whence, the stress tensor Sy is symmetric [38].

For an elastic body, an elastic potential or strain energy function W exists,

measured per unit volume of the undeformed body, which depends on the strain

components y, and has the property

6W = Sij 6Syi (2.19)

The variations 6Yij are subject to conditions

6yi = 6yji (2.20)

so that for a compressible body,
1 _W* W*
Si + ) (2.21)
i 2 ayij ay.i

If the stress vector ot be referred to base vectors gi in the undeformed body,

ot = tj oni gj (2.22)
%where, ti = Si, (6jr + Vj,,) (2.23)

is another stress tensor (first Piola-Kirchhoff stress tensor) measured per unit area

of the undeformed body, depicting the state of stress in the deformed body. The







equations of motion in terms of tf are

tii + Po (F) = P ( f) (2.24)

where t0 is not symmetric but satisfies,

tim gn Gj = tim g G, (2.25)

As a more specific set of stress-strain relations than (2.21), we may assume the

linear version

Sj = Cirs Yrs (2.26)

The stiffness tensor holds the following symmetry relations,

C = Cirs = Cisr = Crsi (2.27)

For a medium having elastic symmetry with respect to the surface x= constant

(z= constant), equations (2.26) reduce to the following form

Sap = CaPyg Yyg + CaP33 Y33
S,3 = 2 Ca3y3 Yy3 (2.28)
S33 = C33y Yyg + C3333 Y33

Equations (2.28) express a similar form to the well-known law of Hooke. For

infinitely small elongations and shears (but not displacements and angles of rotations,

which are in no way restricted by the preceding assumption) not exceeding the limit

of proportionality, the stress-strain relations are linear. However, the above

assumption does not imply a linearized version of the strain-displacement relations.


2.4 Displacement Field

Referring to the undeformed plate, we will define a rectangular Cartesian

coordinate system at the middle surface of each layer as shown in (Fig. 2.1). The

points of the 3-D space of the plate will be referred to by a set of orthogonal







coordinates x;, where xa, (a= 1,2) and x3
x, ,x (r=1,2,3 ... )
denotes the in-plane coordinates and

the normal one to the plane x,=0, -----------2z<-23------.---- x

respectively. The flat plate consists of / -----"
<0>
N layers. A superscript in the brackets
.. -x,< X$
-----------g-h--------------- X1
"<>" associated with a field quantity L ...-<>

identifies its affiliation to the layer of /

the laminate indicated by that index. Figure 2.1: Coordinates System in Each
Layer with Cross-Section in x1 x3 Plane.
The middle layer (core) has index <0 >

and layers above the middle plane have odd indices while those below the middle

plane have even indices. Let 2h be the thickness of layer n where {n:n [O,N-1]}.

Thus we have

<1> <2>
a a a .... .... X
Sv2(n-1)
x" + [h<> +2 h<2r-1> +h<">], odd n_1 (2.29)
S= n/r=1
x3<> [ +2 h<2r-2> +h] even n_2
r=2

We now assume that the components Vi of the displacement vector V are

given in each layer approximately by

v t) = u <> + > x (2.30)

where, ui and *ij are functions of x, y and t only and have the dimensions of length

and slope dimensionlesss), respectively. The displacement components (2.30) contain

six unknown displacement functions for each layer, that is a total of 6*N unknowns

in the plate. These can be reduced in number to 3*(N+1) displacement functions







by implementing the conditions of displacement continuity at interfaces,


Vi >(Xx=h <0>;t) = VlO>(xa,x3=h;t)

Vin> . Vi (xa,x3=H +(-1)"h ;t) (2.31)

= n-2>(X ,3= -)n-"lh;t), n =2,3,..,N-1

where,

X3 = 3 + H<
(n-l)
+[h+2 h<2-l> +h], odd n (2.32)
H<"'> = r
n/2
-[h +2 h <2r-2>+h], even n
r=2

The conditions (2.31) are applied to (2.30) and are simplified, hence

> <2r-l> )
i"i +2 E h <2'r-1>i +h r-l> <"i ">], oddn
r=1 (2.33)
=i n=1
[ 0> 2 <2 h 2-2> <2r-2> +h n>], even n
r=2

Upon substitution of (2.33) into (2.30), we obtain

v +
Vi =i + X3*i
vV(n-1)
i+ 2 hI'<2r-'1> <2r-1> +h <>] >, oddn (2.34)
Ui i i 3 i, odd
V = r=1
i n/2
ui -h 0> -2E h -2> 2> +[x3<> -h <>] i<", even n
r =2

which describe a linear piecewise displacement representation in the thickness

coordinate of the multi-layered plate. Since the strain-displacement relations are

non-linear, the stress quantities are quadratic in the thickness coordinate (Reddy's

dilemma eliminated!)

A special case which will be given a special consideration in the present study

is a three layered plate. For N=3, the displacement field representation is given by








<0> <0> <0>
Vi = ui + X31 i
V<1> <0> + ho> +[x3+> <> (2.35)
Si = Ui + h li +[x3 +h <>]i1
V<2> -h <> +[x<2> -h<2> <2>
i X3

where superscripts 1, 0 and 2 indicate upper face layer, core and lower face layer,

respectively, as shown in (Fig. 2.2). The displacement components in (2.35) are

expressed in terms of the 12 displacement functions
/ <1> <2> (2.36)
(ui ,si ; ,i 't ) ^*Jof

Higher order terms in the

expansions given by (2.35) can be

included. The method of analysis to be

presented is applicable for higher order --

expansions for the displacement
2h ..... .
components. In our approximations, by- -
2h ........--
including second order terms for the

faces in addition to the core in

transverse direction, we have added the Figure 2.2: Rectangular Cartesian
Coordinates in Three-Layered Plate.
capability to allow for compressible

faces as well as a compressible core. In that respect, the present displacement field

represent an extension to that of Ebcioglu [22]. Additionally, interface continuity is

preserved while non-linear distribution through the thickness of the panel and the

differences among the axial in-plane components of the displacements in the layers

are not ignored.







2.5 Hamilton's Principle--The Variational Integral

In dynamics, the counterpart of the minimum potential energy theorem is

Hamilton's principle. It states; The time integral of the Lagrangian function over a

time interval t, to t2 is stationary for the "actual" motion with respect to all admissible

virtual displacements which vanish, first, at instants of time tj and t2 at all points of the

body, and, second, over s,, where the displacements are prescribed, throughout the entire

time interval [391. Accordingly, for the conservative body force and surface traction,

and for an arbitrary time interval t, and t2, the Lagrangian function L has to satisfy


6i Ldt = 0


where;


L = K- (U +A)


U = S ,yijdv
Vo

SA = -po obi ~Vi dv -J oti Vids
Vo so


K Po av, adv
K = 1 fPo-Jdv
Vo

The volume integrals of (2.38),

(2.39), and (2.40) represents,

respectively, the strain energy, the

variation of the potential energy due to

the body forces obi per unit mass of the

undeformed body, and the kinetic

energy, and must be extended over the I
I


(2.38)


(2.39)


(2.40)


Figure 2.3: Undeformed body in
Reference State.


(2.37)





26
volume vo of the body in its undeformed state. The surface integral of (2.39)

represents the work of the surface forces oti (assumed held constant) on the (virtual)

variations in the displacement trajectory [variations from the actual trajectory].

The components of the surface force oti are referred to base vectors in the

undeformed body. Let ot (underlined) be the prescribed value of ot on the boundary .

Then the stress boundary conditions are

t oti = 0 (2.41)
The variational integrals of the stress equations of motion and boundary

conditions consistent with strain-displacement relations (2.14) are obtained from

Hamilton's principle. We can write (2.38) in the following form

U = (S Ya +2Sa3Ya3 + S33y33)dv (2.42)
Vo

Based on (2.19), property of the strain energy function W, it can be shown that

SU = (SaS 6Y# +2S,363 + S336y33)dv (2.43)
Vo

With the use of the strain-displacement relations (2.14) and the divergence theorem,

fFap, dv = Fp onp ds (2.44)
V S

we put (2.43) into the following form

6U = -f S~ls,~ V + (SIP V3,s), V3]dv -S (Sa3~,6. Vs+ S SVO,)d

S333 6 V3dv + (SS 6Va + S, V3, 6V3) bondss (2.45)

+J(S,36V 3 A+S .36V3 n)ds +fs33V3 dS
O so
So so

The variation (2.39) is written as

6A = -f pob 6V +obSV,)dv (ot6 V, +t 6 V3)ds (2.46)
'o $







And, from (2.40) we obtain
t2 t2
6 JKdt = [(f po6iVdv)dt


= (poVi 6V -p oi pSViV) dv)dt
12

[fPO 6Vdv] (f poPt i dv)dt
0 t1 l 0
12
= -(f PoPi sVidv)dt

which simply takes the form,
12 12
6fKdt = f(poV SV+Po36SV3)dvdt (2.47)
'I i Vo
We now substitute the results from (2.45), (2.46) and (2.47) into (2.37), and

Hamilton's principle in a form consistent with plate theory notation is
12



S+ [(SaP V3s,),p + Sa3,a + S3T, + Po(ob V3)] 6 V3 dV (2.48)
+ fdtf [ot (SP on# + S3 on3)] 6V,
t1 So
+ [ot3 (SaP V3'a +S 3)on# -S33 o3] 6V3 )ds = 0

where, the surface forces are

ota = SaP ono + S3 on(2.49)

ot3 = (Sa V3, +SP3)on + S33 on

Here, the vector (on, on3) represents the unit normal associated with the

undeformed body with respect to the base vectors in the undeformed body. An

element of volume dv can be written as

dv = dz dA (2.50)





28

where, z is the coordinate in the transverse direction of the undeformed body where

z= 0 is the middle plane of the core, and dA is an element of area.

The volume integral (2.48) is divided into the number of the layers in the

laminate; and the surface integral is also divided to correspond to the number of

layers in addition to the upper and lower face. Based on (2.50), an integration in the

thickness coordinate may be performed. This procedure leads to the derivation of

the equations of motion and boundary conditions as will be demonstrated in the

following chapters.

A note needs to be made here regarding the expressions in (2.49). These

surface force expressions are extracted from the derivation of the variational integral

(2.48), and hence, are consistent with the components of the strain field given in

(2.14). Since the relations (2.14) are modified non-linear strain-displacement

relations and do not constitute a complete strain tensor as may be defined for the

general non-linear case, traditional tensorial procedures to derive the surface forces

(i.e., Malvern [40]) can not be implemented to derive (2.49).


2.6 The Constitutive Equations

Define the stress resultants in each layer as follows:
N r - f
NaP ,Ma ,K ,aP ,a ,N33 )
* (2.51)
r(S ,<>Z 2 C S ,S )d
# ,a#! ,s>ap z ,,a3 ,"a3 Z ,33 )az


Define body force resultants with respect to the middle planes of the layers

+h
(FatAF[3r>,rF>r<,Mr, r> r o r> r>ob z, )dz (2.52)
r>"<-








and the acceleration resultants

+h
(f 3r> r> ) (j < r>Zd> Z< z > (2.53)
3,3 3 0 a V 3 v. '3
rh


The edge surface force resultants with respect to the middle plane of the layer are

+/h


-h
(s,S> ,L ,L r > (rLt>r> > r tr <> (2.54)



where ot and o3f are prescribed traction forces on the surface of the plate as were

defined in the proceeding section.

From the strain-displacement relations (2.14) and the displacement field

representation (2.34), we obtain the following relations


2
Yap =oYap +z lYap +z 2Yap
(2.55)
Ya3 oYa3 +Z la3

Y33 =oY33
where

1 [
oYap =L a, +Up,a +U3,a U3,
1 r .
1Yaf 23 = a, + ,a +u3,a '3,0 +U3,# 43,p
<0> 1 [,<>
2Yap = 3,2a 3, ] (2.56a)
<0> 1 <> <0>
0Ya3 = [a +u3,a
1
lYa3 =7I3,a

OY33 313









1 <0> <> < <0> <0> > <0> <0> <0> <0> <0>)
oY7a =2[a,# +U,a 3 +U3,a 3, +h >(a, + 1#,a +U3,a 13,0 +U3, 13,a
<0> h >( <>
+h 3,a 93,0 +h ([aA +* fa +U3,a 13, +U3,p 13W,a
+h <" +h <> <) +h ]
(m -1)/2
+hI 3a 11*3 + h3' 13,0 1I13,a *(3, 11301 1




(m-1)/2 (m-1)/2
+ 2r <2r-1> h <2-1 > 3 > -1> 3-1>
r=l s=1
1r . ,, < +> . <"+>
17af + ;2 a, +[11,a +U3,a 13, 3, 3,a +3,0 3, ,3,
(mi -1)/2
+ ,( < )+>\ <2r-l> r i <2r-l> > <2r'-1
+ h v3,a 3,0 ) + h- t [3,a '3,13 +1113,1 3,z
r=1
1 rl >
2Ya > h 3,a 1 3,1 ]
(m-1)/2
1> +h <0> 7<0> +h + h<2r-
oYa3 a U3,a 3, 3,a i3,a 3, E 3,
rr=1
1
1Ya3 = 3,a

oY33 3 (2.56b)

Expressions (2.56b) define quantities for layers above the middle plane superscriptt

m is odd). The quantities for layers below the middle plane superscriptt m is even)

are given by expressions (2.56c) which are obtained from (2.56b) by implementing

the following: (1) replace h<'> with -h where n>0O; (2) change the limits on the

summations to (r=2 to m/2); and (3) replace the summation index 2r-1 by 2r-2.

From (2.55) and the stress-strain relations (2.28), we obtain


r <> 2 /
a#3 = Cpy [l +z lYv +Z 27YYI J + Ca33 oY33
s ,^
Sa3 = 2 Ca303 oYP3 +Z 1Y33

S = rr> 2 /
33 =33a C [0oYa Z Ya +Za 27a# + C3333 of33


(2.57)








Finally, from (2.57) and (2.51), we obtain the following constitutive relations


N n B r>
a/ 'o~~aaY oYVA +l/'YA 1YYL+ +2LaPYa 2YYA + o3a#33 oY33

M D
a -j1a ylP oYvg +2aPVy/ lYyg +3Baf YA 2YyA +l"a33 oY33

RK
a =-2"a#yjv oY/Lg +3a Y#L 1Yyg +4Yafi1y 2Y/LA +22a33 oY33


Qa = 2 [oBa33 oY13 +1Da303 lY3 3

<>
Ta
N
w33 h o"33ar oYae +1j33ae 1Ya# +2B33ae 2Yaf +or3333 o033


where


+h
rmB f Z mdZ
_h


, (m =0,1,2,3,4)


mdz (m=0,1,2)
-Bh?3


n < ~ Z 'dz
,37y3 Ca3y3z r
hr>
+h/ <'>
, = -, n dz
33yt 33yz
-1


, (m=0,1,2)



, (m=0,1,2)


(2.59)


+h
oB = ("dz
Sh h h

Since for the isothermal case, the elements Cil are constants, thus all nBijkl with odd


m will vanish.


(2.58)












CHAPTER III
THEORY OF THREE-LAYERED PLATES AND SPECIAL CASES


3.1 Variational Integral for the Three-Layered Plate

The application of Hamilton's principle to thick, homogeneous and isotropic

plates using nonlinear strain-displacement relations has, according to Ebcioglu [21],

been shown by Herrmann and Armenakas. It has been used by Yu [15], for the case

of small deformation and small angles of rotation, to derive the equations of motion

of sandwich plates. It has been also shown in [21] to lead to the equations of motion

and boundary conditions of sandwich plates for the case of large displace me nts and

large angles of rotation. Ensuingly, the variational integral for a three-layered plate

is derived from Hamilton's principle.

The volume integral of (2.48) is divided into three parts corresponding to the

three layers of the plate. The surface integral is divided into five parts corresponding

to the upper face, lower face, and the edge surface of each layer. In each of the

eight integrals the nine displacement components are substituted using (2.35), and

are now expressed in terms of the twelve displacement functions (2.36). With the use

of (2.50), an integration in the transverse coordinate through the thickness of each

layer is possible in the three volume integrals as well as in the three surface integral

corresponding to the edge surface of each layer. The result is five surface integral

and three line integrals. The surface integrals, thus, are evaluated at the middle

plane of the respective layer. However, based on the first set of relations in (2.29),





33
the surface integrals are grouped and evaluated over the middle plane of the middle

layer, Ao. Similarly, the three line integrals, based on the first set of relations in

(2.29) and since onr> =onp, are grouped and evaluated around the middle plane of

the middle layer. It should be emphasized that the stress continuity conditions at the

interfaces of the layers are completely satisfied in the variational integral. In

addition, the transverse shear stress, Sy, at the upper and lower faces vanishes

automatically, hence satisfying the condition for parabolic distribution of the shear

stress, which vanishes on the surfaces. This condition is usually enforced by

additional equations in the derivation of the variational integral of a layered plate

when a cubic displacement distribution in the thickness coordinate is assumed, as in

Reddy's derivation [29].

Define the following notation for simplicity of terms recognition, the

superscript "i", "o" or "ii" associated with a field quantity identifies its affiliation to the

upper layer, middle layer (core) and lower layer, respectively. Therefore, upon

implementing a lengthy mathematical derivation, carrying out the integration

outlined in the preceding paragraph, and grouping of terms, the variational integral

for the three-layered plate is


t1 {(N i p +Na' +N ,), + (t +ot) + (F +F +F'0 ) -FI +(f +f +f f)} su

+h{(N-N a (-ot)+(Fa-a-F ( a
h ho h h

+hh{(N + -- o-t +F )-( --+fi)}oi
iM Q t Ma i
A N, +' ( (' +,


+ h i(- N + ), -2 -2 +( -F ) -() 6
h h" h" h"








h h'
+t" { [( Np+Nao+ ) u, + N, + h i ( +3
+ h ( 0N 0 ),i 10 -F3)Q +O +Qia) t")+ (Fl+F+Fi)


- (f +f0 +Nf) ( 6uh (N + B -N+ )uo +hof4 h + (1 ) ,'
3,a 3 -a


h0 hI

+h'( +N ),) -h"(- -NA )4 ,a l (0-0- ),o
Ni


-N33 + 3 ot- ) + ( F3 F3 ) 3
+hi[( +NN i)(u 0 jro )+h ( a# +_ To
h hh h h
No" ^ M )m
33 3f M3 .T 6* 0. / 3 M ,\








-+ 2 t + ( M F) -( +F +d f3
+h0{(S'_ +.-S) -(+- + U
ho ,h)0 h N






NM M' m
33 3 ,3 i i ii


+-2t + N-F -f )hD) dAd
h" h" h"



+h f^ (+Su)- ( +-h )i n -( ) + h aU+ oc }( S) ) n
+ 3,{(S+ + -3 h-(N i p + ,--)Na 1 Q
h o h a2 h3 h"










Ni I" in i0
+h( +N-_( h .sih ) hit( m 3
(h(S' +So +S)-(N Np+h +N,"#- n ii


LM M" T
.+I o((S i + s) -h(N a+N"( -)Na"p ),(N i )]Ni
h(-' h'h" +h h

+ fi(si +SO- (Nap -M; +N NIl. o i -+ p) i o
-3 -3 3 -3h 3,)a

+h +
+h i( M + N ) i3,a h i i i 3,3

3h h -NBO ) 3,,72 + (P +

+h ---'-. +N---=. -.h-a -Qi+ ) ]o,n,) *03







L' M'+h K' +h
+h'if( +S_)- [(ap +Np)(u3,a +h ,a)+hi( +2 a +N) ,a
hi h1 h' h'
To' L" M"
+( +Qp)]on)} 6Ii3 + h"{(i -S" -- -N )(u0 -h ,a
--_3 _sii)/-[( -N3)(u-ho<

Kh" _(-2 M .a ( TI on } 61 ds dt = 0 (3.1)
A" h h II

Using the fundamental lemma of the calculus of variations, we obtain twelve

equations of motion and appropriate boundary conditions for any given shape of

three-layered plate.

The stress, body force, acceleration and edge surface force resultants defined

in section 2.6 hold valid for the present case when the appropriate superscript is

used. The stress resultants appearing in (3.1) are given by the constitutive relations

(2.58) with the appropriate superscript. Expressions (2.56) reduce to

0 1 o o o o
oYap = [ua,p +UP,a +U3,aU3,]
0 1 0 o 0 0 o oo0
lYa = 2 [ 2 a, + *#,a +u3,a P3, +U3, 3,a ]
0 1 l o o
oYa3 = a +U3,a
0 1 0
1Ya3 = 23,a
o = o
0Y33 3



+ (u3 +h 03 +h 'qo +h 3)fl *(3,u + h 03i 3 +0
= i( +1 +1 '^aU^ +h (U +h O~ + 3h () +h o
l(-30 a, O3, 3) h (U03+h 3)+3,; +h 03),^]

2i -i i(3.2b)
Yi3 l [ia + (u + h o *0 + ih i *),,
oYa3 = 233+ 3a]

OY33 = 13
0 I13 11(z~ooh a








oY#ii = [ (uo-ho l -h ii^ )'o+ -ho -" hl ,
+(u3 -h --h" )h (u' o--ho -h" "),c ]
ii 1 ii bii u h h o o it itio i I ) a i -0 0 r es h 0 h eu i ins
Ip= 2 *a [,# + *#,a+ 3, (3 ( 1J-h 3,j + *3,0 (U3 3 '-3,h 3
ii 1 ir i (3.2c)
2YzO = 7 [3,, *3,P]
ii 1 hencii eo o m p lan de i
oYa3 = [a +(U 3 I-h 10I3h "*3,a]
ii 1 ii
1Y~r3 = 2 l3,a
OY33 = *3

and must be substituted into (2.58) to obtain the stress resultants.


3.2 Kirchhoffs Hypothesis

Approximate theory, also known as Kirchhoff-Love plate rests on fundamental

assumptions which are direct results of Kirchhoff's Hypotheses. These were first

introduced by Kirchhoff and later presented by Love and extended to thin shells.

The Kirchhoff-Love plate is characterized by the follows ing asumption,: (1)

deflection of mid-plane is small compared to thickness; (2) mid-plane remains

unstrained; hence mid-plane does not experience any in-plane displacements; (3)

plane sections normal to mid-plane remain plane after deformation; thus the in-plane

displacements are assumed to be linear functions of the transverse coordinate; (4)

plane sections initially normal to mid-plane remain plane and normal to that surface

after bending; thus transverse shear strains are negligible and consequently deflection

of the plate is associated principally with bending strains, and, therefore, it is also

assumed that the normal strain is negligible; and (5) normal stress to mid-plane is

small compared to other stress components and may be neglected. It is argued that

the fifth assumption produces an inconsistency in the theory. This inconsistency will







become evident in the following section in deriving von Karman equations and will

be examined then. However, it is appropriate to clarify this point by presenting the

following argument by Boresi and Sidebottom [41, pp 452], which best describes this

dilemma:

In conventional plate theory, it is assumed that the plate is in a state of plane
stress; that is, a,= u,=a =O. For isotropic elastic planes, the relations
ar= = 0 are consistent with Kirchhoff approximation, which signifies that
E,= E = 0. However, the Kirchhoff approximation has been criticized since
it includes the approximation e =0. The condition ez=0 conflicts with the
assumption that az= 0. The condition ,= 0 is incorrect; however, the strain
EZ has little effect on the strains E~, e ey. Thus, the approximation E= 0
is merely expedient. In the stress-strain relations, the condition of plane stress
a==0 is commonly used instead of E =0, and this circumstance is often
regarded as an inconsistency. However, in approximations, the significant
question is not the consistency of the assumptions, but rather the magnitude
of the error that results, since nearly all approximations lead to
inconsistencies. In plate theory, the values of Ezz and azz are not of particular
importance. Viewed in this light, the Kirchhoff approximation merely implies
that Ez has small effects upon a. and a and that ao and ac are not very
significant. We observe further that the Kirchhoff approximation need not be
restricted to linearly elastic plates; it is also applicable to studies of plasticity
and creep of plates, and it is not restricted to small displacements.

Let the two face layers, upper and lower layers, be characterized by only the

fourth assumption of those stated in the previous paragraph. Hence, the assumptions

Ya3=0 and y33;0 when applied to the displacement components (2.35) and noting

the displacement continuity conditions, leads to the relations


i'3 0, 0 0

S= -(u + h (3.3)

S= -(U30 / ),

When conditions (3.3) are applied to the variational integral (3.1), and use is

made of the Generalized Gauss's theorem and integration by parts, i.e.,







SSt.6V,, dA= rfSt.SVn ds f 6VdA (3.4)

where t is a second order tensor, we obtain




ho ho h h
M o


-h "N,), +Q Oa + 2(h' t -h ot ),a t + 0 t + (M F +M + 'F "F ;),)

+ (F' +F3' +Fj') (in. +mj +h 'f hf' h"f I) (f3' +-f +f') .- Su3
Mo KM o


+h {[ (Ni + -a ) +Nl )u3, +h (N a +N) 3 ],,a +(M;, +h N +Mi
h ho
Tao N33

S0 0
+A"N,, ),,, +ho ho+2(h i ^" o-at^i + oti)+ t ot3 + (

+ (F3+(h-F3a-(h-o+hf3+ -f33)}

+ 1 ( (S' +S +Sil) (Ni/ +AN +Ni" )n }6 u0 + h0 {(S' + h -L )

MAof L Ma
(Na', + N, a/i a)/ioi O -a


o" M"
L0 a

-h"{( -S ) -(-_ -N ,~))o }(6u3o,-h1 6 ,a)
h i -a h"

30^ mo
+ {(S +S0+S") -[(N/ i+N NS + 0Na +a )u +h(N+ Mi
-- -3 a +,a ho 3,a
+ (Mi, +h 'N +M h"N ), + Q; + 2(h' ot -h" to)
a i
+ (MO +M +h 'F+ -h "FO ) (rm +m; +h 'f -h "f ) ]0n 5u3
O 0 a/
L3 MM, ii oK ii
+hoa- +h(Nio#+ +0h2 N, *03a

h0
+(Mp +hiN, aO -M +h A ), + +2(h' i +h )

+(Mi-M; +hi'F; +huF;) -(mn -m +hiff +h"f;') ]on} 6o ) ds dt = 0 (3.5)







3.3 Von Karman's Equations of Plate Large Deflection

The equations of equilibrium and boundary conditions for the large deflection

of isotropic elastic plates are derived from the variational integrals (3.1) by

eliminating the core and assuming identical faces (geometrically and materially),

h -. 0 and 2h' = 2h" = t/2 (3.6)

Also, the following assumptions are imposed: (1) Kirchhoff hypothesis holds; (2)

transverse deflection is uniform through the thickness of the plate, hence i*'3- 0 and

3' 0; and (3) static case and zero body forces. Therefore, from (2.35) and (3.3),

the deformation of the plate is now described by

S=u-(zi+h'),a V=uo -h z'<+h'
Vi= u-(zi -hiL)u3, V3 = 3 -h < z"< +h
a3 ,3r V3 U3
Hence, V 0 = u -z uo V3 = u -2hii< z < +2h'

where, u and uO describe the in-plane and transverse displacements of the middle

plane (z=0), respectively. We also assume the following loading conditions on the

upper and lower faces,

t' = t" = t' = 0 ot" = p (3.8)
o-a o- 0-3 0-3 t

Upon imposing these conditions into the general variational integral (3.1), we obtain

j{(N +N: ),N } 6u

+ [(N c+N)uo ,],, + (M +M ,+hN ,-h"N), +p suO dA
+ Jo3.9)

+ (S+S") (Np +N)n 6 (3.9)
CO
+( +S) -[ (NA +N+ )u3 ,a +(MR +M +h N' -h"N ),, ,on, Su

-(Li +L +hiSi -h "Si )-[Ma. +M' +h'N' -h "NA]on S6u,, )dS = 0
aj3 -f3 1313 3,cx







The definitions (3.2) reduce to
i 1 0 0 o o O
oYa = (ua,,p +p, a +U ,a u3 ,) -h u3 ,a
i 1 o o o o o (3.10)
oYa = (u, +u0, u30,0) +h a ("1
i ii o
1Ya = 1Ya = -u3,'a


And the nonvanishing coefficients in (2.59) are



i a Ci t (3.11)
2 a 2 RY 9 6

p33 = lp33 CaP33


Hence, in view of (3.10) and (3.11), and from the constitutive relations (2.58), it can

be easily shown that


N = Cy, oYy, + C vy + C
Nap C~aAP O a13y+ :at 2IYi t- CaP33 oY33

a^ i Caftyi oyY/i + Cafyl 2YIC + ~ Ca#33 oY33

Ma = Ma =- 3C U3 ,yu
+2h'
Np# +Na=f S., dz =Nap

I'N d 'h"N, = -- C^ u3, (3.12)


+2h "
Si + S = otdz =S
-a -a f -a

-2h'
+2+2/h





-2h'

In arriving at the correct form for the two dimensional fourth order isotropic tensor

CO ,, we must derive it from the complete three dimensional tensor by imposing

the conditions of the Kirchhoff hypothesis on the strains. However, this process will





41

lead to the plane strain case, and consequently the flexural rigidity constant D will

differ from that of von Karmin, since he assumed the plane stress case. The

difference is demonstrated by the ratio
Dplanestrain (1-U)2 (3.13)
Dplanesress (1-2u)

For the plane stress case,

Capg = G (S6ay, 6 + 6, 2 6 6 v) (3.14)
(1-u)

We substitute (3.12) and (3.14) into (3.9) and use the fundamental lemma of the

calculus of variations to obtain the following equations of equilibrium and boundary

conditions for the von Kirmin plate type.

N,# =0

Nap u3,aP +p =D u3'a P

On the boundaries,
(3.15)

6u or S =N1P onf

6u30 or S3 = [Nafl3 U,3 D U3o,cr1 ]on,

So Gt3 0 0o
6U3 or L (U3,ap on + u3, o,)
a 6 (l-u)


where

D = E and E = 2G(1+u) (3.16)
12 (1-u2)

and E, G and u are the elastic modulus, shear modulus and Poisson's ratio,

respectively. The stress resultants Nap are due to the middle plane in-plane strains,

and they are unknown quantities. A stress function, c = 0(xa) is introduced, such

that the first two equilibrium equations in (3.15) are satisfied identically by P(x,),







and the stress function is related to the resultants as follows;

Nar = t ( S6,p ,yy P,.P ) (3.17)

Two equations of equilibrium are thus obtained in terms of the stress function p(x,)

and the vertical deflection of the middle plane, u3; and they are identical to those of

von Karmin. When written in expanded form, these are

6'w w2 w -w t[p a w w a _a -' 2 2w ]
+2 + = + + -2 ]
&X4 &x2y2 2 y4 D t y2 2 y2 2 Qy 2 a y
(3.18)
040 + 0 2 = E [(w) ]
Tx4 yT4 &a Y2y2 y -x2 ay2
The first equation in (3.18) is obtained from the third equilibrium equation in (3.15)

by substituting (3.17). The second equation in (3.18) is obtained by enforcing the

strain compatibility condition which is violated by the assumptions on the transit\ erse

strain components.












CHAPTER IV
THE SOLUTION TO THE SANDWICH PLATE
BENDING AND BUCKLING EQUATIONS


4.1 The Assumptions of Eringen's Equations

The problem of bending and buckling of sandwich plates has been formulated

and solved by Eringen [11]. The underlying assumptions for the problem formulation

are summarized as follows: (1) usual small deformation theory; (2) faces are very thin

as compared to the core, but are not membrane; (3) middle plane of the plate

remains unstretched subsequent to loading; (4) displacements of the plate are linear

functions of the distance from the middle plane of the plate; and (5) linear theory

of elasticity. The problem formulation considers all six components of the stress

tensor in the core and the bending rigidity of the face layers.

The simplicity of Eringen's theory is due mainly to its linearity while some of

the generality of the problem is retained. The assumption of a linear distribution of

the deflection components does not extend through the entire plate thickness. More

specifically, it is defined only up to the middle planes of the face layers. This

consideration limits the range of application of Eringen's theory to sandwich plates

of very thin faces as compared to the core.


4.2 The Equations of Equilibrium and Boundary Conditions

The general variational integral and all fundamental equations for the three-

layered plate were derived in the previous chapter. For the static case, with no body




44

forces present, the equations of equilibrium and the necessary and sufficient

boundary conditions are obtained from the variational integral (3.1) using the

fundamental lemma of the calculus of variations. In order to make a comprehensive

and relevant comparison of the results, the face layers are assumed to be identical;

hence h' = h". Kirchhoffs hypothesis is assumed valid in the face layers; thus

conditions (3.3) are valid. In addition, we assume that the middle plane of the plate

(which is also that of the core layer) is unstretched subsequent to deformation, hence

uo = 0. The above mentioned assumptions allow for the comparison with Eringen's

results; however, while Eringen's problem is based on a two-dimensional linear

elasticity formulation, the present is based on a three-dimensional non-linear

elasticity formulation. The major and underlying difference is that the present

formulation assumes the displacement distribution to be linear in each layer rather

than through the entire plate. Implementing the aforementioned assumptions into

(2.35), we obtain

V' = h g+ (zi+h')-(u3 +h O)] V =u3o + h

V VZu 0 (4.1)
Va = -h O + (zi -h )[-(uz -h O )a] V = h 0o

which describe the displacement distribution in the three-layered plate.

The loading on the upper and lower faces of the plate is assumed to be;

t=, a= "t = 0 = p(x) (4.2)

From (3.1), (3.3), (4.1) and (4.2) we obtain







Ao { [h (NO -N ) +Mo ]i, Q } 50

+ [(N +Na +NflN.)ug a +ho(N +ap
h0
+(Mo +h'Ni +Mii -h "N ),aO + Qao +P} 6u

'. O KO
+*{[ (N{ +)' -N ,a)-( + h(Na + +)N} 06 3,O
h 0 ho 8)
T, N3o







{ ('i Li') + hi( +S) (M r -M ) +h'(Na +No ) ]n )h as
+ (Mi +h WN +M -+hN+ ) "N ,,a + -o nP ] d6




















strain-displacement relations. Therefore, we can omit Nay, Mo and Ko in all
a -ah






L'non-lnear terms. Also, in view of (3.3), (2.57) and (2.59), the necessary stress
resultants appearing i (4.3) are obtained from (2.58):
-a-a a -[(Na++N (+N)u3,, (],N M n ~,)h'S 3,,

/ N U3,a h oN 3,a

Lh0

+{(S+ --3 s)- [(Nia + _-Nda )u3, +ho(Nao +N," ) 03,,
ho 3ho ah0
+(M]Ono h o S 0 ) ds = 0 (4.3)
+M +iNhM +h3N.a), (4+3)


The assumption u = 0, according to Ebcioglu [22], allows us to linearize the

strain-displacement relations. Therefore, we can omit NaO, Mo and K0 in all

non-linear terms. Also, in view of (3.3), (2.57) and (2.59), the necessary stress

resultants appearing in (4.3) are obtained from (2.58):









h 0 0
M o 3 caCyA (*yJ +ol,)

Q0 =2hco C'33(r +uo)

3
o 2h0o a
To = -2h Ca s3 38a

N = 2h 0 Co33 (4.4)

N' =hoh'C Y [,, +*.,-2-(u3 +ho a),,,]

M -2 h i3]
Mai =-h Ca pYp [u 3+ho jI

N, =-hoh" Cpy, [ -2- (u -h 0 o3),.
M = 2h ii3 3t,.
MaS = -" Ca [u3 -ho*3]]l'

where, C p, = C4ty, h' = h" and (3.14) is valid.

By substituting (4.4) and (3.14) into (4.3), we obtain

(4hiGi{[l+ G ]h *,+[ 1+v' GOO ]h 2h'
SG'I' 1 -v' G'I'(1 -2v0) (1 -v')
2GOIO ho* 2GHO 02u)ho + [- +N 3 +(N-N)h
G'Ih G'I' {[(N ) -NN)hi,1
+ 4E'h -8Du,'oapp +2GOho o +2Goho A3 +p)6u3S
1-vi2
+ {(N -N" )3+(N +N )h 0, -8Dh o I3,aaP
LO
4GoIOh' o 3a o dA {(S -
(hhO *,a, *hO) -p)h0o693 --a ho -a

[(4 ii2G ,oh o 0 8G'hiv' 4GOhOvO ,,
-[(4G'h'+ )h a(~,a *0 a)+ + h1 *Y 6 a
3 1 -v' 3(1-2vo)
-8G ih' (u + v u3 6) on)]a 0o
1-v'







-{(L +L )+h '(S' -Si) -[4Gih'2h0 ,(I + 2v#,a ,v 6a)

- 8D(1 -v) (u3, + ,u ap Su {(L -L )+hi(S +S-)
+ Vi o 0 + f o s +'.
1-v
-[8Di(1-v')ho(0,a + 3,v alo + {(Sa +o+S i)
1-v'



0 0 .N:i 0 )
[ (N +N )U -N)ho 3,ar -8Du,0o +4G'h hoa
LO
+ 2G2oh o(4 + )]on -o +( -S 0 -[ 3,a(N







3(1-vi2) (1-2vo) 3

By using the fundamental lemma of the calculus of variations, we obtain from

(4.5) the following four partial differential equations of equilibrium in cartesian

coordinates;
8 D1 u^O 2 h G (u3^ +^) (1 -v)
S(1-v)

=p +(N.a +N )u3ao +(Np -N i)ha O3,


Dn o 4GO1Oh' 0 ,o3 3a0 ,o1]
8Dho3,aap 4GI h3,a- 3

= -P + (N Na ) u3 p + (Npa + Na ) ho 3,a

GI o 1+vi G a-1)hOa
(1+ 1)ho a + V+ Gh OO(aOopa
G'I' 1 v' G'I'
(4.7)
S2Goh2 o o 2h' 0o
-G'I( +3,a ,- v3 0
G'I' 1 -V'


and the boundary conditions are specified from








65,=0, or S -S"+ = (4Gih "2G )(ho h ho ,
ha 3
8G'h'iv' 4GOhOvo
1-v' 3(1-2v)
i2o v +Vi uo0 Onp
-8 h G'2 (u3,ap 6a-u3,yy] ) In
1-v'


6u5 =0, or Li+L +hi(S' -S)= 4Ghl"2ho(w as+,p +-- 6p 'Vo)

8D (1 -v') (u v 6- uv)j on


S, =0, or L'-Lu+hi(S' +S")=8DhO(1-v')(ra* S 6 v) on
3,a- -a a 13 1--v 3


6u3 =0, or S~+S + = [(N +N)u +(N' -N )h -8D u3,o

+4Gi hii2h(*,a+ v *a./) +2Go o(h 0 + ) on
1-v'

L0 r
603=0, or S +-- =(N -N)u +(Ni +N" )hO
6 ha -a3 a -a -a+
(4.8)
8 D 3 h O3,aap + 2 3ho ,# on3

The above partial differential equations are comparable to those derived by

Ebcioglu [22] under the same assumptions. They represent four coupled but linear

fourth order partial differential equations to be solved for the four unknown

displacement functions

(uT3, e3 *, ) (4.9)
The boundary conditions are given in (4.8) and must be satisfied by the solution.







Three special cases of the above equations are derived by Ebcioglu [22] for:

(a) no core, h -,0;

(b) bending moment of the core is negligible, D = 0; and

(c) D --0 and --0, which leads to the equations given by Hoff [9].


4.3 Applications: Numerical Results and Discussion

Consider a rectangular sandwich plate simply supported along face edges x = 0,

x=a andy= 0,y=b. It is submitted to an arbitrarily distributed transverse load p(xy)

on its face and a uniformly distributed axial compressive load (-Nx) along the face

edges x=0 and x=a. The problem is solved by solving the four partial differential
equations (4.7) with Ni = N = -N and N<" =N<'> =0,. The boundary
-xV --XX -xy -yy

conditions specified in (4.8) become

along x= 0 and x=a:

3 = o3 = I2 = 0

Gho 2 6G'h' o v 0
6 GOhO(l-v) (*1,1 + v'2),2 + -2 (*1 2) + 1,
6 G h 0(1-v') 1-2v0o
Gihi2 o 0
( + v' 3,22) =
1-v'

G 'h '2h o
G (h ,1 +vi 2,2) -D1 (u3,11 + u322) = 0
1 v'

-D h( i 22)= (4.10a)
along y= and y= 22
alongy=0 andy=b:











6G(' 1 (hi p i 2 +,1) V 1 ,2 o
Goho(1-vi) 22 1 1 -2vo
2I -2\
(u3z22 + v' u3n) = 0
1- v
1V1 u2vuh=


G'hi2ho ,o I ,o0

-Dih (~, 2 + v' ,)
-D h ha 3,22 + 3,11)


-Di (u3,22 + viu11) = 0


(4.10b)


It may be seen that the following series satisfy all of the boundary conditions:


m=1

03 (x,y) =
m=1


01(xy)=
m=l n=l
0 0
(x y)=1E E
m=l n=l


o s mrx nry
U3, sinm" sinm
a b

,o mrX n y
3mnn sm-r sin b

a b
4YOu cos Ta sinn7ry
mn a b


o mrx
*2,,, smin
a


where uM, 3m,, *Oln and i2r, are unknown constants.

The foregoing series satisfy differential equations (4.7) if p(xy) is expanded

into the Fourier series


p(x,y) = Pmnn
m=l n=1


sin mx si n y
a b


P 4 = r b
ab Jo


a p(x,y) sinmrx sin5 ry dy
Jo a b


(4.13)


Substitution of (4.11) and (4.12) into equations (4.7) yields four simultaneous

algebraic equations for the unknown coefficients u,, i',,, I* and ,,,,,, (see


U30 = 03 = 0I =0


GOh 02
6


+ 112,2


(4.11)


nry
cos n
b


where


(4.12)





51

Appendix B). The displacement functions for any point (xy) can thus be obtained

by means of series (4.11), substituted into (4.1) to obtain the components of

displacement at any point of the 3-D space of the plate.

The equations presented by Eringen are solved for the deflections of the

sandwich plate in the case of a simply supported rectangular plate under the actions

of a uniformly distributed transverse load on the upper face of the sandwich plate

and a uniformly distributed edge compressive load along the edges x= 0 and x=a.

In this case p(xy) = p = constant and equation (4.13) reduces to

Pm= -16p m,n = 1,3,5,.. (4.14)
7r mn

The deflections of the middle plane are computed based on two approaches,

Eringen's theory and the present, for several different geometrical and material

configurations. While maintaining as constants the following: (p=100 psi; Nx=0;

a= 100 inches; E= 2*107 ksi; and h =t+ 2tf= 10 inches, where tc= 2h0 and tf= 2h') and

with b/a = 1, several analyses are conducted by varying the ratios Ef/Ec and tf/t. The

number of Fourier terms in the series was taken to be 19. The results of these

analyses are conveniently plotted in comparable groups in Figures 4.1 through 4.4.

The deflections of the middle plane of the plate are plotted for half the span length

in the x-direction along the line y=b/2, which, due to the loading and geometrical

symmetry of the problem, is representative of the deflected shape of middle plane

of the plate in other perpendicular directions. In Figure 4.1, the face to core

thickness ratio was held at 1 to 10 which indicates very thin faces as compared to the

core. The deflected shape predicted by the two approaches for an elastic modulus

ratio of 1 to 10 are almost identical; however, as the elastic modulus ratio increases,








Distance in x-direction along line y=b/2


a/2


3a/4


Figure 4.1: Vertical deflection comparisons for tf/tc= 1/10.


Distance in x-direction along line y=b/2
a/2 3a/4


Figure 4.2: Vertical deflection comparisons for tf/t,=1/4.








Distance in x-direction along line y=b/2


a/2


3a/4


Figure 4.3: Vertical deflection comparisons for tf/tc= 1/2.


Distance in x-direction along line y=b/2


a/2


3a/4


Figure 4.4: Vertical deflection comparisons for tf/t,= 1.





54

indicating a weaker core material, Eringen's results are slightly higher than the

present results. The maximum difference at the center of the plate is of the order

of 9% which is still within an acceptable comparison range. Figures 4.2 and 4.3

present the results for face to core thickness ratios of 1 to 4 and 1 to 2, respectively.

The pattern is similar to that in Figure 4.1, where again the difference between the

two predicted deflected shapes tends to increase as the core becomes weaker. The

variation between the two predicted deflected shapes seems to widen with the

increase of the face to core thickness ratio as well. The results become incomparable

when the thickness ratio becomes unity, that is the face and core layers are of equal

thicknesses.

The results of Eringen's equations start diverging from the present results and

become unreliable by over-predicting the deflected shape enormously as the face

thickness approaches that of the core, as well as when the core material tends to

weaken. In order to justify the just stated conclusion, consider the following

situation. The three-layered plate in the previous analysis is subjected to the

following (b/a = 10, a/h = 1, and E/E, = 1) which simply states that the face and core

layers are assumed to be made of the same material. By holding the thickness of the

plate constant, the results may be compared to that predicted by the classical plate

theory. The maximum deflections at the center of the plate are computed based on

the present formulation and based on Eringen's equations for several ratios of face

to total thickness of the plate and the results are plotted in Figure 4.5 for the

maximum deflection versus the thickness ratio. The values compare very well for

small thickness ratios; however as the ratio increases, more specifically beyond 1/3,

the maximum deflection predicted by Eringen's equations increases rapidly. The








0.00E+00-
a)
+-3
-1.00E-02- Ef/Ec=l
b/a=I
S-2.00E-02- b/a=l

2 -3.00E-02-

o -4.00E-02-

-56.00E-02-
0
-6,00E-02- \
U \
S-7.00E-02-
CD
-8. 0E-02-
U Present
H -9.00E-02-
4- --- Eringen

S -1.00E-01- I I I I
0 0.1 0.2 0.3 0.4 0.5

Face to total thickness ratio, tf/h

Figure 4.5: Comparison of vertical deflection at the center of a simply supported
square plate subject to a uniformly-distributed transverse load.



maximum deflections for a homogenous plate of the same dimensions and thickness

are obtained from the equations of the classical plate theory, and the ratios of the

maximum deflections from the present and from Eringen's equations to that of the

classical plate theory are plotted versus the thickness ratio in Figure 4.6. The ratios

of the present formulation to the CPT deflections are very close to unity, indicating

a good prediction by the present equations regardless of face to total thickness ratio.

They are not exactly equal to unity for the simple reason that in the present

formulation the transverse shear strains are not neglected. The ratio of Eringen's to

CPT deflection starts diverging from unity when the face to total thickness ratio

reaches 1/3.








2 ---

19- Ef/Ec=l
E-
S 1. 8 b/a=l

> 1 7- h/a=l/10
0
S 1. 6 -

S1. 5-

o 1.4 /
H
S 1. 3-
(1 /
S1.2

S1-1
CO
U j

-Present
>----- E- ringen
0 8 I I I
0 0.1 0.2 0.3 0.4 0.5

Face to total thickness ratio, tf /h

Figure 4.6: Comparison of the max vertical displacements ratio for varying face to
total thickness ratios.
(Note: MLPT = Multi-Layered Plate Theory; CPT = Classical Plate Theory.)


4.4 Stability Analysis of the Sandwich Plate

The transverse displacement of the plate is assumed to vary linearly in the z-

coordinates, thereby allowing the core to experience a change in thickness. This is

usually referred to as the flattening of the core (i.e., the approach of the two faces).

It is measured as the difference between the transverse displacements of the two face

layers. It was first examined by Eringen [11] as to its effect on the overall instability

of sandwich plates. The present theory predicts the flattening of the core, and the





57

results are in good agreement with Eringen's for the same ranges stated earlier in the

case of overall plate deflections.

For the particular example of an arbitrarily distributed transverse load, it can

be easily proved that, owing to the in-plane compressive forces Nx, all the coefficients

uo,, as well as all the coefficients inn,, increase; hence deflections of a compressed

plate are larger than those of the equally loaded identical plate without in-plane

compressive loads applied at the face edges. It is seen also that by a gradual

increase of compression we approach a value of Nx for which one of the coefficients

u,,n or one of the coefficients ,,mn approaches infinity. The smallest of these values

of Nx is called the critical value.

From the solution to the four simultaneous algebraic equations for the

unknown coefficients u,,, Jon, 0O, and i0,n, we obtain expressions for the two

coefficients um,, and r~,, as the following (see Appendix B)

o P =Pmn (4.15)
mn Bmn

Here, buckling occurs when either Amn,,- or Bn-*O, depending upon whichever gives

the smallest critical load. These critical values are Nc,1 and NCR2, and they are

positive increasing functions of n. Thus, n = 1 makes NcaR and NC2 minima. The

critical stresses a c and ac2 may be obtained by dividing NcA and NCR2 by face

thickness (2h'). The critical stress acI represents the buckling of the plate as a

whole and ac,2 represents the buckling due to flattening of the core.

To better demonstrate the effect of the compressibility of the core (see Figure

4.7), consider the sandwich plate having faces made of aluminum alloy materials

(Ef= 107 psi) and the core made of soft isotropic materials (Ec=2000 psi). We

















BUCKLING WITH CORE FLATTENIIIIGI


Figure 4.7: Buckling modes of a sandwich plate.


observe from the computation of acR and aCR that (a) acRl and acR.2 have

minimums at some a/mb, and (b) the number of half-waves (m) for which (acR.1)min

occurs is much smaller than that for (acR2)min.

The values of (acRl)min and (acR2)min are plotted for various face thickness to

width ratios t/fb as well as for various core thickness to width ratios tc/b, as shown

in Figure 4.8. Similar curves are generated in Figure 4.9 based on Eringen's theory.

The obvious conclusion from Figure 4.8 is that, except for the case of

tf/b = 1/1000, flattening of the core does not influence the overall buckling of the

plate until the core thickness to width ratio exceeds 1 to 5 which indicates a very

thick core of a very narrow plate (thus sandwich beam-column analysis may be more

appropriate). For face thickness to width ratios of the order of 1 to 1000 and

smaller, buckling occur due to flattening of the core when tc/b exceeds 1/10. These

results greatly under-emphasize the effects of core flattening predicted by Eringen

shown in Figure 4.9. A statement to that effect may be made, that while the


EULER (OVERALL) BUCKLING





























0 0.04 0.08 0.12 0.16
Core thickness to width ratio Tc/b


0.2


Figure 4.8: Minimum Critical Stresses of the present theory for both types of buckling
(plate as a whole & with the flattening of the core).


0 0. 04 0. 08 0.12 0. 16
Core thickness to width ratio Tc/b


0.2


Figure 4.9: Minimum Critical Stresses of Eringen's theory for both types of buckling
(plate as a whole & with the flattening of the core).





60

26

24

2 22
en
20
OL
M 18 -
16 PRESENT
U3









H i4
( ERINGEN
o 12 -

10 REISSNER


-- ---- ------------- ^
al 6 .......-----"--


2

0 --I I I \i -
0 0.002 0,004 0.006 0.008 0.01

Face thickness to width ratio. Tf/b

Figure 4.10: Comparison of the critical stresses for te/b= 1/20.


compressibility of the core may be a critical element in the instability analysis of

sandwich plates, it does not become so until face to core thickness ratio tf/tc is very

small.

In general, while predicted (aCR2)min are tremendously different, the overall

buckling or minimum acR (the lesser of (aclj)min and (aCR2)min) compares well

between the present and Eringen's theory. In Figure 4.10, these values based on the

present and Eringen's theory are compared with the minimum aCR from Reissner's

theory [10] for core thickness to width ratio of 1 to 20. The ratio of the values

increases as tf1b increases or as tC/tf decreases. As indicated by Reissner, this may





61

be due to the fact that for thin sandwich plates (small tc/tf) overall buckling occurs

with small wave lengths for which the theory developed there becomes inadequate.


4.5 Reduction to the Classical Plate Bending and Buckling Equations

When the core is eliminated (h= 0), the loading is assumed to be a uniformly

distributed transverse load only, and, making N,= 0, we may derive an expression for

the vertical deflections of a homogenous plate having thickness t = 4h and dimensions

a and b. Implementing h= 0 into the four simultaneous algebraic equations for the

unknown coefficients u, ,, 3, Jra and ir,, and solving lead to the only non-

vanishing coefficients
P
0 _mn
u ,n 2(4.16)
U3mn -8 D [(mR)2 (m1)2]2 (4.16)
a b
where

S16p m,n = 1,3,5,..
r2 m n

The vertical deflection is obtained by substituting the coefficients (4.16) into

the first of the series (4.11) and noting that the second series in (4.11) vanishes for

every term in the series. Thus
mrx nry
sinm x sinnrY

7~"(8 D) m mn [(m)2+ )2

where

t 3
2E'h3 2E ( Eit3 D (4.18)
8D 1 = 8 3E-2 = 8t3(-l 2 =12 = Dc_1r
33(1-v') 3(1-vi2), 12(1-v'2)





62

The deflections V3(xy) given in (4.17) are identical with the one obtained in the

classical plate theory [42].

We now assume that there is a compressive force, Nx, acting on the edges of

each of the face layers (i.e., 2Nx acting on the plate edges) and that the transverse

load may be arbitrarily distributed. Implementing h = 0 into the four simultaneous

algebraic equations as before and solving lead to the only non-vanishing coefficients;

0o Pmn
U3mn = (4.19)
S8D, (mr) + (nr)2] 2Nx(Tr)2


The critical buckling load is the smallest value of Nx for which one of the coefficients

umn becomes infinite; thus it is necessary to take n = 1, thereby obtaining

(2Nx)c = (8 D) 2 m+1 a2 (4.20)
a2 mb2

which is identical with the one obtained in the classical theory [42].












CHAPTER V
CONCLUDING REMARKS


A variational theorem of three-dimensional elasticity has been used in

developing the fundamental equations of the theory of multi-layered plates in terms

of a reference state. The theory has been demonstrated for the case of a three-

layered plate as far as deriving the equations of motion and boundary conditions.

Special cases were considered and studied by solving the differential equations for

given loading and geometry conditions and comparing the results obtained

numerically to those of the literature.

The theory presented considers all stress components in each layer; and the

assumed displacement components were defined separately for each layer. Hence,

it can be stated that the procedure of this derivation without making any further

assumptions can be extended to a layered plate of more than three layers. The

mathematical derivation, however, is more involved. Furthermore, it may not be a

worthwhile task at the present time to attempt such a derivation unless means for

numerical implementation exist. It is obvious that the theoretical development in this

field is more advanced than the availability of techniques for solving the system of

non-linear partial differential equations characterizing these structural elements.

This lack of solving techniques incorporates both analytical as well as numerical

aspects. However, in this age of rapid advancement in the area of computational

utilization, which has yielded the introduction and implementation of numerous





64

sophisticated numerical techniques in several areas of the engineering domain, it is

well conceivable to find in the near future some mechanism for solving these highly

non-linear systems.

The general theory presented has the advantage of producing a complete set

of fundamental equations consistent with various stages of linearization in the general

strain-displacement relations. The literature abounds in such intermediate theories,

and the present work is hoped to have shed some light on these special cases as well

as cleared the way for a systematic development of plate and shell theories directly

from the three-dimensional theory of elasticity in terms of a reference state.












APPENDIX A
NOTATIONS OF SOME EARLIER WRITERS


(a) Notations in general theory appearing in chapter II:

Green and
Field Quantity Adkins [38] Fung [39] Malvern
(Present) [40]
Position of undeformed body xi ai Xi
Position of deformed body yi Xi xi
Displacement components V u ui
Green's deformation tensor GO, Ci
Symmetric Strain tensor Yij Ei Ei1
Unit normal in undeformed position oni Voi Ni
Symmetric stress tensor per unit area
of undeformed body (second Piola- Si Si Ti
Kirchhoff stress tensor)
Unsymmetric stress tensor per unit
area of undeformed body (first Piola- t T- 7
Kirchhoff stress tensor)
Stress vector per unit area of
undeformed body associated with ot t
surface in deformed body


(b) For Si :


Novozhilov


Eringen


(i=x,y,z; j=x,y,z)

,pseudo-stress


C, Kirchhoff-Trefftz stress


Novozhilov


T" pseudo-stress, Piola stress


Eringen


(c) For ti :


a,9 (i=x,y,z; j= ,r7,()












APPENDIX B
SOLUTION TO PDE AND LISTINGS OF COMPUTER PROGRAMS


Define the following geometric and material constants

S4E'h i2ho =6D
1-vi2 h'

4GoIohoh' 2GOh02
2 = i 3

G010 G oho
773=ho 1+-GI =ho 1+ ho
G'I' = 6G'h' (B.1)

1=h +vi G I 10a0-1) ho 1+v' Goho (ao-1)
S 1-v' G' l 1-v' 6G'h'
2GOh 02 Go
SG'Il 2h'G'
2hi
1-v'

The four simultaneous algebraic equations for the unknown coefficients in Fourier

series (4.11) may be written in matrix form as follows

E11 0 E13 E14 U3mn 1
0 E22 0 0 3,,n, C2 (B.2)
E31 0 E33 E34 *lmn C3
E41 0 E43 E44 n.2mn C4
where

= +P. C2 = -Pn 3 = 4 = 0

('1 = 8D, (m 4+2(m-)2 n )2 +( 4 +2hG ( mr)2+( )2 i2N r )2
a a b b I.a b a








13, = 2hoGo( r (m ( mr n (
a a a b

E14 = 2hoGo( nEr ) 71mi )2(r).( nr )3
b a b b

S8D, m)4 +2(7r)2(nr )2+(r )4+72 (Mr)2.(mn )2 3a (Mrh)2
322 = 8Dh (--- ) ~)2 --f -) +(--f 2 o 2Nxho a 2
a a b b a b h2 a

31 = -l5 a + 76 ( )3 +( )( )2


E33 = -73 (-- f ) (- -5)

E34 = -4(--)(--)
a b

E41 = -5( ) +'16 ( )2( )H( )3
b [a b b
mr nr
E43 4 -7r4(-)(
a b
m/r" 2 /n7r 22 /n7/ 2
-44 -,3 (a) +(- B) 574(-*-) 75
a b b
The algebraic system (B.2) is thus solved for the unknown constants (u3,,,,,

l3mn,, TImnw 1r2n), and these constants are in turn substituted into their respective

series in (4.11) to obtain an expression for the displacement functions at any point

(xy). The constants (u3,,n,,, l,,,,,,,, im 2;,,,,) are actually functions of the series indices

m and n; therefore, the simultaneous algebraic system of equations is solved as many

times as the number of terms in the series. A complete computer program

implementing the procedure outlined in chapter IV and the foregoing paragraph is

presented at the end of this section.

In order to determine the critical load Nx it is necessary to obtain an

expression for 3,,,, and I3mn and then set the denominator equal to zero. It can be

shown that this is obtained by solving the following two equations for N,





68

E22 = 0
3 3E13(~ ~44-34 4) 4(E43 31 33 41) (B.3)
(E33 E44 E34 E43)

where the constants appearing in (B.3) are defined above. Note that the term N,

appears only in the definitions of ,, and E22 (LHS of equations B.3).








Listing of a computer program written in FORTRAN language to compute and print
the complete defomation data of a simply supported rectangular sandwich plate
under uniformly distributed transverse load, and to evaluate the critical stresses for
instability, based on the present theory.

C Program to solve 4 pde's using Fourier series
C Navier Method!
C
C iout = 1: buckling critical stresses
C 2 : fixed Ef, h, b/a, varying tf/tc, Ef/Ec
C 3 : maximum deflection to screen only
C else : complete deformation output at (x,y,z) & p(x,y)
C -------------------------------------------
dimension vcl(30,30,5),vc2(30,30,5),vc3(30,30,5),hfd(3)
dimension u3(30,30),s3(30,30),sl(30,30),s2(30,30),p(30,30)
dimension cm(4,4),bm(4),um(4),scfac(4),aa(4,5),
+ pmn(19,19),u3mn(19,19),s3mn(19,19),slmn(19,19),s2mn(19,19)
common pi,a,b,hc,hf,vc,vf,ec,ef,po,rnx,nft
open(unit = 1,file = 'seql.dat',status = 'old')
open(unit = 2,file = 'seql.out',status= 'old')
data hfd/.25,.1,.05/
read(l,*) a,b,hc,hf
read(l,*) vc,vf,ec,ef
read(l,*) iout,rnx,po,npx,npy,nft
pi = 3.14159
C --------------------------------- ----------
if(iout.ne.1) goto 105
write(2,101) a,b
101 format(5x,'Table Generated for a= ',f6.1,' and b= ',f6.1/
& /6x,'tf tc',7x,'tf/b tc/b',6x,'m Scrl.min',
& 2x,'m Scr2.min Scr.min'/)
do 104 i= 1,3
hf=hfd(i)
do 102 j = 1,11
hc=real(j-l)
call screva
102 continue
write(2,103)
103 format(/)
104 continue
goto 999
C ---------------------------------------- --
105 if(iout.ne.2) goto 111
hc= (10-4*hf)/2
count= 1
106 ec= ef/(10*jcount)
if(jcount.eq.3) ec= ef/50
111 do 120 m=1,nft
do 120 n= 1,nft
pmn(m,n)= 0.
u3mn(m,n) =0.
s3mn(m,n) =0.
slmn(m,n) =0.








s2mn(m,n)= 0.
120 continue
C ---------------------------------------------- -------
call coefs(pmn,u3mn,s3mn,slmn,s2mn)
do 200 i= ,npx+1
xa= (i-l)/real(npx)
do 200 j= 1,npy+
yb= (j-1)/real(npy)
u3(i,j)=0.
s3(ij) =0.
sl(ij) = 0.
s2(ij)= 0.
p(ij)=0.
do 140 m= l,nft,2
col = sin(m*pi*xa)
co2= cos(m*pi*xa)
do 140 n = ,nft,2
co3 = sin(n*pi*yb)
co4= cos(n*pi*yb)
u3(ij)= u3(ij) + u3mn(m,n)*col*co3
s3(ij) = s3(ij) + s3mn(m,n)*col*co3
sl(ij) = sl(ij) + slmn(m,n)*co2*co3
s2(ij) = s2(ij) + s2mn(m,n) *col*co4
p(ij) = p(ij) + pmn(m,n)*col*co3
140 continue
do 150 k =1,3
zh= (k-2)*hc
vcl(ij,k)= zh*sl(ij)
vc2(ij,k)= zh*s2(ij)
vc3(ij,k)= u3(ij) + zh*s3(ij)
150 continue
200 continue
C --------------------------------- ---------
if(iout.ne.2) goto 230
write(2,210) b/a,hf/hc,ef/ec
write(2,220) (100*vc3(i,2,2),i= 11,21)
210 format(5x,'b/a =',f5.2,6x,'hf/hc =',f6.2,6x,'Ef/Ec =',f6.2)
220 format(lx,ll(f7.4)/)
if(jcount.ge.3) goto 999
jcount=jcount+ 1
goto 106
C -------------------------------- ----------
230 if(iout.eq.3) goto 666
do 300 i=1,npx+1
do 300 j= ,npy+1
write(2,600) ij,p(ij)
write(2,610) (k,vcl(ij,k),vc2(ij,k),vc3(i,j,k),k= 1,3)
300 continue
goto 999
600 formaL(5x,2i6,5x,e 10.2)
610 format(3(i4,3el2.4/))
666 write(6,*) (vc3(2,2,k),k= 1,3)
999 end
C ------------------------------------------








subroutine coefs(pmn,u3mn,s3mn,slmn,s2mn)
dimension cm(4,4),bm(4),um(4),
+ pmn(19,19),u3mn(19,19),s3mn(19,19),slmn(19,19),s2mn(19,19)
common pi,a,b,hc,hf,vc,vf, ec,ef,po,rnx,nft
gc = ec/2/(1+ vc)
gf=ef/2/(1+vf)
dl= 4*gf*hf**3/3/(1-vf)
ac=2*(l-vc)/(1-2*vc)
tl= 8*gf*hf**2*hc/(1-vf)
t2=2*gc*hc**2/3
t3 = hc*(l + gc*hc/(6*gf*hf))
t4 = hc*((1+ vf)/(1-vf) + gc*hc* (ac-1)/(6*gf*hf))
t5= gc/(2*gf*hf)
t6 = 2*hf/(1-vf)
do 160 m= 1,nft,2
ftl=m*pi/a
do 160 n = ,nft,2
do 110 i= 1,4
bm(i) =0.
um(i)=0.
do 110 j= 1,4
cm(ij) =0.
110 continue
ft2= n*pi/b
t7=ftl**2+ft2**2
t8= ftl**4+2*(ftl**2)*(ft2**2) + ft2**4
cm(1,1) = 8*dl*t8 + 2*hc*gc*t7-2*rnx*ftl**2
cm(1,3) = (2*hc*gc-tl*t7)*ftl
cm(1,4) = (2*hc*gc-tl*t7)*ft2
cm(2,2) = 8*dl*hc*t8 + t2*t7 + 2*ac*gc-2*rnx*hc*ftl* *2
cm(3,1) = ftl*(t6*t7-t5)
cm(3,3) = -t3*t7-t4*ftl**2-t5
cm(3,4)= -t4*ftl*ft2
cm(4,1) = ft2*(t6*t7-t5)
cm(4,3)=-t4*ftl*ft2
cm(4,4) = -t3*t7-t4*ft2**2-t5
pmn(m,n) = 16*po/pi* *2/m/n
bm() = pmn(m,n)
bm(2)= -pmn(m,n)
if(hc.ne.0.) goto 140
u3mn(m,n) = bm()/cm(1,1)
s3mn(m,n) =0.
slmn(m,n) = 0.
s2mn(m,n)=0.
goto 160
140 call simeq(4,cm,bm,um)
u3mn(m,n) = um(l)
s3mn(m,n)= um(2)
slmn(m,n) = um(3)
s2mn(m,n) = um(4)
160 continue
return
end
.-................................................................








C Subroutine to Evaluate the Critical buckling loads
C
subroutine screva
common pi,a,b,hc,hf,vc,vf,ec,ef,po,rnx,nft
gc= ec/2/(1+ vc)
gf=ef/2/(l+vf)
dl= 4*gf*hf**3/3/(1-vf)
ac=2*(1-vc)/(1-2*vc)
tl=6*dl*hc/hf
t2=2*gc*hc**2/3
t3 = hc*(1 + gc*hc/(6*gf*hf))
t4 = hc*((1+ vf)/(1-vf) + gc*hc*(ac-1)/(6*gf*hf))
t5 =gc/(2*gf*hf)
t6 = 2*hf/(1-vf)
do 150 i= 1,2
m=1
scrm = 10.e + 9
101 ftl=m*pi/a
ft2=pi/b
t7=ftl**2+ft2**2
t8=t7**2
if(i.eq.2) goto 102
cl = 8*dl*t8 + 2*hc*gc*t7
c13= (2*hc*gc-tl*t7)*ftl
c14= (2*hc*gc-tl*t7)*ft2
c31= ftl*(t6*t7-t5)
c33= -t3*t7-t4*ftl**2-t5
c34=-t4*ftl*ft2
c41= ft2*(t6*t7-t5)
c43=-t4*ftl*ft2
c44= -t3*t7-t4*ft2* *2-t5
tmp= cl3*(c31*c44-c41*c34)-cl4*(c43*c31-c41*c33)
scr = (cll-tmp/(c33*c44-c43*c34))/4/hf/ftl**2
goto 103
102 c22=8*dl*hc*t8+t2*t7+2*ac*gc
if(hc.eq.0.) goto 130
scr = c22/4/hf/hc/ftl**2
103 if(scr.ge.scrm) goto 130
scrm= scr
m=m+l
if(m.lt.50) goto 101
write(6,120) i,m
120 format(lx,' In SCR(',il,') exceeded max m = ',i4)
stop
130 if(i.eq.1) then
scrml = scrm
ml= m-1
else
scrm2 = scrm
m2=m-1
endif
150 continue
scrmin = min(scrml,scrm2)
write(2,200) 2*hf,2*hc,2*hf/b,2*hc/b,m l,scrml,m2,scrm2,scrmin








200 format(2x,2f7.2,lx,2f10.5,2(2x,i2,2x,f9.0),2x,f8.0)
return
end
C ---------------------------------------------
C Subroutine to Solve Simulteneous Equations
C
subroutine simeq(ne,cm,bm,um)
dimension cm(4,4),bm(4),um(4),scfac(4),aa(4,5)
do 20 i=1,ne
do 10 j=1,ne
aa(ij)= cm(i,j)
10 continue
aa(i,ne + 1) =bm(i)
20 continue
do 50 i= 1,ne
big= abs(aa(i,1))
do 30 j= 2,ne
anext = abs(aa(i,j))
if(anext.gt.big) big= anext
30 continue
if(big.lt..000001) then
write(6,40) i
40 format(lhl,//,'** Elements in Row ',i2,' are Zeros **')
stop
end if
scfac(i) = 1./big
50 continue
do 60 i= 1,ne
do 60j= ,ne+1
aa(i,j) = aa(i,j)*scfac(i)
60 continue
do 110 i=1,ne-1
ipvt= i
ipl=i+l
do 70 j =ipl,ne
if(abs(aa(ipvt,i)).lt.abs(aa(ij))) ipvt=j
70 continue
if(abs(aa(ipvt,i)).lt..000001) then
print *,'solution not feasible.',
& a near zero pivot was encountered'
stop
end if
if(ipvt.ne.i) then
do 80 jcol= ,ne+1
temp = aa(ijcol)
aa(i,jcol) = aa(ipvt,jcol)
aa(ipvtjcol) = temp
80 continue
end if
do 100 jrow = ipl,ne
if(abs(aa(jrow,i)).eq.0.) goto 100
ratio = aa(jrow,i)/aa(i,i)
aa(jrow,i) = ratio
do 90 kcol= ipl,ne + 1








aa(jrow,kcol) = aa(jrow,kcol)-ratio*aa(i,kcol)
90 continue
100 continue
110 continue
aa(ne,ne +1) = aa(ne,ne + 1)/aa(ne,ne)
do 130 j =2,ne
nvbl=ne+ 1-j
1= nvbl+ 1
temp = aa(nvbl,ne +1)
do 120 k=l,ne
temp = temp-aa(nvbl,k) *aa(k,ne +1)
120 continue
aa(nvbl,ne + 1) = temp/aa(nvbl,nvbl)
130 continue
do 140 i=1,ne
um(i) = aa(i,ne +1)
140 continue
return
end







Listing of a computer program written in FORTRAN language to compute and print
the complete defomation data of a simply supported rectangular sandwich plate
under uniformly distributed transverse load, based on Eringen's theory.

C BENDING OF A SIMPLY-SUPPORTED RECTANGULAR SANDWICH PLATE
C UNDER A UNIFORMLY-DIST. TRANSVERSE LOAD ON THE UPPER FACE
C AND A UNIFORMLY-DIST. AXIAL COMPRESSIVE LOAD ALONG FACE
C EDGES X=0 AND X=A.
C
DIMENSION UP(30,30,5),VP(30,30,5),WP(30,30,5)
DIMENSION U(30,30),V(30,30),W1(30,30),W2(30,30),P(30,30)
COMMON PMN(19,19),UMN(19,19),VMN(19,19)
COMMON W1MN(19,19),W2MN(19,19)
COMMON PIA,B,H,TC,TF,VC,VF,GC,GF,RI1,RIC,RIF,NFT,PO,RNX
C OPEN( UNIT= 1, FILE= 'ERINGEN1.DAT', STATUS ='OLD' )
OPEN( UNIT =2, FILE='ERINGEN1.OUT', STATUS ='OLD')
A=100.
B=100.
TC= 10.
TF=2.
VC=.3
VF=.3
EC =2000.
EF= 10000000.
RNX= 10.
PO = 100.
NFT=9
NPX=4
NPY=4
PI=3.14159
cc jcount= 1
cc READ(5,*) TF
cc TC= 10-2*TF
C READ(1,*) A,B,TC,TF,VC,VF,EC,EF
C READ(1,*) PO,RNX,NPX,NPY,NFT
C
cclO ec=ef/(10*jcount)
cc if(jcount.eq.3) ec=ef/50
H=TC+TF
RI1=TF**3/12.
RIC=TC**3/12.
RIF=TF*H**2/2.
GC=EC/2./(1+ VC)
GF=EF/2./(1+ VF)
DO 100 M=1,NFT
DO 100 N=1,NFT
PMN(M,N) =0.
UMN(M,N)= 0.
VMN(M,N) =0.
W1MN(M,N)= 0.
W2MN(M,N) =0.
100 CONTINUE
CALL COEFS







C
DO 200 I=1.NPX+1
XA= (I-1)/REAL(NPX)
DO 200 J=1,NPY+1
YB= (J-1)/REAL(NPY)
U(I,J)= 0.
V(I,J) = 0.
W1(I,J)= 0.
W2(I,J)=0.
P(I,J) = 0.
DO 120 M= 1,NFT,2
CO1= SIN(M*PI*XA)
CO2= COS(M*PI*XA)
DO 120 N= 1,NFT,2
C03= SIN(N*PI*YB)
CO4= COS(N*PI*YB)
U(I,J)= U(I,J)+ UMN(M,N)*C02*C03
V(I,J) = V(I,J) + VMN(M,N)*CO1*C04
W1(I,J) = W1(I,J) +W1MN(M,N)*CO1*C03
W2(I,J)=W2(I,J)+W2MN(M,N)*C01*CO3
P(I,J) = P(I,J) + PMN(M,N)*CO1*C03
120 CONTINUE
DO 130 K= 1,3
ZH= (K-2)*TC/2
UP(I,J,K)= ZH*U(i,J)
VP(I,J,K)= ZH*V(I,J)
WP(I,J,K)= W1(I,J) + ZH*(2/h)*W2(I,J)
130 CONTINUE
200 CONTINUE
cc write(2,210) b/a,tf/tc,ef/ec
cc write(2,220) (100*wp(i,2,2),i= 11,21)
cc210 format(5x,'b/a = ',f4.2,5x,' tf/tc = ',f5.2,5x,' Ef/Ec = ',f5.2)
cc220 format(lx,ll(f7.4)/)
cc if(jcount.ge.3) goto 999
cc jcount=jcount+l
cc goto 10
c write(6,*) wp(2,2,2)
DO 300 I = 1,NPX+1
DO 300 J =1,NPY+1
WRITE(2,600) I,J,P(I,J)
WRITE(2,610) (K,UP(I,J,K),VP(I,J,K),WP(I,J,K),K= 1,3)
300 CONTINUE
600 FORMAT(5X,216,5X,E10.2)
610 FORMAT(3(I4,3E12.4/))
999 END
C
SUBROUTINE COEFS
COMMON PMIN(9,1),N(19,19), N(19,19),VMN(19,19),WlMN(19,19),W2MN( 19,19)
COMMON PIA,B,H,TC,TF,VC,VF,GC,GF,RI1,RIC,RIF,NFT,PO,RNX
TO=RNX/GC
T1= GC*RIC/GF/RIF
T11=RIC*TF/RIF/TC
T2= 2*(1-VC)/(1-2*VC)
T3= (1-VF)





77

T33 = GF*RI1/T3/GC/RIC
DO 100 M= 1,NFT,2
T4= (M*PI/A)
T5=(T4**2)*TO*TC
DO 100 N=1,NFT,2
T6= (N*PI/B)
T7= (T4**2+T6**2)*TC**2
T8= (2/T3+ T1*T2)*T7/12. +T1
D1MN= T7*(-T1+ (T33*T7/3. + 1.)*T8)-2.*T5*T8
D2MN=T7*(T33*T7/6. +T11) + 12.*T2*T11-T5
PMN(M,N) = 16.*PO/PI**2/M/N
UMN(M,N)= -(PMN(M,N)/2./GF/D1MN)*(RIC/RIF)*(M*PI*TC/A)*H
VMN(M,N)= UMN(M,N)*A/B
W1MN(M,N) = (PMN(M,N)*TC/GC/D1MN)*T8
W2MN(M,N)= PMN(M,N)*TC/2./GC/D2MN
100 CONTINUE
RETURN
END








Listing of a computer program written in FORTRAN language to evaluate the
critical stresses for instability of a simply supported rectangular sandwich plate, based
on Eringen's theory.

DIMENSION TFD(3)
COMMON A,B,VC,VF,EC,EF
DATA TFD/.5,.2,.1/
VC=.3
VF=.3
EC =2000.
EF= 10000000.
A=100.
B= 100.
OPEN(UNIT= 3,FILE = 'ER IN-1.OUT',STATUS ='UNKNOWN')
WRITE(3,101) A,B
101 FORMAT(5X,'Table Generated for a= ',F6.1,' and b= ',f6.1/
& /6X,'tf tc',7X,'tf/b tc/b',6X,'m Scrl.min',
& 2X,'m Scr2.min Scr.min'/)
DO 104 I =1,3
TF=TFD(I)
DO 102 J= 1,11
TC=2*REAL(J-1)
CALL SCREVA(TC,TF)
102 CONTINUE
WRITE(3,103)
103 FORMAT(/)
104 CONTINUE
STOP
END
CC
SUBROUTINE SCREVA(TC,TF)
COMMON A,B,VC,VF,EC,EF
PI=3.14159
GC=EC/2/(1+VC)
GF=EF/2/(1+VF)
C1= (PI**2*EF)/(12*(1-VF* *2))
C2= (1-VF)/(PI**2)*(GC/GF)
C3= (1-VF)*(2*(1-VC)/(1-2*VC))/12*(GC/GF)
C4= 12*(2*(1-VC)/(1-2*VC))/PI**2
DO 150 I= 1,2
M=1
SCRM= 10.e+9
101 X1=A/B/M
X2=TC/B
X3=TF/B
SCRF= C1*(X3**2)*(1/X1+ X1)**2
IF(I.EQ.2) GOTO 102
ETA1= (3*C2*X2/X3 '3)/(1+ 1/X1**2+C2*X2/(X3*(X3 + X2)**2
& +C3*X2**3))
SCR=SCRF*(1+ETA1)
GOTO 103
102 IF(TC.EQ.0.) GOTO 130
ETA2=C2*(X2/X3)**3*(1+X1**2+ C4'(X 1/X2)**2)/








& (1/X1+X1)**2/(X2+X3)**2
SCR = SCRF*(1+ ETA2)
103 IF(SCR.GE.SCRM) GOTO 130
SCRM=SCR
M=M+1
IF(M.lt.50) GOTO 101
WRITE(6,120) I,M
120 FORMAT(1X,' In SCR(',I1,') exceeded max m = ',14)
STOP
130 IF(I.EQ.1) THEN
SCRM1= SCRM
M1=M-1
ELSE
SCRM2= SCRM
M2=M-1
ENDIF
150 CONTINUE
SCRM IN = MIN(SCRM1,SCRM2)
WRITE(3,200) TF,TC,X3,X2,M1,SCRM1,M2,SCRM2,SCRMIN
200 FORMAT(2X,2F7.2,1X,2F10.5,2(2X,I2,2X,F9.0),2X,F8.0)
RETURN
END












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83

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[43] H. L. Langhaar, Energy Methods in Applied Mechanics, First Edition, John
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BIOGRAPHICAL SKETCH


Aref Altawam was born on September 21, 1963 in Damascus, Syria. Upon

graduating from high school in 1980, he attended the University of Damascus and

later the Mississippi State University where he received a Bachelor of Science in civil

engineering in 1984 and in 1987 received a Master of Science in structural

engineering. In August 1987 he entered the University of Florida to pursue his Ph.D.

in engineering mechanics. He received his doctorate degree in the Fall of 1990. He

plans to pursue a career in academia.







I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.



"Tira-im K. Ebcioglu, Chairman
Professor of Aerospace Engineering,
Mechanics and Engineering Science

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.



Lawrence E. Malvern
Professor of Aerospace Engineering,
Mechanics and Engineering Science

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.



Cliff, Hays, J
Prof ssor of Civil Engineering

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.



Bhavani Sankar
Associate Professor of Aerospace
Engineering, Mechanics and
Engineering Science







I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.



David Zi m an
Assistant P oessor of Aerospace
Engineering, Mechanics and
Engineering Science

This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.

December 1990

Winfred M. Phillips
S Dean, College of Engineering




Madelyn M. Lockhart
Dean, Graduate School







































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