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I.

.‘z

""*’ LIBRARY

Michigan State

University

//l in]; mm

”WW/m,
Mm.-. ‘._..__ 1 a, 7

    
   
   

This is to certify that the
thesis entitled
A NEW METHOD

FOR THE SOLUTION OF
ANISOTROPIC THIN PLATE BENDING PROBLEMS

presented by

Benjamin Chin-wen Wu

has been accepted towards fulfillment
of the requirements for

__.Eh.D..__ degree in mechanics.

 

Date February 14, 1980

 

0-7639

 

MAY 1 2 203‘

 

OVERDUE FINES:
25¢ per day per item

RETUMIIKS LIBRARY MTERIALS:

Place in book return to remve
charge from circulation records

 

 

 

A NEW METHOD
FOR THE SOLUTION OF
ANISOTROPIC THIN PLATE BENDING PROBLEMS

BY

Benjamin Chin-wen Wu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics, and Materials Science

1980

ABSTRACT

A NEW METHOD
FOR THE SOLUTION OF
ANISOTROPIC THIN PLATE BENDING PROBLEMS

BY

Benjamin Chin-wen Wu

A numerical method for the solution of thin plate pro-
blems is presented. With the conventional assumptions for
thin plates implied, plates of arbitrary plan form, sub-
jected to arbitrary loading and boundary conditions, and
made of anisotrOpic material are considered. This method
is developed from the concept of the indirect boundary in-
tegral method.

The indirect boundary integral method uses the Green's
function of a clamped circular plate of isotrOpic material.
To solve an isotropic thin plate problem, the first step
is to embed the real plate into the fictitious clamped cir-
cular plate for which the Green's function is known. Along
the embedded contour, N points are prescribed, at which the
boundary conditions for the original problem are specified.
The numerical solution of the problem is then to find the
magnitude of the set of N line forces and N ring moments
imposed along the embedded contour such that the boundary
conditions at the N boundary points are satisfied. With
this method, problems with clamped and simply supported
boundaries can be easily solved. For a free edge, however,
due to the logarithmic nature of the Green's function and
the fact that fourth order derivatives must be taken for
the fictitious ring moments in the boundary condition
equations, there are second order singularity difficulties
during the numerical integration along the embedded contour.

In this thesis, three major modifications are intro-
duced. These are (1)the set of fictitious moments are re-
placed by an additional set of fictitious forces, and the
entire set of fictitious forces is located outside of the
embedded contour, (2)the numerical integration is replaced
by a simple summing process, and (3)the Green's function
for a clamped circular plate is replaced by the Green's
function of an infinite plate. With these modifications,
significant improvements in solution accuracy and compu-
ting efficiency have been achieved. The second order sin-
gularity difficu1ties associated with free edges are avoided,
and due to the simplicity of the new method, the computing
costs are reduced by about sixty percent. Since the Green's
functions for orthotropic and anisotropic infinite plates
are also available, the new method is readily extended to
orthotropic and anisotropic thin plate bending problems.

ACKNOWLEDGEMENTS

The author wishes to take this opportunity to express
his gratitude and deep appreciation to his major advisor
Professor Nicholas J. Altiero for his able guidance and
sincere encouragement during the course of this thesis
research, and for his spending many hours with the author
in the preparation of this thesis. Special thanks are due
to other members of the graduate guidance committee,
Professor William A. Bradley, Professor George E. Mase,
Professor Larry J. Segerlind, Professor David H. Y. Yen,
and Professor David L. Sikarskie of the University of
Michigan for their helpful advice, suggestions, and dis-
cussions. To the Department of Metallurgy, Mechanics, and
Materials Science of Michigan State University, the author
is grateful for its support on computing costs.

The author also wishes to thank his parents for their
continuous encouragement during his studies and research
for the past many years. Finally, and most importantly,
the author wishes to acknowledge his wife Wei Sheng.
Without her genuine understanding, encouragement, and moral
support in the past several years, this work could never
have been done.

ii

LIST OF TABLES . .
LIST OF FIGURES. .
LIST OF APPENDICES.

INTRODUCTION.

CHAPTER I

CHAPTER II

CHAPTER III

CHAPTER IV

APPENDICES

BIBLIOGRAPHY.

TABLE OF CONTENTS

PRELIMINARIES ON PLATE THEORY. . .

1.1
1.2

1.3

GOVERNING EQUATIONS . . . .
BOUNDARY CONDITIONS . . . .

THE GREEN'S FUNCTION METHOD .

INTEGRAL EQUATION APPROACH. . . .

11.1
11.2
A NEW
111.1

111.2

THE BOUNDARY INTEGRAL METHOD .
THE AUXILIARY BOUNDARY METHOD.
METHOD . . . . . . . .
ISOTROPIC PLATE PROBLEMS . .
ANISOTROPIC PLATE PROBLEMS. .
III.2.1 ORTHOTROPIC PROBLEMS .

III.2.2 ANISOTROPIC PROBLEMS .

CLOSURE . . . . . . . . . .

iii

Page

iv

12

30

Q6

.127

Table

10

11

LIST OF TABLES

Comparison of numerical and exact results
for a square plate, edge one free, edge
two and four simply supported, and edge
three clamped . . . . . . . . . .

Comparison of numerical and exact results
for a square plate, edges one and three
free, edges two and four simply supported.

Comparison of computing costs. . .

Comparison of results of a simply supported
triangular isotropic plate. . . . .

Comparison of results of a clamped square
plate, double-looped fictitious forces at
am and 6m away from the plate boundary. .

Comparison of results of a clamped square
plate, double-looped fictitious forces at
1m and 3m away from the plate boundary. .

Comparison of results of a clamped square
plate, double-looped fictitious forces at
0.5m and 2.5m away from the plate boundary

Comparison of results of a clamped square
plate, single-looped fictitious forces at
um away from the plate boundary . . .

Comparison of results of a simply supported
orthotrOpic square plate, Ex=2.068X10 MPa,
Ey=Ex/15, vx=0.3, h=0.01m, p=0.1. . . .

Comparison of results of a simply supported
orthotrOpic square plate, E =2.06BX10 MPa,
Ey=EX/1S, vx=0.3, h=0.01m, §=1.o. . . .

Comparison of results of a simply sup orted
orthotrOpic square plate, E =2.068X10 _MPa,

X
Ey=Ex/15, vx=0.3, h=0.01m, p=10.0 . . .

iv

Page

28

29

36

42

U3

US

56

57

Figure

LIST OF FIGURES

Problem of interest. . . . . . . . .
Problem for which analytic solution is known
Fictitious problem . . . . . . . . .

A square plate with an auxiliary integration
contour O O O I O O O O O C O O O

A square plate with two sets of fictitious
forces . . . . . . . . . . . . .

A simply supported equilateral triangular
isotrOpic plate . . . . . . . . . .

Comparison of results of a triangular plate.
A simply supported orthotropic plate . . .

Flow chart for the computer programs . . .

Page

16

20

20

26

32

35

37

55

66

LIST OF APPENDICES

Appendix ' Page

A Derivatives of the Green's function for
a clamped circular plate . . . . . . . 67

B Computer program for the boundary inte-
gral method . . . . . . . . . . . 80

C Computer program for the point-force
method . . . . . . . . . . . . . 90

D Derivatives of the Green's function of
an infinite orthotropic plate . . . . . 95

E Computer program for an orthotrOpic problem. 100

F Derivatives of the Green's function of
an infinite anisotropic plate . . . . . 107

G Computer program for an anisotropic problem. 112

H Computer program for the verification of
equations used for anisotropic problems . . 119

vi

INTRODUCTION

The thin plate bending problem is one of the most
common problems in structural engineering. Design engineers
encounter it daily. Plate theories and methods of solution
can be traced back to the early eighteenth century. Famous
names like Euler, Bernoulli, Lagrange, Navier, and Kirch-
hoff were all involved in the development of plate theories.
In this century, Nadai, Love, Huber, Timoshenko, Lekhnitskii,
von Karman, and Reissner, to name a few, are well-known for
their work related to plate problems.

Mathematically, the thin plate bending problem is a
typical boundary value problem. Since solving the problem
reduces to finding a solution satisfying the governing fourth
order partial differential equation and all the boundary
condition equations, exact solutions are available only for
special cases. In addition to many common methods for a wide
range of problems shown in [1], the method of complex va-
riables has been successfully applied [2,3,u,5,6] solving
many additional problems. However, for a generalized pro-
blem, numerical techniques such as finite difference and
finite element methods must be employed, [7,8].

In this dissertation, a different numerical method is
introduced. Developed from the concept of an indirect boun-
dary integration equation method [9], the new numerical
method employs a known Green's function, the scheme of em-
bedding the real plate in a fictitious plate for which the
Green's function is known, and the imposition of fictitious
forces so that all the boundary conditions are satisfied.
This method is very effective because of the simplicity of

2
its formulation. It can solve constant thickness plate
problems with arbitrary plan form, arbitrary loading and
boundary conditions, and anisotrOpic material properties.
Since the fictitious forces are located far away from the
plate boundary, this method gives more accurate results
near the boundary than does the boundary integral method.

The procedure of this method can be summarized in
three steps. The first is to find a Green's function of
a certain type of plate problem. Many of them are avail-
able. Take isotropic problems for example. The Green's
functions for a clamped circular plate [1,3,10], and for
an infinite plate [1] are two possible choices. The second
step is to embed the real plate in the aforementioned fic-
titious plate for which the Green's function is known, and
prescribing a set of N boundary points. Since there are
two boundary condition equations associated with each boun-
dary point, a set of 2N fictitious forces are placed around
the plate boundary. Solution of the problem, therefore,
involves the determination of this set of 2N fictitious
forces such that the 2N boundary condition equations are
satisfied. The third step is to superimpose these 2N fic-
titious forces onto the actual loadings of the plate and
compute the deflections and bending moments inside the
plate.

There are four chapters in this dissertation. The
governing fourth order partial differential equations for
isotropic, orthotrOpic, and anisotropic plate problems
and their associated boundary condition equations are
reviewed in Chapter I. Chapter II introduces the indirect
boundary integral method originally derived by Altiero
and Sikarskie [9]. Their method is modified by moving the
integration contour to the outside of the plate boundary.
In so doing, the second order singularity difficulties for
the free edge boundary conditions, encountered in their
work, are avoided. In the meantime, a significant improve-

3

ment in the solution accuracy is noticed. Chapter III
illustrates the new point-force method for the solution

of a generalized thin plate bending problem. The point-
force method contains three major alterations over the
boundary integral method, though the basic concept remains.
The three changes are (1)the set of fictitious moments are
replaced by a second set of fictitious forces, (2)the inte-
gration is replaced by an algebraic summing process, and
(3)the Green's function for a clamped circular plate is
replaced by the Green's function of an infinite plate.
These changes have made the original method more effective.
A saving of sixty percent for computing costs is realized.
More importantly, it is due to the successful use of the
Green's function of an isotrOpic infinite plate and, the
Green's functions for orthotropic and anisotropic infinite
plates are readily known [11,12,13], the new method can be
extended to solve general orthotropic and anisotropic plate
problems as well. Chapter IV presents several comments
regarding to this new point-force method.

In order to verify the method, several test cases have
been solved. The results are tabulated and graphed, and the
computer programs are included in the Appendices. Though
these computer programs are specifically designed for the
example problems, they can be easily revised to accommodate

general problems.

CHAPTER I

PRELIMINARIES ON PLATE THEORY

I.1 GOVERNING EQUATIONS

Following the assumptions involved in the well-known
Kirchhoff—Love small deflection plate theory, all of the
stress components within the plate can be expressed in
terms of the vertical deflections, w(x,y). Therefore, for
static equilibrium, the governing differential equation
can be derived in terms of the deflection function and the
two independent coordinate variables x and y. For the three
material types, namely isotropic, orthotropic, and aniso-
trOpic, the derivation of the governing differential equa-
tions can be found in most texts on plate theory [1,7,1u].

Assuming constant thickness, for isotropic plates, the

governing differential equation is

3“w(x,y) + 2 3“w(x,y) + a“w(x,y) = q(x,y) (1)
3x“ 8x23y2 By D

where w(x,y) is the vertical deflection of the plate after
bending, q(x,y) is the load in the vertical direction, and
D is the flexural rigidity of the plate defined by

3

D = Eh
12(1-v2)
where E is the Young's modulus, v is the Poisson's ratio,

and h is the plate thickness.

5
For an orthotropic plate which has its geometric coor-
dinates aligned with the principal material directions, the

governing equation is

 

 

kw” 2H—1(x—'Y—)-+D——3"—(l‘—’-X)—-q(x,y), (2)
3x 3x28y Y 3y“
where,
E ~h3 E -h3
= X . = x . = E
Dx 12(1-vxvy)’ Dy 12(1-vxvy)’ and H vay + 12

The subscripts x and y indicate the principal directions of
the material constants, and G is the modulus of rigidity.
Finally, the governing equation for an anisotropic plate,
with one plane of material symmetry parallel to the middle

surface of the plate is

D __W(_Xr_.‘L)_ + DD _w_<><_lx)_+ 2(D12+2D6)M

ax ax3ay axzay
to to

., 140263—1341)— + Dnmlxl = DD“), (3)
axay3 3y“

where Dij are associated with the material constants, and

are determined as follows:

 

 

 

 

 

 

D = ha (ayaGG’afiz) . D = ha (alLa 66'3152 ) ,
11 12 det. ' 22 12 det. '

D = ha (311912-3112) . D = h3 (aleazi-anass) .
56 12 det. ' 12 12 det. '

D = h3 (31232;:322316), D = ha (aizais‘aiiaze) ,
'5 12 det. ' 26 12 det. '

 

 

 

 

 

 

 

 

 

 

 

and,
a11 a12 a16
det. = a12 a22 a26 ,
a16 a26 a66
aij is the material constant matrix, i.e.,
r r j r ‘
6x 1 a11 a12 a16 0x
16y > =1 a12 a22 a26 i 10y 1
nyJ La16 a26 a66J LTny
The coefficient, a , are:
v n n
a =—1_p a =-—-§;a =M=M
11 12 16 G E ’
x x xy x
1 1 n x nx
a = ___ ; a = ___ ; a = _XL_X._ __XLX
22 E 66 G 26 G E ’
Y XY XY Y
where n and n are called the coefficients

xy,x’ nxy,y' x,xy' ny,xy
of mutual influence of the first kind and the second kind, res-
pectively, [15,16]. Physically, it is clear that they represent
mutual influences between shear strains and normal stresses and

between normal strains and shear stresses.

1.2 BOUNDARY CONDITIONS

Only the three major types of boundary conditions, namely,
(1)clamped, (2)simply supported, and (3)free, are considered in
this dissertation. Many others such as elastically-supported
edges can also be handled with minor changes. The equations

7

associated with these three major types of boundary conditions
are

(1)Rigidly clamped edge (BC):

w(x,y) = 0 ; gfléfiLXL-- O . (Ha)

(2)Simply supported edge (BS):

 

w(x,y) = 0 ; Mn = O (ab)
(3)Free edge (Bf):
ant
.M = O ; N 4- n = 0 (HC)
n n as ,

where 5% is the derivative along the contour arc, Mn is the
unit edge bending moment, Nn is the unit edge shear force,
th is the unit edge twisting moment, and the subscript n
means acting along the normal direction of the edge. These
values can be written in terms of their components in the

x and y directions as

M = M -n 2 + M -n 2 + 2H -n on (5a)
n x x y y xy x y
= — O O O 2 — 2
th (My Mx) nX ny + ny (nx ny ) (5b)
N = N 'n + N -n (5c)
n x x y y

where nx and ny are direction cosines of an outward normal to
the contour arc. For an anisotropic problem, the bending moment,
twisting moment, and shear force in the x and y directions can

be written in terms of the deflection function w(x,y) as

 

 

 

 

 

82w 32w 32w
M = - (D -—— + D ——— + 20 ) (6a)
x 113x2 123y2 165x5y
_ 32 82w 32w
My - (D123x2 + D223y2 + 20265??? ) (6b)
32 32w 82w
H = - (D + D + 2D ———— ) (6c)
xy 163x2 268y2 663x3y
3 a 3
Nx = - [D113e5 + 3D16 a w + (012+ 2D66) a w
3x3 szay axay2
3
4. D263 W ] (6d)
8y3
33w 33w 33w
N = - [D ——— + (D + 2D ) + 3D
y 163x3 12 66 axzay 263x3y2
3
+ 022—3 w ] (69)
Bya

Substituting Eqs. (5) and (6) into Eq. (a), the boundary
condition equations for anisotropic plates can be written
explicitly as

w(x,y) = 0 on Bc + Bs (7a)

fiéralm + 91.92am = D on D (7D)
x X By y c

32W(XIY)

 

O 2 O O O 2
2 [D11 nx + 2D16 nx ny + D12 ny]

3x

32W(X,X)

+ 3x3y

 

[2D1 -n2 + an °n -n + 2D

. 2
6 x 66 x y 26 my] +

2
2 X Y Y 5

By 26 x 22 f
(7C)
3
3.119111). . 2 .s- ..2
3x3 [D11 nx(1+ny )+2D16 ny D12 nx n ]

3
+ 3 W(XLY) [4D °n +D °n (1+n 2)+tlD -n 3-D -n 2'1':

 

3x23y 16 x 12 y x 66 y 11 x y

3
- ..2é_v_1l>;:_y_>. . . 2 .a
2D26 nX ny ]+ axayz [L1D26 ny+D12 nx(1+ny )+uD66 nX

a
.. . . 2- . 2. M . 2
D22 nx ny 2D16 nx ny] + 3y3 [D22 ny(1+nx )
O 3- O 2. =
+2D26 nX D12 nx ny] O on Bf (7d)

BC is the clamped portion of the boundary B, B8 is the simply

supported portion of B, and Bf is the free portion of B. Clearly,
B=B¢+BS+Bf.
D66 being replaced by Dx’ Dy' and Dk' respectively; D12 by vny

For orthotropic plate problems, with D11, D22, and

or vyDX; and D16=D26=0; Eqs.(6) and (7) can be reduced to
82w(x y) 32W(X X)
Mx = -D [ ' + v ' ] (8a)
X 8X2 y 8y2
82w(x y) 32w(x y)
M = -D [ ’ + v ' ] (8b)
y y Syz X 8x2
32w(x )

xy k 3x8y

10

 

 

 

 

2 2
Nx ___ __ 5%[Dx a w(x,y) + H3 w(x.1)] (35)
3x2 3Y2
2 2 '
N = _ §3_[H a w(x,y) + D 8 w(xly)] (8e)
Y Y 3x2 y 3Y2
where,
_ G-h3 _
Dk _ 12 , and H — vay + 2Dk
w(x,y) = O on BC '1' BS (93)
3w(x,x>.n + BELEIXL.D = 0 on B (9b)
8x X By Y C

2 2
a—EifiLXL(D -n 2 + v D -n 2) + 3 W(X'~X)(D -n 2 + v D °n 2)
3x2 X X X Y Y 3Y2 Y Y X x

2

 

 

 

. . = + B 9
+ Bxay k nx ny) 0 on BS f ( c)
83w(x,y) . 3 . . 2 -
3X3 [Dx nx + DX nx ny (2 vy)]
+ 83w(x,y)[v D -n (1+n 21+un -n 3-D -n2«n 1
axzay x y y x k y x x y

 

3
+ 3 “(x'¥l[v D 'n (1+n 2)+un
x x y

3 2
on -D on on ]
axay2 X Y X y

k

33w(xIY)
aya

+ . 3+ D on 2on 2-v = 0 on B (9d)
[Dy ny y x y( X”

11

For isotropic plate problems, these two sets of equations
_D(1-v)

can be further reduced by having Dk——_2_—— , vx=vy=v, and
D =D =H=D:
X Y
2 2
M = _ D[ 8 W(XIY) +\) a w(xri)] (10a)
X 8x2 3y2
2 2
M = - D[ M+VM] (10b)
y 3y2 8x2
2
H = - D(1-V)§_El§LXL (10C)
xy 8x8y
Nx = - ”3—3)? [ V2w(x,y)] (10d)
_ 3 2
Ny — - D5; [V w(x,y)] (10c)
where,
VZW(X,Y) = BZW(Xpi) + 82w(xlz)
8x2 3y2
and;
w(x,y) = O on BC + Bs (11a)
EELELXL°n + agiiLXl’n = O on B (11b)
3x X By y c

 

12

3 3
a w(x,y) n [1m 2(1_D)] + 2.13211 D [1+n 2(1-v)]
8x3 X y 3y3 y x

a
+ g—ELELXL n [(2v-1)n 2 + (2-v)-n 2]
3x23); y x Y

3
+ M n [(Zv-1)n 2 + (Z-V)°n 2]: 0 on B (11d)
axay2 X y X

1.3 THE GREEN'S FUNCTION METHOD

Solving a plate problem is, mathematically, to find the
deflection function w(x,y) such that the governing differen-
tial equation as well as the prescribed boundary conditions
are all satisfied. For the special problem of a concentrated
force applied at an arbitrary location, the solution is
called the "Green's function" of the problem, and is often
written as G(x,y;€,n). That is, with the prescribed boundary
conditions, a Green's function will provide the deflection
at any point (x,y) when there is a concentrated force located
at some point (€,n). The deflection function for a distri-
buted load q(x,y) over a region R inside the plate can be

written, using superposition, as:
w(x,y) = ffRG(X,y;€,n)°q(€,n)d€dr1 (12)

This is called the Green's function method or the influence
function method, [1]. The Green's function can be either in
closed form or in infinite series form, and varies with the
problem. For isotropic problems, there are many Green's
functions available. Some examples are given here.

The Green's function for a clamped circular plate is
[1,3,10],

13

G(x,Y:E,n) = -—1———wua2-x2-y2)(aZ-aZ-nz)

16nDa2
(13)

+[a2(x-€)2+a2(y-n)2]ln .a2(x-€)2+a2(y-n)2
(az-xz-yz)(az-Ez-n2)+a2(x-Efiga2(y-nV

where a is the plate radius.

There are two well known Green's functions for a simply
supported rectangular plate, and both of them are of the
infinite series type. The one with double trigonometric series
is called the Navier's solution, while the single trigonometric
series function is named after Levy, [1,7]

 

 

 

 

 

singleE sinflgxisinggé sing;11
G(X:Y:E:n)= 11 Emil 2 2 (1“)
Dn ab ( m + n )2
3.2 32
m=1,2,3,4,...w; n=1,2,3,4,...m
and,
2 B y B y 8 n B n
_ a _ m m __ m m
G(x,y,g,n)— D."3 1% (1+BmCOtth TCOth—b TCOth—b )
B n B y
sinh-g— sinh—g— sinfllé sing-g—é
x a (15)
masinhB
m

where, Bm=flg2, and m=1,2,3,ll,...co ; if y<n, replace n by

b-n ; if yin, replace y by b-y.

There is also a Green's function for an infinite plate, DJ

(x-€)2+(y-n)2

a2

G(X.y:€,n)=13%§-Hx-€)2+(Y-n)2]£n (16)

where a is an arbitrary reference radius.

14
There are also Green's functions for orthotropic and
anisotropic plates [11,12,13,1fl], though not as many. They
will be discussed later.
The major limitation to the Green's function method is
that one must know the Green's function of a specific plate

shape and boundary conditions before Eq. (12) can be employed.

CHAPTER II

INTEGRAL EQUATIONS APPROACHES

II.1 THE BOUNDARY INTEGRAL METHOD

Several investigators have applied boundary-integral
techniques to thin plate problems. Jaswon, et al,[17,18,19],
have developed a "direct" approach. Altiero and Sikarskie
[9] have developed an "indirect" version of the boundary
integral equation. The indirect approach is outlined here.
The problem of interest is a thin plate of arbitrary plan
form and arbitrary boundary conditions, subjected to an
arbitrarily-distributed load, Figure 1. However, due to
numerical difficulties in the evaluation of second order
singularities for a free edge, plates with free edges
were excluded in [9].

A plate problem is solved similar to an elasticity
problem [20,21]. The plate of interest is embedded in a
fictitious plate of the same material for which the Green's
function is known. In order to satisfy the boundary con-
ditions of the original problem, a set of fictitious line
forces and a set of fictitious ring moments are introduced
along the boundary of the embedded plate. The problem is
therefore solved if the magnitude of these fictitious
forces and moments can be determined such that the original
boundary conditions are satisfied.

Knowing that the influence function for a point moment
is simply the derivative of the Green's function for a
point force with respect to the direction at which the
moment is oriented, the influence function for the moment

15

16

q(x,y)

 

Figure 1. Problem of Interest

17

can be easily derived:

H(X,Y:€,n) = - BG‘X'Y’E'“ = - 39m - 39m (17)

35 3X X By y

where G(x,y;€,n) is the known Green's function of a point
force located at (€,n); H(x,y;€,n) is the derived influence
function of a point moment located at the same place (€,n)
which is oriented along the direction 5, and nX and n are
the direction cosines of the outward normal vector n with
respect to the x and y coordinates, respectively. The
negative sign is for convention only.

Let P* represent a set of fictitious forces, Mn* re-
present a set of fictitious moments, and q(x,y) be the
given distributed lateral load. By superposition, the de-
flection at any point (x,y) can therefore be computed from
the following deflection equation.

w(x,y)=ffq(€,n)G(x,y;£,n)d£dn+¢P*(€,n)G(x,y;€,n)ds(€,n)
R 3*
(18)

 

 

* _ 8G(x,y;€,n)_ 86(X,y:€,n) ,
:fiMn(£,n)[ nx(€.n) a: ny(€,n) an lds(c.n)

where R is the region over which the distributed load q(x,y)
is prescribed, B is the boundary of the embedded plate, and
nx(§,n) and ny(£,n) are the direction cosines of the unit
outward normal to the plate boundary at the loading point
(€,n). Values of P* and M; are to be determined from the
boundary condition equations. The boundary condition equa-
tions are shown in Eq. (4), and they can be written in terms
of the x and y coordinates as shown in Eq. (11).

The Green's function chosen in [9] is the Green's
function for a clamped circular plate, Eq. (13). See
Figure 2. Substituting this Green's function into the boun-

dary condition equations, one therefore can determine the

18
set of fictitious forces and moments from a set of two
boundary conditions. However, since both plan form and
boundary conditions are arbitrary, a numerical method must
be employed. By modeling the plate with an N-sided polygon,
and assuming the fictitious force and moment remain constant
along each of the straight edges of this polygon, one can de-
termine a set of N fictitious forces and N fictitious moments
such that the boundary conditions at the N mid-points of the
N-sided polygon are satisfied. Take the zero deflection
boundary condition, for example, which occurs on both

clamped and simply supported edges. One will have

 

 

n *
k2-?1Pk(Qk)[ékG(x,y;E,n)ds(€,n)]
’ (19)
n * 3G(x.y:€.n)
+ Z M (Q )[-n (Q )f dS(€,n)
k=1 “k k x k 5k 3:
_ 3G(x,y:€,n) _ _ .
ny(Qk)é an ds(€,nn - £{q(€,n)G(x,y.€,n)d€dn

where Qk represents the centerpoint of the kth side of the

polygon B, and S is a coordinate along the kth side, i.e.,

k
OfSkSASk. Now, if the boundary conditions on B are forced

to be satisfied at each of N locations (xB,yB), Eqs. (19)

lead to a system of 2N linear algebraic equations for 2N
*

nk'
can be expressed in matrix form by

* .
unknowns Pk and M k=1,2,3,...N. This system of equations

RM -——- = RL (20)

where RM is a 2NXZN matrix, the elements of which are the
line integrals of Eq. (19), and RL is a 2NX1 column matrix
consisting of the area integrals of Eq. (19). Once Eq. (20)
is solved for the unknown fictitious forces and moments,

* .
P* and Mn' the displacements at any internal p01nt can be

19
found using

w(x,y)=ffq(£,n)G(x,y;€.n)d€dn+g P;(Qk)[fG(x,y;€.n)ds(€,n)l
k=1 5
k

4.
k

 

11 M5

Mn;(ok>{-nx(Qk)IBG‘xL§g5'”)ds(a.n)
1 S (21)

 

_ BG(x y;€,n)
ny(Qk)ka LOT] —dS(€pT1)]

This method works satisfactorily if free boundary con-
ditions are not included. For a free edge, the two boundary
condition equations are as shown in Eq.(11). It can be seen
that, associated with the fictitious moments, there are
eight terms involving fourth order derivatives of the Green's
function, Eq.(13). When integrating along the kth side of
the polygon B, there will be difficulties in the evaluation
of the second order singularities. That is, there will be
terms such as

 

f 1 dSk(€rn)
(x-€)2+(y-n)2

Sk

which pose difficulties when €+x and n+y. In addition, like
other boundary integral methods, the errors in the region
near the boundary can be substantial. Due to these two di-
ficiencies, an "auxiliary boundary" method has been deve-

loped. This method is presented in the following section.

11.2 THE AUXILIARY BOUNDARY METHOD

The only difference between the current method and
the boundary integral method discussed in the previous
section is that an integration path, 8*, is chosen dif-

ferent from the plate boundary, B; see Figure 3. In so

20
/lLL////

LO _ a

(£1 7])

Figure 2. Problem for which Analytic Solution is Known.

///L/////

 

 

 

 

Z

Figure 3. Fictitious Problem

21
doing, the singularities which arise in the integrand during
numerical integration for the free boundary condition are
avoided. Since integration is now carried out along the fic-
titious integration path, B*, there is no need to model the
plate with an N-sided polygon. Instead, there are N boundary
points prescribed on B where boundary conditions are to be
satisfied. This, combined with the fact that the fictitious
forces and moments are now located away from the boundary,
provide significant improvements in the solution accuracy.
For example, at the center point of a clamped rectangular
plate, the errors for displacement and bending moments are
reduced from 1.8 and 8.0 percent, shown in [9], to 0.0“ and
1.6 percent, respectively.

Following the aforementioned procedure but integrating
along B* instead of B, plate problems with mixed boundary
condition of all the three types can now be solved. Writing
the boundary condition equations more explicitly, Eq.(11)

become

P P*(€,0)G(x,y;§,0)d5(5,n)

 

 

 

 

D.
+:*Mg(g.n>[-nx<g,n)36(X'%g§L”’-ny(a,n)BG‘X'§;5'”?ps(a,n>
= ‘fi{q(€:n)G(x,y;£,l)d€dn (22a)
if (x,y) is on BC + BS;
2 P*<a.n)[nx(x.y)BG‘X'§;§'”’+ny<x,y)36(x'§;§'“’1ds(g,n)

8*

2 0
+8 Mfi(a.n)[-nx<g.n)nx(x.y)3 G§:S§.a.n>
B'k

32G(X,y;§,n) )BZG(X11;gIT])

 

 

 

-nx(€:n)ny(X:Y) ayag -ny(g'n)nx(x'y 3X30
-n (g )n (x )BZG'X'Y’g’”)]ds<€ n)
y In y :Y ayan ' (22b)

22

8G(x,yg§,n)

 

 

=-ffq(f;,n) [nx(x,y) ax’ +ny(x.y) aG‘xg§‘€'”’ ldadn
if (x,y) on BC;
826(x.y;€.n) 326(x.y:€.n)

 

 

é P*(€,n)[n;(x,y)

3* 3x2 +2nx(x,y)ny(x,y)

Bxay

2 0
+1.12 (X,Y)a G(x’y’€’n)]d5(€,n)
Y 3Y2

3 O
+¢ M;(E;,n)[-nx(€,n)n;(x,y)a G(X'Y'5'”)
3* 3x236

a3G(x,y:€,n)
-2nx(€.n)nx(xvylny(x'y) axayai

336(X1Y1510)
szan

33G(x.y:€.n)

-n (€,n)n2(x,y)
X y 3y23£

2
-ny(€,n)nx(x,y)

33G(XIY7€IU)
Bxayan

-2ny(€,0)nx(xpy)ny(xry)
336(XIY7SIU)]dS(€’n)

_ 2
ny(€.n)ny(X.y) Byzan

 

 

2 , 2 ,
a G(X.y.§.n)+2nx(x'y)ny(x'y)3 G(xLy,€,n)

 

__ 2
- [RIQ(§171) [nX(XIY) 3x2 3X3)!
2 .
+n;(X,Y)3 G<xLy’€Ln)]d£dn (22C)

8y2

if (x,y) is on BS:

23

826(X1Yiim)

g*P*(€In){[n;(XIY)+Vn;(XIY)] 2

8x

 

- 32G(le7$'n)
+[2(1 v1nx<x.y>ny(XvY’] axay

2 .
18 G(XIYI€IH) }dS(E'n)

+ n2 x, + n2 x,
[ y( y) v X( y) 3y?

3 0
+2 M;(€:n){-nx(€,n)[n;(x,y)+vn2(x,y)]a G<Xry:§:n)
3* y 3x285

33G(X.y:€rn)

-nx(g.n)[2(1-v1nx(x1y)ny<X'Y’] axayaa

le;gln)

-nx(€,n)[n2(x,y)+vn;(x,y)]836(

3 O
-n (€,n)[n§(x,y)+vn2(x,y)]3 C(X'Y'gLfl)
y y axzan

33G(le;§ln)

-ny(g,n)[2(1-v)nx(x,y)ny(x'y)] axayan

 

3 .
-ny(€rn) [n;(X.Y)+vn;(X:Y)] 3 G(x,y, g'n) }d3(51n)
Byzan

2 .
= —ffq(a.n){[n;(x,y)+vn2(x,y)]3 G‘X'Y'éefl)
R y 3X2

(xiy;§inl

+[2(1-v)n (x y)n (x y)]32G
x ' y ' axay

 

2 . .
+[n2(x.y)+vn2(x,y)]3 G(x’y'§'n)}dgdn (22d)
y X 3Y2

if (x,y) is on Bf;

2U

336(le;grn)

¢ P*(£.n){[n;(x,y)+(2-v)nx(X.y)n;(x.y)] 3

8* 3X

 

3 O
+[(2v-1)n2(x,y)n (x,y)+(2—V)n3(x’y)]a G<X.y.€.n)
X y y axzay

]83G(X:Y;€,n)

+[(2'V)n3 (X:Y)+(2v-1)n (XIY)n2 (XIY)
X X Y axey2

3 .
+[(2-v)n2(x,y)n (x,y)+n3<x,y>]a G(x'y'€'n)}ds(€,n)
X y y ay3
3“G(X.y:€.w)

+ M* p - I 3 I 2- I 2 I
g* n(€ n){ nx(€ n)[nx(x y)+( v)nx(x y)ny(x y)] axaay

3~G(XIX7 €177)
8x28y3€

 

-nx(g,n)[(2v-1)n;<x,y)ny(x.y)+(Z-v)n;(x.y)1

3“G(x,y:€,n)

-n (€.n)[(Z-v)n3(X.y)+(2v-1)n (x,y)n2(x.y)]
X x X y axayzag

u C
’nx(5'”)[(2‘V)ni(X.y)n (x,y)+n3(x,y)]a G(XLY.£,n)
y y ayaaa
- 3 ’ 2 3“G(XIX:£,n)
ny(€pn)[nx(X,Y)+(2 v)nx(x,y)ny(x,y)] axaan
8~G(Xfl;£,n)

-n (€.n)[(2v-1)n2(X.y)n (x,y)+(2-v)n3(x,y)]
y X y y szayan

'y)]8“G(x,y;§,n)

-n (§,n)[(2-v)n3(x,y)+(2v-1)n (x,y)n2(x
Y X X y Bxayzan

u 0
]8 G(leI§ln) }dS(g,T‘|)

- - 2 3
ny(€.n)[(2 v)nx(x,y)ny(x,y)+ny(x,y) ay3an

25

aza<x.y:€.n)
3

 

- _ 3 _ 2
_ ffq(g,n){[nx(x,y)+(2 v)nx(x,y)ny(x,y)]

R 3x

]336(xLy;€gn)

- 2 _ 3
+[(2v 1)nX(X.y)ny(X.y)+(2 v)ny(x,y) axzay

)] 336(X0y75rn)

+[(2-v)n;(x,y)+(2v-1)nx(x,y)n;(X.y 2
Bxay

3 O
+[(2-v)n;(x,y)nv(x,y)+n;(x,y)]a G(X'y’g'n)}d€dr‘. (22e)

if (x,y) is on Bf. The derivatives of the Green's function
are listed in Appendix A.

The solution is then obtained using the same nume-
rical procedure shown in the previous section. However,
the plate is no longer modeled with an N-sided polygon.
Instead, N points are assigned along the boundary, and
two of the above equations are satisfied at each of these
N points. This is done by adjusting the magnitudes of the
unknown fictitious forces P* and moments Mn* at the N
meshes along the fictitious integration path 3*. Thus,
solving a set of ZNXZN linear algebraic equations for
the set of fictitious forces and moments on B*, and sub-
stituting into the deflection equation Eq.(21), the pro-
blem of Figure 1 is solved.

A plate problem of arbitrary plan form, arbitrary
lateral load, and arbitrary boundary conditions has now
been solved. The complete computer program is shown in
Appendix B.

Two example problems are illustrated here. In both
cases uniformly-loaded square plates are considered. The
dimensions of the plate and the fictitious contour 3*
are shown in Figure a. The plate is embedded in a fic-
titious circular plate with a radius of 80m. Other

26

 

 

 

 

 

 

 

 

 

Edge 2

 

Edge 3

 

* 0 Field Point
Y
0 Boundary Point

A Loading Point

Figure u. A Square Plate with an Auxiliary
Integration Contour-

27
constants are E=2.068HX105MPa, v=0.3, h=0.01m, and q=1N/m?.
In the first example, edge one is free, edges two and four
are simply supported, and edge three is clamped. In the
second example, edges one and three are free while the
other two are simply supported. The boundary conditions
are satisfied at forty points spaced at a distance 1m
from each other and .5m from the corner; see Figure u.
The contour B* is located am away from the plate boundary
and is divided into forty intervals. Integration over an
interval B* is done by simply multiplying the value of
the integrand at the center of an interval by the interval
length. Integration over R is accomplished by subdividing
R into 100 equal divisions and multiplying the value of
the integrand at the centerpoint of each division by the
area of that division. This area subdivision is also shown
in Figure u.

Results of displacements and bending moments at each
of the fifteen field points shown in Figure u are presented
in Tables 1 and 2. When compared with the exact solutions
from [1], the average errors are less than three percent
for both deflections and bending moments. It may be noted
that locations one, two, and three are on edge one of the
boundary and between two adjacent boundary points where
the boundary conditions are forced to be satisfied. There-
fore, accuracy of results at these locations are expected
to be the worst. Yet, the errors are less than four per-
cent. Thus, the method using an auxiliary integration path
is a great improvement over the previous indirect boundary
integral method, especially when results near the boundary
are needed.

mm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

oP~.~ mmo.ou smm.ms mas.m am=.Fu mPa.Fu mam.. am~.o =m~.o m.
o.s.~ m.m.=u ~om.=- ama.a com..- awe..- mom.P =P~.o P.~.o 3.
moo.m amm.P- -~.P- hma.o. >m~.o- s:~.on mmm.. mmo.o nmo.o m.
ooa.m.-. has.o m.m.o ooo.o omm.~ oom.~ mom._ mom.P mam.F N.
.m~.P.- maa.o mmm.c Pomp.on oom.~ =em.~ osm.. Pa~.F Ps~.F P.
m>~.o.- mmm.o mam.o mmmm.ou mmm.P mmm.P Nua.. mpm.o =om.o o.
For... s=~.~ =mn.~ mmo.. apo.m m.o.m oma.. =~o.m oom.~ a
Pom..- smm.~ smm.~ som.o ooa.3 ~.a.= mam.P m~=.~ m~=.~ m
=mo.mu mmo.P moo.. mop.o mam.~ oom.~ sma.. smm.o mam.o a
ooh.o Fps.~ ~mo.~ ams.. om~.m omo.m o==.~ sam.s mm=.a o
omm.o mom.~ am~.~ mao.r aao.h mma.o =mm.~ o.s.m m_o.m m
Pom.. oso.. omo.. mom.P mm_.m map.m m:m.~ o=:.. mo:.. 3
ooo.o ooo.o ooo.o osm.~ ama.a oms.a mm..m moo.m .mm.m m
ooo.o ooo.o ooo.o Pao.m Pwm.m mm~.m mso.m mom.a oms.= m
ooo.o ooo.o ooo.o .>~.a mam.m smm.m .m~.m spa.P 0mm.F .
E\Euz E\E|z E\Enz E\Elz .EE .EE
uouumu A.s=:o>z imauuosz HOHHma i.escoxz Amsuuoxz “chum“ ..e=coz Amounts .uooa

 

 

 

 

 

 

 

 

 

 

A: .mfimv meEMHU mouse wmpm can .Umuuommsm xameflm usom new 039 wmmpm .wmuh mco mmom
.wumHm mumsvm m HON muasmmm uomxm cam HmoflumEDZ mo COmfiummEoo

P manna

 

mm

 

 

 

 

 

 

 

 

 

 

 

mmm.ml Nam.~ =o>.~ mum.w oba.mp o=N.mF mmm.m oso.h mmm.o m
m~:.ml m=N.N mom.~ hom.P mh=.op mmN.oP omm.m mon.m Fhm.m m
me.ml mmm.o omo.P moo.P oom.= mm:.= m=N.N ~o~.m =mP.N n
bao.o hmp.m mme.m wmm.P mao.mp Poa.mp amm.~ mmm.h 530.5 m
mpm.o Pam.P 5mm.P mmm.P mmm.ow mwa.op mh:.m Phw.m mmh.m m
moo." Pom.o ham.o mum.P omm.= mha.= Nam.~ hmm.m FFN.N a
ooo.o ooo.o ooo.o abh.m ama.mp oop.mp moo.m mmo.m mmm.n m
ooo.o ooo.o ooo.o mmm.~ Nmm.PF oPo.PP mso.m mhm.o mmm.m N
ooo.o ooo.o ooo.c omn.m mpw.: mm>.: mmo.m mam.~ hma.m w
E\Elz E\Enz E\Enz E\Euz .55 .EE
HOHuMR A.E:cvwz Amsnuvmz Houumx A.Edcvxz Awsuuvxz HOHHMR A.ESCV3 Amsuuvz .uooq

 

 

 

 

 

 

 

 

 

 

A: .mflhv pmuuommnm wamfiflm much can 039 mmmpm .mmum mouse man one mmmpm

wumam mumsvm m How madamwm uomxm cam HmoHHwEsz mo GOmAHMQEOU

m magma

 

CHAPTER III

A NEW METHOD

III.1 ISOTROPIC PLATE PROBLEMS

Though the use of the boundary integral method to
solve isotropic plate problems has been proved successful,
the boundary condition equations, Eqs.(22) and Appendix A,
are quite lengthy, especially for a free edge. To sim-
plify the formulation, it is reasonable to consider re—
placing the set of fictitious moments by a second set of
fictitious forces. In so doing, there is no need to eva-
luate all the derivatives of the Green's function with
respect to E and n, since they are associated with the
fictitious moment Mn* only. With this simplification, the
length of boundary condition equations, Eqs.(22), can be
reduced by about fifty percent.

In practice, there are two simple ways that one can
enter twice as many fictitious forces P*. One can either
double the number of meshes along the integration contour,
or define a second integration contour. Tests indicate
that the latter provides somewhat better results. On the
other hand, during numerical integration, the fictitious
force is assumed constant along each mesh. It is there-
fore logical to replace this evenly distributed line
force by a concentrated point force and place it at the
center of the mesh. With this change, together with the
elimination of the fictitious moments, the deflection
function w(x,y) of Eq.(21) can be reduced to the following

form.

30

31

w(x,y) = If G(x.y;€.n)°q(€.n)d€dn
R

2n

+ Z G(X,Y7€pn) 'Pk*(§rfl)
k

k=1,2,3,... (23)

Following a numerical procedure similar to that of
section 11.2, this simplified method was tested using
the same square plate. The radius of the fictitious clamped
circular plate was kept at 80m. The first set of fictitious

forces were located at four meters away from the plate boun-

dary, and the second set were located at two meters away from

the first set, Figure 5. The results were almost identical

to those obtained using both fictitious forces and moments.
Thus far the boundary integral method has been mo-

dified somewhat, in that the boundary integration has been
replaced with an algebraic summing process. Though the eli-
mination of the fictitious moments has been successful,

the method is still tedious if free boundaries are involved.
In order to further simplify the method, the well known
Green's function for an infinite plate, [1], is introduced.
It is

 

- 2 _ 2
G(X'Y75vn) = Tg%5’Ux-€)2+(y-n)2]£n(x g) :(y n) ,
a

(24)

Mathematically, this is the fundamental solution to the
isotropic plate problem with a unit force at (€,n). The
beauty of this Green's function is that the denomenator

in the logarithmic term is a constant, a2, where a is an
arbitrary reference radius at which the deflection is zero.
When derivatives are evaluated, this new Green's function
gives a much shorter form than that obtained from the

32

5-69 ® ® GD 69

 

 

 

 

 

 

 

 

 

 

 

 

}—@ CD (2) CD
®
69
33383» ‘39
8838—!— ®
161$} 1m ——4m—-+¢2m
‘ 1H 15 ha
P7 1C5) .L
’X
J R B
80m

 

4 0 Field Point
Y

(:3 Concentrated Fictitious
Force

Figure 5. A Square Plate with Two Sets of
Fictitious Forces.

@ © @

t©-'—®

33

Green's function of a clamped circular plate. Consider

3 .
g§¥ for example. The new Green's function gives

SET = unD 12 (x-£)2+(y-n)2 q(g,n)d€dn

 

33w 1 {f,(x-€)3+3(y-n)2(x-€)

2N(x-€)3+3(y-n)2(x-E)

+ E (x-£)2+(y-n)2

P§<a,n)} (25)

while the old Green's function produces a very tedious
expression.

In order to assess the advanteges of using this new
Green's function, the previously-solved example problems,
i.e., square plates, are repeated. The results are un-
changed. However, the saving of computing time is large,
approximately sixty percent, as shown in Table 3.

As an illustration of the capability for solving
plate problems with odd plan forms, a simply supported
and uniformly loaded equilateral triangular plate has
also been treated; see Figure 6. For this triangular plate,
each side is ten meters long and discretized into ten boun-
dary points. There are, therefore, thirty boundary points
in all. To simulate the evenly distributed loading con-
dition of one Newton per square meter, one hundred o.u333
Newton concentrated forces are placed at the centroids of
the one hundred little equilateral triangles which form
the plate. The sixty fictitious forces are equally spaced
along two contours four and six meters away around the
plate boundary. The results are compared with the exact
solutions obtained from [1] shown in Table 4 and Figure 7.
The errors are quite small.

With the new Green's function and all the other sim-
plifications, the formulation of the new method becomes
neat and simple. All the boundary condition equations can

be written explicitly. Similar to Eqs.(22), we now have

am

.umusmfioo whelumnhu 000 m :0 mucoumm mu coflusomxm*

 

 

 

 

om *mmm.o *oam.o N .02 mmmu
mm *me.o *hmm.o P .02 mmmu
a monumz venue:

mmCH>mm sz was HmummucH wumpcsom

 

 

 

HDOm paw O3» wmmpw cam

mucflom gamut m, can

 

.pmmEmHo mum mmopm usOm may Had

 

.wwouom HmcuwucH cop

.pwuuommsm kaEHm mum

“ovum mum wwunu was one wmpm

.muCHom

"N .02 mmmu

.— .oz wmmu

humpcsom o:

pmoq omusnfluumflo >HEHOMHGD Hops: mumam mumswm oflmouuomH

mumoo ocflusmsoo mo COmflnmmEou

m magma

35

  
    

’/

Om Om 0:: 0w ON

 

04

  
 

Ooo

 

  
 

Om

 

      
   

o
D
b
o;
:0

  
 

    

‘ A
_I

 

 

 

‘- 10!“ 7|

0 Field Point
A Loading Point
0 Boundary Point

Figure 6. A Simply Support Equilateral Triangular
Isotropic Plate.

mm

.Houum mandamnm Human pan .uouum mmmuamonom mmumq*

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

* mmpoo. mumorm. . omoo.ou m1mmmm. * summmm.n ooo.o PF
.m~.o smP.P amP.P mom.m PFm.o .ms.o mmo.~ map.o ss..o op
G-.ou mos.. ohs._ mrm.ou mam.. omm.. mm~.P hm~.o mmm.o m
smo.o- mmm.F oom.F Fom.P mms.F ~ms.. m-.P mom.o mom.o m
:Gm.ou mam.F smm.P arm.ou som.F mpm.P Pm~.P mm~.o sm~.o p
om:.o- FFF.P .PF.F moo.~ sha.s smm.. Gom.F m=~.o ~:~.o G
mom.~- smm.o osm.o mmo.o- «as.F oom.F mam.P mmF.o GGF.o .m

. mmmoo.u smaoo. mam.~ .sm.. mam.F mm».s mmmo.o oomo.o :
smo.m mom.ou mam.o- mmm.o amp." mNP.F as..~ ssmo.o smmo.c m
5mm.: mo=.ou Nam.o- mmm.s oom.o «Fm.c mam.s mmmoo.o mimoo.o m

* F~oo.o mumarm. * mmPo.o mumm.~.- * summmo.n ooo.o F

E\E|z E\Elz E\Euz E\E|z .85 .EE
Monumn A.Ed:vwz Amsuuvwz Monumu A.Edcvxz Amsuuvxz HOHHMR A.En:v3 Amauuvs .uooq
AG munmamv

madam UHQOHuOmH umaamcmaua pounommsm mamfifim n no madammm mo somwummsoo

: OHDMB

 

moos‘
o..oouv

0.003 0
0.002 .

Y

0.001 4

0.00054.

Displacement (mm.)

IL
j!

0.00001 .

V

V

0.000005

 

Displacement

 

O Exact Solution

numerical Solution

 

A 4 A A

1

1

>2.0

 

 

Figure 7.

3 5 ‘3 6
Location

‘31
O
‘04 b
..a
O
-D
—0

Comparison of Results of a
Triangular Plate.

Bending Moment- (N-I/fl)

38

 

2N _ 2 _ 2
Z P:(€:T1)[(X-§)2+(y-n)2]gn(x E) +<y n)
i=1 a2

 

_ 2 _ 2
= -ffq(€.n)[(x-€)2+(y-UV]2n(X 5’ :‘Y ”’ dadn <26a>
R a

If (x,y) is on Bc + BS;

 

N _ 2 _ 2
igl P:(€,n)[1+£n(x €)a:(y n) ][(x-€)nx+(y-n)ny]

(26b)

 

- 2 _ 2
= -ffq(€.n)[1+£n(x a) +(y n) ][(x-€)nx+(y-n)n ]
R a2 Y

if (x,y) is on BC;

7 2 2 2
N P:(E,n){n2[ 2(X—E) + 1 +2n(x-g) +(Y'n) ]

1 X (X-€)2+(y-n)2 a2

 

ll MN

1

+ ann 2(x-E)(xfn)
y (X-E)2+(y-n)2

+ n2[ 2(tn)2 +1 +£n(X-€) 2+( _n)2]}
Y (x-€)2+(y-n)2 a2

2(x-EV‘ + 1 +£n(x-E)2+(x-n)2]
(X-€)2+(y-n)2 a2

'IIQ(€IU){n;[

x (26c)
Y (x-€)2+(y-n)2

‘ 2
+ n2[ 2(y-nV + 1 +£n(x-E)2+(y-n) ]}dadn
Y (x-€)2+(y-n)2 a2

if (x,y) is on BS;

39

2
P:(€,n){[n;+(2-V)nxn2](x-i)[(x-g) +3(y-n)2]
1 y [(X'€)2+(y—n)2]2

_ 2
+ [(20-1)n2n +(2_V)n3](Y'n)l(Y'n)2'(X €)4l
X Y [(x-€)2+(y-n)2]2

2
+ [(2-v)n3+(2v-1)n n2](x-€)[(x-g)2-(y-n) ]
X X [(x-£)2+(y-n)2]2

+ [(Z-v)n2n +n3](Y'n)[(Y-nY%3(x-§)2l}
x y y [(x-g)2+(y-n)2]2

- _ 2 _ 2
= -fo(€,n){[n;+(2-v)nxn2](x €)[(x a) +3(y n)_l
R Y [(x-€)2+(y-n)2]2

2
+ [(2v-1)n;n +(2-V)n3](y-n)[(y-n)2-(x-g) ]
y [(X-§)2+(y-n)2]2
1(X-€)[(x-€)2-(y-n)21.

+ [(Z-v)n3+(20-1)n n2
X X [(x-€)2+(y-n)]2

 

2 _ 2
+ [(2-v)n:n +n31'Y'“)[(Y'”) +3'X 5) ]}dgdn (26d)
Y Y [(x-E)2+(y-n)2]2

if (x,y) is on Bf; and

N _ 2
P:(E,n){(n;+vn2)[ 2(x 5) + 1 +£n

1 y (x-§)2+(y-n)2 a

(x-€)2+(y-n)2]
2

I! MN

1

+ 2(1-v)nxn 2<X'5)(Y“”)
Y (x-€)2+(y-n)
2(y-n)2 + 1 +Rn(x-E)2:(x-n)2]} =

+ (n2+vn2)[
y (x-€)2+(y-n)2 a

X

00

 

= -ffq(€.n){(n2+vn2)[ 2(x-g)2 + 1 +£n(X-€)2+(y-n)2]
R X Y (x-a) 2+<y-n>2 a2

2(x-€)(y-n) (26e)
(X-€)2+(y-n)2

+ 2(1-v)n n
X Y

2 2
+ (n2+vn2)[ 2(y-n)2 + 1 +£n(X-€) :(y-n) ]}d€dn

(X-€)2+(y-n)2 a

if1(x,y) is on Bf. Note that the common constants such as
TEFB’ involved in both sides of the equations have been
deleted.

With the new Green's function and boundary condition
equations, and following the same numerical procedure shown
previously to solve a set of ZNXZN linear algebraic equa-
tions for the unknowns of 2N fictitious forces, a general
isotropic plate problem with arbitrary plar form, loading
and boundary conditions can be solved. Although there are
many improvements from the original boundary integral method
[9], there are added numerical questions to be studied. In
the original method, the radius of the fictitious plate
involved in the Green's function is the only value to be
chosen before analysis. An imprOper selection of this radius
will result in poor solution accuracy. Fortunately, it has
been found that good results can be obtained for a wide
range of values of this radius. Take a 10m square plate for
example. No change in solution has been noticed for values
of this radius selected between 80m and 8000m.

For the new point-force method, on the other hand, in
addition to the reference radius a in Eq.(Zu), the locations
of the fictitious forces must also be determined. It has
been observed from the numerical tests of this 10m plate
problem that the fictitious forces must be placed within

a1
a narrow band 1m to 10m away from the plate boundary. Less
accurate solutions will result if they are placed within
1m from the boundary, and no solution can be obtained if
they are located farther than 10m away. It is conceivable
that, like the boundary integral method, when the fictitious
forces are too close to the boundary, it is impossible to
get good results for those field points near the boundary.
This is simply due to the fact that the boundary condi-
tions are not satisfied everywhere along the boundary, but
at those discretized boundary points only. On the other
hand, when these fictitious forces are placed too far from
the plate boundary, the influence due to each individual
fictitious force is so weak that together with computing
truncation errors, the RM matrix may become ill-conditioned.

Some results for a 10m clamped square plate are shown
in Tables 5 through 8 to illustrate the change of solutions
when fictitious forces are placed at different locations.
Tables 5, 6, and 7 are for double-looped fictitious forces,
and the double 100ps are at am and 6m, 1m and 3m, and 0.5m
and 2.5m away from the plate boundary, respectively; see
Figure 5 for reference. Table 8 is for a single-looped
approach. That is, all the 2N fictitious forces are distri-
buted along a single contour surrounding the plate. This
contour is am away from the plate boundary; see Figure u
for reference. The results are compared with the exact so-
lution published in [1]. The five locations indicated in
these tables correspond to locations 1,6,10,13 and 15 shown
in Figure 5. The discrepencies between MK and My of the
exact solution were due to the fact that they were obtained
from truncated infinite series solutions.

The computer program for isotropic plate problems with
arbitrary plan form, loading, and boundary conditions using
this simple point-force method is shown in Appendix C. It
is believed that for a plate of any size and shape, good
solution accuracy can be achieved if the locations of fic-
titious forces are selected properly. The determination of

ma

 

 

 

 

 

 

 

th.rl =m~.m Pam.m omm.P| =mm.~ omm.m ooo.o =mm.o =mo.o m

mmw.PI mmm.P mum.w om>.Ft mmm.P 35¢.P ooo.o ohm.o mnm.o :

hmp.ml mPP.P PmF.F one.mn mPP.F FmP.P mno.o: Pom.o Pom.o m

oom.mF| mar.o Fmr.o oom.mpl mar.o Pmp.o omp.ol hme.o hmp.o N

PNN.oP th.ou omm.on o:>.m Pum.OI hmm.on ooo.o owo.o meo.o P
E\Euz E\Elz E\E|z E\Elz .28 .EE

MonumR A.Escvmz Amsuuvwz HOHHMx A.Escvxz Amsuuvxz HOHHMR A.Es:v3 Amouuvz .uooq

 

 

 

 

 

 

 

 

 

 

mumocsom wpmHm may Eonm >m3< Em pom E: um mwouOh msoflufluoflm Ummooqlwansoa

wumHm mumsqm UOQEMHU m mo muasmmm mo COmHHmQEOU

m manna

 

m:

 

 

 

 

 

 

 

mmm.FI mmm.~ Pm~.~ Po.P| mm~.~ omm.m osm.ou =mm.o amo.o m

0mm.PI mmm.P mum.P omm.P| mmm.F ano.P Pmo.ou ohm.o ohm.o :

mPP.mI oFP.P FmF.F mmo.mu oFP.P PmP.P m=P.on oom.o Pom.o m

ohm.hpn map.o pr.o Nmm.hFI map.o Fmp.o mo.ou hmw.o hmw.o N

om~.PF mhm.ou mmm.o: arm.op mnm.ou hmm.01 ooo.o opo.o mwo.o P
E\Elz E\Elz E\Etz E\Elz .58 .EE

HOHHMR A.Eocvwz Awsuuvwz HOHHMx A.E:cvxz Amsuuvxz uouumR A.Escv3 Amsuuvz .uooa

 

 

 

 

 

 

 

 

 

 

humocsom mumHm on» Scum >m3€ Em cam Er um moonom maofluauOflh vmmooglwansoo

madam mnmsqm owdEMHU o no muasmmm mo cowfiummeou

@ OHQMB

 

=3

 

 

 

 

 

 

 

who.N| maN.N PmN.N amo.N| maN.N omN.N onw.ou mmw.o =wm.o m

5mm.N| mNm.F mum.P mom.NI mNm.P :N¢.— Npm.o1 onm.o mnm.o :

mom.m: moP.P FmP.P omh.mn moP.P FmP.P Pmo.Pn owm.o Pom.o m

MNm.hPI map.o Pmp.o =P=.NPI o=P.o Pm—.o mmP.—I mmp.o hmp.o N

:oP.m bom.OI mmm.o: omm.m hmm.on 5mm.0I oom.NI w—o.o meo.o P
E\Euz E\EIZ E\Elz E\Enz .58 .EE

Honumu A.EDGVMS Amsuuvwz HOHMMR A.Escvx2 Amsuuvxz HOHHMx A.Ed:v3 Amsuuv3 .uooq

 

 

 

 

 

 

 

 

 

 

wuwocsom wumam on» Eoum >m3< Em.N new Em.o um mmuuom mDONuwuowm pmmooqnmansoo

mumam mumsvm ommfimao M NO muasmmm mo conflummeoo

a manna

 

m:

 

 

 

 

 

 

 

mmm.P| mmN.N PmN.N mmm..u mmN.N omN.N ooo.o cow.o 300.0 m

mow.eu osm.F mum.w w>N.PI oam.~ sum.r ooo.o ohm.o ohm.o a

mmo.m| oPP.P PmP.P omo.ml wPP.P FmP.P mmo.o: Pom.o Pom.o m

www.mpn map.o PmF.o omp.mpu osp.o FmF.o Nmp.o| nmp.o bmp.o N

MNm.wF mmm.ou omm.on, moo.mp mmm.o: hmm.o1 www.ml mFo.o mpo.o P
E\Elz E\Euz E\Enz E\Elz .28 .EE

Monuma A.Escv>£ Amsuuvwz uouum: A.Escvxz Amsuuvxz HOHHMx A.Edcv3 Awsuuvz .uOOA

 

 

 

 

 

 

 

 

 

 

whopcsom madam mnu Eoum >m3< Ea um mmouom mooflufluOAm @mmooqlmamcflm

mumHm mumsvm omQEmHU m mo muasmmm mo conflummfiov

m OHQMB

 

Q6
of these locations may not be an easy task, and this nu-
merical question requires further study.

On the other hand, for multiply connected plates,
seemingly there will be difficulties if the "holes" are
small and many boundary points are prescribed at the holes,
since placing many fictitious forces in a small area inside
a hole will certainly lead to numerical problems. Further
study is needed in the search for optimum locations for

fictitious forces.

III.2 ANISOTROPIC PLATE PROBLEMS

Due to the increased use of composite and multilayered
plates for strength and weight reduction, anisotropic plate
problems are becoming more and more important. With mid-
plane symmetry of the material properties, the governing
equation is Eq.(3), and the boundary conditions are given
by Eqs.(S) and (7). They are far more complicated than when
the plate is made from an isotrOpic material. Finite dif-
ference and finite element methods are generally used to
obtain a solution. In this dissertation, a new numerical
method is introduced. Using the Green's function for an
anisotropic infinite plate and the same point-force tech-
nique shown previously, solution of an anisotropic plate
problem with arbitrary plan form, loading, and boundary con-
ditions is obtainable.

Since general anisotropic problems are very difficult
to solve, they are often reduced to orthotropic problems
through coordinate transformation or approximation, whenever
possible. Therefore, orthotropic problems will be discussed
first. For an orthotropic material, there are three mutually
perpendicular planes of symmetry with respect to the elastic
properties of the material, and the problems are greatly
simplified compared with general anisotropic problems. In
practice, it appears that orthotropic problems are more

07
common than the general anisotropic plate problems. Rein-

forced decks in civil, marine, and aerospace engineering,
and plates made of layered composite materials are typical

examples.

III . 2 . 1 ORTHOTROPIC PROBLEMS

For an orthotropic problem, the governing differential
equation, Eq.(2), is a special form of Eq.(3), the equation
for anisotropic problems, with D16=D26=0. The boundary con-
dition equations are also simpler than those for their aniso-
tropic counterparts; see Eqs.(u) and (8). It is due to these
simplifications that solutions are obtainable for many pro-
blems. Bares and Massonet [22] used a beam and grid analogy,
Vinson and Brull [23] used a power series expansion, and
Rajappa [24] tried a Maclaurin's series. In addition, the
application of finite difference method is clearly presented
by Szilard [7], and the theory of finite element method is
explicitly shown in Zienkiewicz's text [8]. For classic
approaches, texts [14,26] of Lekhnitskii and Huber, res-
pectively, are probably the most important.

For an orthotropic material, if the geometric coordi-
nates are aligned with the principal material directions,
the governing differential equation for equilibrium can be
shown in Eq.(2). There are four material constants, namely
Ex’ Ey’ Vx' and ny, where Ex and By are the two Young's
moduli evaluated along the x and y directions, respectively;
Xx is the Poisson's ratio in the x direction due to normal
stress in the y direction; ny is the shear modulus. The
other Poisson's ratio vy is related to Xx by Betti's reci-

procal theorem

48
and therefore is not an independent material constant.

In order to apply the new point-force method, the first
requirement is to find the Green's function of some appro-
priate problem. There are two Green's functions readily
available for a simply supported rectangular plate, namely
Navier's double series solution and Levy's single series

 

 

 

 

 

 

 

 

 

 

solution:
“b3.».nsin:2€ sinmg: sing-g-Il sin§%X
C(XIY;€IU)= Ind (27)
n“a Dx (——$’+2H(n% m)2+ Dyn
m=1,2,3,... , n=1,2,3,
and
G(le;gln)= M31) D .(82_AITI§ n3 X
KY
(28)
Bsinhnnxéa-E)si nhnglx Asinhnngm-g) sinhnflgx]
x[ -
sinhnnga sinhnflga

n=1,2,3,

for 0§x<£; and substitute x by (a-x) and (a-fi) by E for
Efxfa, where B and A are the roots of the characteristic
equation (which will be discussed later), a and b are the
dimensions of the rectangular plate, and i and n are the
location of the point force. It is known that the Navier's
double series solution converges slowly. However, due to its
simplicity in higher order derivatives, it was also tested
along with Levy's single series solution. Before the full
development for orthotropic problems, these two Green's
function were evaluated for their efficiency in isotropic
problems. For a square plate under uniformly distributed
load, the results were disappointing for both approaches.
For a solution accuracy greater than ninety percent, more
than one hundred terms of Levy's series were needed, and the
number is even higher for Navier's series. The computing

49
costs were formidable. Therefore, the idea of using either
approach was abandoned.

Since the fast converging Levy's series failed to yield
satisfactory results in the application of the point-force
method, it was clear that the Green's function to be adopted
for the method must be in closed form. One of the currently
existing Green's function in closed form is given in [12]
for an infinite plate. Depending on inter-relationships
among the material constants, this Green's function contains
a group of three independent equations. These equations are
derived in terms of several new material parameters. There-
fore, it is necessary to introduce these new material
parameters prior to the presentation of the governing equa-

tions. Let

D
— —— and a“ = .33 (29)
D

then, the governing partial differential equation, Eq.(2),

can be re-written in the form

W + 2 €28“w(x’ ) +El’auw(xl ) = (XI ) 30
o —————-X—- -—————X—- 9———X— ( )
any BXZQYZ 3X1. Dy 0

Since the Green's function is the solution for the Dirac
delta loading function of this equation, and this equation
can be integrated in its homogeneous form, i.e., q(x,y)=0
for (x,y)#(€,n), we can write the Green's function symbo-

lically as

D1DZD3DQG(x.y;€.n)=O. (31)

where the D's are linear differential operators, in the form

50

of
.. 3 .2.
Di - 3y ri 8X (32)
and ri are determined as the roots of the characteristic
equation

r“ + 2p€2r2 + E“ = 0 (33)

These roots are either complex or pure-imaginary as shown
by Lekhnitskii [27]. That is, the roots are in the form of

= til . (3“)

Depending on the value of 0, either greater than, equal to,
or less than unity, the values 8 and A can be easily deter-
mined by using either one of the following three equations,

8 = E/p+\/;T. A= END-fa}: (356)

B = E , A = 8 (35b)

for O>1;

for 0:1; and

B = 01+ i02 , A U1“ iU2 (350)

for p<1, where

 

51
It is clear that, depending on the material constants,
each of the three criteria must be considered. For an in-
finite plate, Mossakowski [12] has derived three different
Green's functions for these three different material types.
With

we have

(x-£)2+A2(y-n)2

a2

1
8NDO(BZ-x2

 

G(X.Y:€,n)= ){8[(x-£)2-)2(y-n)fiin

-uAB(x-€)(y-n)[arc tglifiggi ’ arc théx:2)]

n(x-£)2+BZ(y-n)2

a2

-) [ (X-E) 2-82 (y-:n)2]£

-3(B-))[(x-€)2+AB(y-n)2]} (36a)
for p>1;

. .. 1 (x-g)2-t-e:2(y-n)2 (x-g)“+2p8(x-§)2(y-n)2+€umn)u
G(XIYI€In)-32nDO{ U1 fin a“

_2[‘(X'€)2-EZ(y-n)2]+ arc tg 2U11Jz (I'mz
U2 (X’€)2+D€2(Y’n)2

_2e2(x-§)(y-n)inplqy‘n)2+l(X-€)-p2(y-n)]2
”1'” u12(y-n)2+[(X-€)+112(Y’VH 2

_6[(x-€)2+ez(y-n)fl,} (36b)
01

for p<1; and

52

 

_ 2 2 _ 2
G(X.y;€.n)=Tg%EE;{[(x-§)2+52(y—n)2]gn(x 5) :: (y n)

- [3(X-£)2+€2(y-n)2]} (36C)

' for 0:1.

The second order derivatives of these equations are
also given in [12]. Since they are needed not only in the
boundary condition equations for a simply supported or free
edge, but also in the determination of bending moments after
fictitious point forces are computed, they are worth in-
cluding in the following. Other derivatives are listed in
Appendix D. There are three sets of equations, one set for
each Green's function.

 

 

 

 

 

 

 

32 _ 2 A2 _ 2 _ 2 2 _ 2
a ? =WD (;T_)\2T[B’Q’n(x 5) +2 (X n) _A£n(x 5) +5 (y T1) 1
X 0 a a
326 E2 _ 2+82 _ 2 _ 2+A2( _ )2
3Y2 =HNDO(32-A2)[B£n(x 5) a2 (y n) _A£n(x 5) a2 y n 1
32G _ 22 B( -n) A( -n)
$§F§‘EEDO(ez-AZ)[3X° t9"T¥:ET ’ arc t9 x-E ] (37a)

for o>1:

 

2 =1st 1 ‘ u

2 I, 2 _ 2 _ 2 lo _ lo
a G 1 L&JF<X-g) +206 (x E) (y n) +8 (y n)
O a

3x

 

2u1u2(y—n)2 ]

-22 ICC tg 2 2 2
(x-E) +06 (y-n)

LI

326 :2 1 (x-a)“+2pe’(x-€)2ly-n)2+6“(y-n)“ +
ayz -i6flDo[TJ—l£n a“

53

2U1U2(y-n)2
(x-€)2+pez(y-n)2

2
+ ——— arc t
02 g

326 -52 1112(y-n)2+[(x-€)-U2(y-n)]2
- 2n (37b)
BXBY '6"Do“1“2 ulz(y-n)2+[(x-€)-02(y-n)]2

for p<1; and

 

 

 

 

 

 

 

32c _ 1 (x-€)2+sz(y-n)2 282(y-n)2
- BNED [2n - 1
3x2 0 a2 (x-£)2+62(y-n)
326 = 8:0 [2n(x-€)2+€2(y-n)2 + 262(y-n)2
3Y2 0 a2 (x-€)2+62(y-n)
826 _ 1 e(x-€)(y-n)
Bxay “ND 1 (37C)

0 (x-E)2+ez(y-n)2

for p=1.

Following the same numerical procedure as in isotropic
problems, orthotropic plate problems of arbitrary plan form,
loading, and boundary conditions can now be solved. The
added complexity is that depending on 9&1, there are three
sets of equations to be concerned with. In order to verify
the results for all the three possibilities, three sample
cases have been solved. Consider a simply supported 10m
square plate, and let Ex=2.O6BX105 MPa, Ey=EX/15, vx=0.3,
and h=0.01m. Varying p from 0.1 to 1.0 and 10.0, the accuracy
of all theses three sets of results of deflections and
bending moments are excellent when compared with the double
series solution shown in [1a], taking “00 terms. It is due
to the fact that the changes are minimal when more than 100
terms are taken in the double series solution, the “GO-term
double series solution is believed very close to the exact

50
one. The comparisons are tabulated in Tables 9 through 11.
The nine locations of field points are shown in Figure 8.
The large errors for the bending moments in the y
direction of Table 11 are somewhat misleading, because
their magnitudes are small in comparison with the bending
moments in the x direction. The computer program for these

examples is shown in Appendix E.

III . 2 . 2 ANISOTROPIC PROBLEMS

An anisotropic thin plate is considered as a plate made
with a material which has the mid-plane of the plate as the
only plane of material symmetry. It is due to the complexity
of its governing and boundary condition equations, Eqs.(3)
and (7), that efforts are always made to reduce anisotropic
problems to orthotropic problems. That is, methods such as
coordinate transformation are often tried to eliminate the
two material constants a16 and a26 in Eq.(3). This is, how-
ever, not always possible. In general, neglecting these cons-
tants often times will lead to large errors, [28]. Therefore,
though techniques to the solution of orthotropic problems
are more important than that of general anisotropic problems,
methods for the latter must also be developed.

Since a large number of anisotropic problems are related
to man-made layered composite materials, it is wise to review
a few references that will provide a better understanding of
composite anisotropic materials: [28] presents basic concepts,
fundamental equations, and many interesting illustrations;
[29] introduces many exotic materials, their mechanical pro-
perties and applications; [30] illustrates many matrix systems
and their characters; and [31] gives many studies of the
applications and their significant contributions to the aero-
space industry.

The solution of anisotropic plate problems is again

55

 

 

 

 

 

 

 

 

 

 

 

2m
_L_
A R 1153 .2},

80m

 

 

Y' 0 Field Point

O Concentrated Fictitious
Force

Figure 8. A Simply Supported Orthotropic Plate.

om

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=mo.:1 aNm.o :mm.o mrm.o www.m— Nm>.mw mop._ oPN.m ONP.m m
mon.m1 ome.o ho>.o mFo.P meo.rp mma.PP "PF.P sho.m Foo.w m
th.mP1 th.o omN.o bsm.P chm.a mnm.a mam.o th.N mam.N h
omm.NI mao.P mno.e moo.r omo.NP opm.PP maN.P mop.h mpo.h m
th.m| Nam.o m5m.o hmP.P o=P.oP mNo.oP mwN.P omn.m >o>.m m
man.wrl mom.o mam.o mmP.N mP=.: MNm.s ham.P mMN.N mON.N a
me.ol bom.o mpm.o mmm.P wMN.m hmP.m mwh.P hop.m mmo.m m
NmN.F1 mmh.o mhh.o :mn.~ mmm.s um.: PNm.F mam.N hm=.N N
bwm.wl oom.o mNm.o mph.= mmN.N NmP.N mmm.P omm.o mnm.o P
E\Elz E\Elz E\Elz E\Elz .85 .EE
MOHHMx A.Es:v>z Amsuuvwz HouHMm A.Edcvx2 Amouuvxz uouuma A.E:cvz Amsuuvz .DOOA
P.ouo
eso.onc .m.o x> .mp\xmu>m .mmz mocxmmo.~uxm

madam mumsqm camouuonuuo pmuuommsm SHQEHm M MD muHSmOm mo :OmHHMQEOU

a magma

 

5m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

030.01 5F0.0 000.0 0P0.0 0N5.0 000.0 000.0 mNm.m P55.m m
0=N.51 003.0 0mm.0 300.0 00N.0 00N.0 0N0.0 m=5.= 000.: 0
mNP.5pw 05P.0 0PN.0 50N.P P50.m 3N0.m 0:0.m 5mw.r 055.? 5
mm0.:1 000.0 mm5.0 000.0 mNm.0 N5N.0 000.: m00.a 005.: 0
500.01 500.0 m00.0 505.0 0NP.5 N50.5 0m0.P m00.= 5N0.= m
0mm.:P1 00N.0 P=N.0 :==.P NMN.m 00P.m 050.9 050.5 P00.P :
00N.N1 FF0.0 0N0.0 000.0 0F0.m 05m.m 5Nm.P 00P.N 5NP.N m
030.31 0F0.0 Fm0.0 maP.P NOP.m 0ar.m 000.9 055.. N35.P N
mmm.m~1 009.0 mNN.0 50m.N 3P0.P 05m.F 050.? 000.0 :00.0 P
E\Enz E\EIZ E\Euz E\E1z .85 .EE
uouumR A.Escv>z Awsuuvwz uouum: A.Esc0xz Amouuvxz MonuMK A.Esc03 Amouuvz .uooa
0.Hua
2.0.ou: .m.oux> .ms\xmusm .mmz mopxmeo.~uxm

mumHm mumsqm Osmouuonuuo pmguommom hamfiflm m mo muHSmmm mo cemwummfioo

or manna

 

00

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

500.001 mmmr.0 055.0 000.51 m0N.N 00N.N 000.0 053.5 503.5 0
003.001 005.0 505.0 000.51 M00.N 5NP.N 500.0 0PN.F 00N.P 0
00N.00l 0N0.0 000.0 305.51 005.5 PFN.F 000.0 000.0 503.0 5
500.0NI 0NP.0 055.0 000.51 N00.P 000.5 N05.0 :3N.5 :MN.P 0
500.001 N05.0 005.0 500.5I 005.5 005.5 030.0 0N0.F 0N0.P 0
500.001 NN0.0 000.0 505.NI NNO.P 330.5 000.0 003.0 500.0 a
500.5NI 355.0 505.0 0mm.:l 000.0 030.0 N~0.0 000.0 0N0.0 m
005.le 000.0 Nmp.0 500.31 005.0 005.0 000.0 500.0 503.0 N
005.051 050.0 000.0 050.01 003.0 000.0 005.0 055.0 055.0 P
E\Elz E\Etz E\Euz E\Elz .EE .EE
uouuma A.Escvwz Amnuuvwz uouuma A.Esc0xz Amsuuvxz HOHHMx A.Esc03 Amsuuvz .uOOA
o.oPna
epo.oun .m.onx> .mexxmusm .maz meexmmo.~uxm

wumHm mumsvm camouuonuuo Umuuommsm xamEHw m 00 muasmmm mo acmflummeou

PP OHDMB

 

59

obtained using a known Green's function for an infinite
anisotropic plate [11,13] and applying a set of fictitious
forces surrounding the plate boundary such that all the
boundary conditions are satisfied. Similar to the previous
problems, the numerical procedure is to solve the 2NX2N
algebraic boundary condition equations for the unknown mag—
nitude of fictitious forces. The only added work is in the
determination of the complex roots of the characteristic
polynomial equation. IMSL computer subroutine ZPOLR has been
conveniently employed for this purpose.

The characteristic equation for the homogeneous solu-
tion of Eq.(3) is [13,14],

D

U
_I
—I

+2D D
12D 66 r2 + 4 16 r +

22 2

D
r“+u—2-6-r3+2

D22

0 (38)

 

m)
U

n)

n:

where the roots ri are involved in the four linear differen-
. 3 3 . .

tial operators 3; - ri 5;, the same as in the orthotropic

formulation. Solving this fourth order algebraic equation,

the roots can be determined in the form of

= . 0 = i .
r1'2 a i 18 , r3,“ Y 1)

They are all complex values as proved in [27].

For an infinite plate, the Green's function shown in
[13] is

, _ 1 (a-xlz-(Bz-AZ)D ,
G(X'Y'€'n)_8n022¢1¢2{ B “1(x,y.€.n)

(a-Y)2+(82-XZ)

+ A R3(X:Y;€:T1)

+ 4(c-y)[S1(x.y;€,n)-S3(x,y:£.n)]} (39)

60

where,
¢1 = (G’Y)2+(B'A)2 3 ¢2 = (G'Y)2+(B+A)2 3

R1(x,y:€.n)={[(x-€)+a(y-n)]2-82(y-n)2}

x {Rn1(x-€)+a(y-n)12+82(y-n)2 _ 3}

a2

- 4B(y-n)[(x-€)+a(y-n)]arc tg 8(X'”' :

(x-€)+a(y-n)

S1(X.y;£,n)= B(y-n)[(x-£)+a(y-n)]

x {Qn[(x-€)+a(y-n)12+82(y-n)2 _ 3}

a2

+{[(X-€)+a(y-n)]2-Bz(y-n)0arc tg 8(y-n) :

(x-€)+u(y-n)

and R3(x,y;€,n) and S3(x,y;£,n) are obtained by replacing
a and B by y and A, respectively.
As with the orthotropic Green's function, the first
order derivatives are quite lengthy. The second order deri-
vatives, however, can be reduced to very compact forms.
Since they are the most important derivatives, they are listed

here. Others are shown in Appendix F.

Q)

26 1 {(a-y)2-(ez-AZ)

 

L1(XIY;€IW)

(a-Y)2+(82-A2)

+ A L3(x,y;g,n)+4(a-y)[N1(x.y;E.n)-N3(X,y:€.nn}

 

(40a)

61

325 = 1 (02+82-2ay)(a2+82)+(a2-BZ)(Y2+A2)
ayz ”"D22¢1¢2{ B L1(X.y:€.n)

2 x2-2 2 2 2' 2 2 2
+ (Y + ay)(j +A );(Y A )(a +8 )L3(x,y:€.U)

 

- ala<y2+A2)-v<a2+82>1[N1(x.y;€.n)-N3(x.y;g.n)]} (uOb)

 

326 1 {(a-Zy)(d2-Bz)+oijz+lz)
B

Bxay- UND22¢1¢2 L1(XIY7€IU)

-. 2A2 22
+ (Y 2&)(Irf‘ );Y(a +8 )L3(X,y;€.n)

+ 2(a2+82-Y2-12)[N1(x,y;€.n)-N3(x.y:€,n)]} (u0c)
where,

£n[(x-E)+a(y-n)]2+82(y-n)2

a2

L1(XIY7€IU) =

8(y-n)
(X-€)+a(y-n)

N1(X.y:€.n) arc tg

and L3(x,y;€,n) and N3(x,y;€,n) are obtained by replacing
a and B by y and A, respectively. It is worth noting that
Eqs.(39) and (40) can be easily reduced to Eqs.(36) and (37)
by making a=y=0 for p>1.0; a=u2, Y=-U2. B=A=p1 for p<1.0;
and a=y+0, B=A+e for p=1.0.

Following the same numerical procedure shown in the
previous two sections, using the new point-force method,
the solution of an anisotrOpic thin plate problem with
arbitrary plan form, loading, and boundary conditions can

62

be obtained. For the verification of results, however, due

to lack of exact solutions available for anisotropic problems
to be compared with, a different approach must be taken. An
orthotropic plate problem will become apparently anisotropic
if the geometric coordinates are made different from the
principal material directions. Therefore, solutions of ortho-
tropic plate problems can be used to validate the equations
for general anisotropic problems. This approach can be summa-
rized in four steps as shown in the following. First, an
angle of rotation for the geometric coordinates is chosen ar-
bitrarily, and corresponding to the new coordinate system,
locations of the boundary points, the fictitious forces,

the field points, and the unit outward normals of the boundary
points are determined. The second step is to compute the six
flexural rigidity constants Dij used in Eq.(3), [14]. The
next step is to employ the Green's function of the infinite
anisotropic plate to solve the pseudo-anisotropic problem.
The final step is to determine the displacements and bending
moments at the prescribed field points in the original coor-
dinate system using coordinate transformation, and then make
comparison with the orthotropic solutions.

With this validation method, orthotropic plate example
problems shown earlier with all the three types of p, i.e.,
‘greater than, equal to, and less than unity have been tested
against four coordinate rotation angles, namely 15, 30, Q5,
and 60 degrees. The discrepencies of results were Within one
percent and were believed to be due to truncation errors
during the added numerical processes. The computer program
for this validation is shown in Appendix H, while the pro-
gram for a general anisotropic plate problem is shown in
Appendix G.

It must be noted that the two flexural rigidity constants
and D26
of Eq.(3). In order to investigate the influence due to

D16 are based on the two material constants a16 and

a
26
these two material constants, several sample problems using

a simply supported square plate have been tested. Since the

63
the material constant matrix shown in Eq.(3) must be positive
definite, a16 and a26 were selected to be less than a11 and
a22, respectively. Under this condition, take four typical
cases with a16=0’1/Ex’ a26=0.1/Ey; a16=0.9/Ex, a26=0.9/Ey;
a16=0'1/Ex’ a26=0.9/By; and a16=0.9/Ex, a26=0.1/Ey for example,
it has been found that the differences in displacements and
bending moments were smaller than one percent. However, when
the shear modulus ny is small in comparison with EX and By,
a16 and a26 can be made greater than a11 and azz; their con-
tribution to the solution may become significant.

CHAPTER IV

CLOSURE

Starting from a boundary integral equation method for
an isotropic thin plate problem with boundary clamped or/and
simply supported, a very efficient numerical solution to
problems with arbitrary plan form, arbitrary loading and
boundary conditions, and anisotropic material, has been
develOped. The method uses the known Green's functions of
isotropic, orthotropic, and anisotropic infinite plates. The
problem is solved after the real plate is embedded in the
fictitious infinite plate, and the boundary conditions at
the N prescribed boundary points are forced to be satisfied
with an imposed set of 2N calculated fictitious forces located
somewhere outside the plate boundary. Though no efforts have
been made to compare with the two leading numerical methods,
the finite element and the finite difference methods, it is
believed that the new method has the following two advantages:
(1)since the Green's function is the exact solution to a point
force problem, and there are no assumed polynomials for results,
high solution accuracy is expected; (2)due to the fact that
the equations are simple, and the modeling is for the plate
boundary only, the current method is easier to use.

Large percentage errors indicated in all tables are some-
what misleading. Take the simply supported triangular plate
problem for example. Percentage errors shown in Table u are
huge at certain locations. However, the real errors are small
as shown in Figure 7.

During the deve10pment of the current method, it was found
that though series type Green's functions were easy for

6Q

65
formulation, they were not suitable for the current method.
This was due to the fact that large number of terms of the
series were needed to provide acceptable solution accuracy,
and this would lead to formidable computing costs.

Though the current method is efficient, numerical ques-
tions remain. An imprOper choice of the locations for the
fictitious forces may result in poor solution accuracy or
no solution at all. Therefore, in order to take full advan-
tage of this method, some further studies should be made so
that the locations of fictitious forces chosen will bring
optimum results. In the meantime, due to the involvement of
many "looped" summing processes, Eqs. (19), (21), (22), and
(26), it is also necessary to do sensitivity studies to
minimize the numbers of boundary points, fictitious forces,
and internal forces for least computing cost.

All the five computer programs developed for this thesis
research are shown in Appendices B, C, E, G, and H. The first
is for the boundary integral method. It uses the Green's
function of a clamped circular plate. The second is for the
new point-force method for isotropic problems. The third
and the fourth are for orthotropic and anisotropic problems,
respectively. The fifth is a method employed to validate the
equations for general anisotropic problems, using exact
solutions for orthotropic plate problems. All these computer
programs are coded in FORTRAN, and their flow chart is shown

on the next page, Figure 9.

 

66

 

Do a coordinate transformation
with an arbitrary rotation

 

Read input values
and create the model.

 

 

Appendix H only

Appendix G only-1

y

angle and create a pseudo-
anisotrOpic problem from the
orthotropic problem.

 

 

 

 

I
Set up the fourth order poly-

 

 

 

 

Determine the values
involved in the RL vector

nomial characteristic equation
and use IMSL subroutine ZPOLR
to compute complex roots.

1

 

 

 

 

 

with the known load dis-
tribution function.

 

 

 

 

Set up the RM matrix

and solve the ZNXZN
linear algebraic equation
using IMSL subroutine
LEQT1F.

 

 

 

 

I

 

Use the determined fictitious
forces (and moments) to com-
pute the deflections and
bending moments at the pre-
scribed field points.

 

-Appendix H only

 

Transform the bending moments
back to the original coordinate

 

 

 

system and compare with the
known orthotropic results.

 

 

 

 

 

End

Figure 9

The flow chart for the programs
shown in Appendices B, C, E, G, and F.

APPENDICES

APPENDIX A

DERIVATIVES OF THE GREEN'S FUNCTION
FOR A CLAMPED CIRCULAR PLATE

APPENDIX A

DERIVATIVES OF THE GREEN'S FUNCTION
FOR A CLAMPED CIRCULAR PLATE

The Green's function shown in Eq.(10) can be written as

2

 

 

G(X.y;€,n)= 75%5{(1-r12)(1-r22)+r122£n 2r12 2 2}
(1-r1 )(1-r2 )+r12
2 2 2 2 2 2
where, r2: .x_+_X_, r2: _€_:_D_' and r = (x-E) +(y-n) .
1 a2 2 a2 12 a2

For simplicity, from now on the variables x, y, E, and n
are all made non-dimensional. That is, the variables x, y, E,
and n shown in the following equations are actually the ratios

x E .
of 3' g, 3, and 2, respectively.

 

 

as a2 r122(x'r22'€)
-— = -—ix-r 2- g -
3X 8WD 2 (1'3? 2) (1_r 2)+r 2

1 2 12

2
r
+(x-£)£n 2 12 2 2}
(1-r1 )(1-r2 )+r12

as a2 2 r122‘y'r22’”)

 

_ 2 _ 2 2
(1 r1 )(1 r2 )+r12

67

68

2
r
12 }

- 2 - 2 2
(1 r1 )(1 r2 )+r12

 

+(Y-n)£n

r122(§-r12-x)

 

3: ”5%6{5°r1 ' X “
(1-r12)(1-r22)+r 2

 

 

 

 

 

 

 

 

 

12
2
r
+(E-x)£n 2 12 2 2 }
(1-r1 )(1-r2 )+r12
2 . 2-
39 - -—3—{ . _ r12 (” r2 Y)
an - 8wD n r2 y - (1-r 2)(1-r 2)+r 2
1 2 12
2
+(n-y)£n r12 }
(1-r1 )(1-r2 )+r12
- _ . 2 _ 2, 2
325 = a2{r 2 + 2(x-€)2+ “(X €)(€ x r2 ) r12 r2
2 8ND 2 2 - 2 - 2 2
8 x r12 (1 r1 )(1 r2 )+r12
2r 2(E-x-r 2)2 r 2
+ 12 2 2 2 2 3 2n 2 12 2 2 }
[(1-r1 )(1-r2 )+r12 ] (1-r1 )(1-r2 )+r12
- - . 2 - _ , 2
326 = a2{2(x-§)(y-n) + 2(Y ”)(E x r2 1+(X €)(n y r2)
Bxay 8111:) 2 _ 2 _ 2 2
r12 (1 r1 )(1 r2 )+r12

2_.2 _.2
+ 2r12 (E x r2 )(n y r2 )

 

- 2 - 2 2
(1 r1 )(1 r2 )+r12

.. _,2_ 22
“(Y n)(n y r2 ) r12 r2

 

2 _ 2
v - $113“ —Y-D—2‘ ’ +
- 2 - 2 2

69

 

 

 

 

2r 2(n_y.r 2)2 r 2
+ 12 2 22 2+2n 2 12
_ - 2 2
[(1-r1)(1-r2)+r122] (1 r1 )(1 r2 )+r12
2 -. 2 . 2-
326 a2 2(x-€)2 2r12 (X 5 r1"X r2 5) 2
§§§E= -8fiD{ 2 + 2 2 2 +2nr12
r [(1-r1)(1-r2)+r122]

12

(1-ZXE)(1-r12)(1-r22)-2(x-€)(x-€°r12)-2(x-€)(x'rzz-E)

 

 

 

 

 

 

 

+
2 _ 2 2
(1-r1 )(1 r2 )+r12
- £n[(1-r12)(1-r22)+r122]}
82G _ _ a { 2(y- €)(x-n) + 2‘122‘Y'”'r12’(x’r22‘5’
Exam- 8TD 2
r12 [(1-r12)(1- r2 22)+r12 1
(-2xn)(1-r12)(1-r22)-2(x-€)(y-n-r12)-2(y-n)(X°r22-€) }
+
2 _ 2 2
(1-r1 )(1 r2 )+r12
O 2 — O 2-
32G _ _ a2{ 2(x-€)(y-n) + “122”"g r1 ) (y r2 “)
ayag 8wD _ 2 2
r122 [(1-r12)(1 r22)+r12 1

(-2y€)(1-r12)(1-r22)-2(y-n)(x-£°r12)-2(x-E)(y-rZZ-n)

_ 2 _ 2 2
(1 r1 )(1 r2 )+r12

 

 

2 , 2 ‘ 2_
a2G _ __ a 2(X‘”)2 2r12 (y-n r )(y r2 n) 2
5y5n 8ND{ 2 + + Rnr12

r12 [(1-212)(1- r2 22)+r12 )2

(1-2yn)(1-r12)(1-r22)-2(y-n)(y-n-r12)-2(ysn)(y'rzz-n)

 

_ 2 _ 2 2
(1 r1 )(1 r2 )+r12

7O

- £n[(1-r12)(1-r22)+r122]}

 

 

 

 

 

 

 

 

 

 

. - 2 _ 2 -
336v: - a2{ 3(x_€) - 2(X-§)3 - X(1-2X €)(1r2)+€(1 r1 )(1r22)
3x285 “ND r122 r122 (1-r12)(1-r22)+r122
_ (x-a)<1-2xa)-2(a-x-r22)+(x-g)r22 _ 4r122(x-€r12)(€-X°r22)2
(1-r12)(1-r22)+r122 [(1-r12)(1-r22)+r122]3
(1-ZXE)(E-X°r22)(1-r12)(1-r22)-u(x-€)(x-Sr12)(£-x-r22)
+
[(1-r12)(1-r22)+r122]2
2(x-5)(E-x-r22)2+r122r22(x-Er12)-r122(1-2x5)(i-x-rzz) }
+
[(1-r12)(1-r22)+r122]2
2 2 _ , 2 2
83G : _ a2{ (y-n) _ 2(x-g)2( —n) _ “r12 (Y'nr1 )(g X r2 )
8x23n ““D r122 r12“ [(1-r12)(1-r22)+r122]3

x(-2xn)(1-r22)+n(1-r12)(1-r22)+(y-nr12)+(x-€)(-2xn)+(y-n)r22

 

_ 2 _ 2 2
(1:3 )0 r2)+r12

+ (-2Xn)(E-x-rzz)(1-r12)(1-r22)-4(X-€)(y-nr12)(€-X'r22)+2(y-n)(E-X°r22)2

 

[(1-r12)(1-r22)+r122]2

+r122r22(y-nr12)-r122(-2xn)(E-x'rzz)

 

71

2 _ 2 _ .
33c; _ ‘az {(x-g) _ 2(y-n)2(x-£) _ “‘12 (X gr1 ”” Y ’22)2

ayzag “"9 r122 r12“ [(1-r12)(1-r22)+r122]3

y(-2y€) (1-r22)+§(1-r12) (1-r22)+(X-€r12)+(y-n) (-2y€)-i~(x-€)r22

 

_ 2 _ 2 2
(1 r1 )(1 r2 )+r12

(-2y€)(n-Y'rzz)(1-r12)(1-r22)-u(y-n)(x-Er12)(n-Y'r22)+2(x-€)(n-y°r22)2
+
[(1-r12)(1-r22)+r122]2

+1.2122r22bc—Er12)+r122 (-2y€) (n-y-rzz)

 

2 2 _ . 2
33G - - a2 {3LY'T1) - 2(y_n)3- “I12 (Y'UI'1 )(T‘l Y r2 )

2 (MD 2 u _2_2 23
By an r12 r12 [(1 r1 )(1 r2 )+r12 ]

 

 

y(1-2yn)(1-r22)+n(1-r12)(1-r22)+(y-nr12)+(y'-n)(1-2yn)-2(n-Y'r22)+(y’-n)r22

_ 2 _ 2 2
(1 r1 )(1r2)+1:‘12

(1-2yn)(n-y'r 2)(1-r 2)(1-r 2)-4(y-n)(y'-nr 2)(n-y°r 2)+2(y'-n)(n-y-r 2)2
+ 2 1 2 1 2 2
[(1-r12)(1-r22)+r122]2

22_2__2- -.2
+r12 r2 (y nr1 ) r12 (1 2yn)(n y r2 )

 

ur122(xr§r12)(€-x'r22)(n-Y°r22)

 

 

 

336 = _ _§:,{x:g _ 2(xP€)21ybn) _
axayag MD r122 r12“ [(1-1‘12) (1-r22)+r122]3

(€-X°r22)(1-2yn)(1-r12)(1-r22)-2(€-x'r22)(yhn)(ybnr12)+2(§-x-r

+

72

2 .
2 )

[(1-r 2)(1-r 22)+r12 )2

-(y-n) (n_y.r22)-2(x-§) ‘Y‘W’ (”'Y'r22)*2‘“'y'r22)”"122

 

x(r22-1)(1-2yn)+2xn(ybn)+(€-X°r 2

+

 

(1-r12)(1-r22)+r

33G = 32 {3(X‘€)_

 

 

3X3 “TTD

2(x-EV

)

2 }
2

12

3(5-X°r22)-3r22(x-£) Hr 2(E-x-rzz)3

+ + 12

 

 

2

 

 

 

 

 

 

2 2_2 23
r12 r12 (1-r12)(1-r22)+r12 [(1-r1 )(1 r2 )+r12 ]
- . 2 2 _ , 2
+ 6(5 x r2 ) (x- i) -3r12r 2 (E x r2 ) }
[(1-r12)(1-r22)+r122]2
. 2 - 2 _ 2 _ , 2 3
836 = a2 {3(ybn) - 2(y_nyg+ 3(n-y r2 ) 3r2 (y n) + ur12 (n y r2 )
ay3 “"D r122 r12 (1-r12)(1-r22)+r122 [(1-r 2)(1-r 22)+r12 13

- . 2 2- 2 2 _ , 2
6(y n)(n-y r2 ) 3r12 r2 (n y r2 )

}

 

+

33G __ 32 {y-n _

[(1-r1 2) (1 -r2 2)+r

 

4ND

axzay r

2

12 r

2(x-a) 2 (x-n) +

2
12 2]

n-y-rzz-r22(ybn)

 

u _ 2 _ 2 2
(1 r1 )(1 r2 )+r12

“(XP€)(€-x-r22)(n-y°r22)-r122r22(n-y~r22)+2(ybn)(E-X°r22)

 

+

2 2 2
[(1-r12)(1-r2 )+r12 ]

+ur122 (E-X°r22)2(n-Y°r22)

 

[(1-1:1

2 _ 2
)(1 r2 )+r12

2]3

73

 

.ifki_._éf.{§:§__ 2(ybn)2(xr§) + €‘X°r22-r22(x-a>+ ur122(fi-y-r2 2)2(g.x.r22)
(“TD 2 I. (1-r12) (1-1:22)-1-1:'_122 ”1.1.1 2) (1-r2 2 )+r

 

 

8x8y2 r

3
12 I12 12 ]

“(y-n)(n-y-r 2)(€-x'r 2)-r 2r 2(5-x-r2 2)+2(x-€)(n- °r 2)2
+ 2 2 12 2 Y 2

 

[(1-r12)(1-r22)+r122]2

2 2 _ , 2
a“c _ .43: 1_;_,+ 8(x-g)“ _ 12(x—€)2_ 2“r12 (“’521 "2 X 22 )

8x335 r 2 r126 r12“ [(1-r12)(1-r22)+r

 

 

12 1“

-(1-r22)(1—2xg)+uxg(1-r22)-2(1-2xg)+2g(x-g)—3r22

 

_2_2 2
(1 r1 )(1 r2 )+r12

_ _ 2 _ , 2 _ _ _ 2 _ 2 _ , 2 _ _ , 2 ,
+ 4X(1 r2 )(5 x r2 )(1 2x5) ug(1 r1 )(1 r2 )(E x 1'2 ) 6(5 x r2 )

°(x-€r12)-8(€-X°r22)(x-E)(1-ZX€)+6(€-X°r2 2)-12r22(x~€)(€-x-r 2)

2

[(1-r12)(1 -r2 2)+r12 2]2

“2(1-r 2)(1-r2 2)(1- -2xg)+6r2 2(x~€)(x— -§r1 2)+2r 2(1-2xg)+2r1§g(g-x-r22)

122122

u<1-2x5)(a-x-r22>2(1-r12)(1-r22)-2u<x-€)(x-ar12)<a-x-r 2) +

2
2]3

+

 

2
[(1-r12)(1--r2 )+r12

+8(x-£)(§-x-r2 2)3+12r2 r2(x gr12)(g-x r 22)- 8r1 22(1-2xg)(g-x-r22)2

12r2

 

7’4

2 - O
.226 - - a2 {-61x-a)<ybn) + 8<x~51212~n1 _ 2“22 ‘Y‘r12”"€ x 222)

“77D 1. 6
r12 r12

 

[(1-r12)(1-r22)+r122]“

8x2n(1-r22)(é-X°r22)-fln(1-r12)(1-r22)(n-x'r22)-6(ybnr1)(E-X°r22)+

 

+

+12xn(x-€)(E-x-r22)+2r22xn(1—r12)(1-r22)+6r22(x+£)(y-nr12)-6r22(y*n)°

 

[(1-r12)(1-r22)+r122]2

°(E-x-r22)-uxnr122r22+uxn(x—E)(E-x'r22)+2nr122(E-x-r22)

 

-8xn(£-X°r22)2(1-r12)(1-r22)-24(XP€)(ybnr12)(E-x-r22) +
+

 

[(1-r12)(1-r22)+r122]3

+8(y-n)(i-x'r22)3+12r122r22(y-nr12)(a-X°r22)+16r122xn(E-X°r22) }

 

- 2 _
a“G _ _ a2 {-6(Xv§)(y~n) + 8(xeg)(y.n)3 + 6y€(1 r2 )+4y€+2€(y n)

3y33€ “"9 r12“ r125 (1-r12)(1-r22)+r12

 

 

2

8y2£(1-r22)(n-y-r22)-4€(1-r12)(1-r22)(n-y'r22)-6(x~€r12)(n-y°r22)+

 

+

+12y€(y~n)(n-y°r22)+2y€r22(1-r12)(1-r22)+6r22(y~n)(x-Er12) -

 

[<1—r12)(1-r22)+r12212

-6r22(x-€)(n-y-r22)-uy£r122r22+uyg(ybn)(n—y-r22)+2£f122(n-y-rzz)

 

75

+ -3y€(1-r12)(1-r22)(n-y-r22)2-2u(y-n)(x—Er12)(n-Y'r22) +

 

[(1-r12)(1-r22)+r122]3

__,23 22_2_.2 2 .2
+8(x €)(n y r2 ) +12r12 r2 (x €r1 )(n y r2 )+16r12 y£(n-y r2 )

 

_ 24r122(x-£r12)(n-Y'r22)

 

[(1-r12)(1-r22)+r122]“

_ 2
.ifigi__.=z- _E:.{- 6(X-€)(y-n) + 8(x—£)3(y-n) + 23"“1 r2 )+2Y€
axzayag MID r12“ 1.126 (1'1'12) (1-r22)+r122

 

 

 

-2x(1-2x€)(1-r22)(n-y°r22)-2€(1-r12)(1-r22)(n-y-r22) -
+

-2(x-€r12)(n-y°r22)-2(x-€)(1-ZXE)(n-y°r22)+u(€-X°r12)(n-y°r22) -

 

[(1-r12)(1-r22)+ r122]2

-2 (x—51r22 (n-y-r22)-2y(1-ZXE) (a-x-r22) (1-r22)+8y€(x-€) (a—x-r22)+

 

+2(y-n)r22(Xrir12)-2r122r22y£-2(ybn)(1-ZXE)(E-X°r22)

 

+

“(1-2xfi)(E-X°r22)(1-r12)(1-r22)(n-y°r22)-16(x-£)(x-Er12)(€-X°r22)°

 

-(n-y°r22)+8(x-€)(é-X°r22)2(n-y-r22)+ur122r22(x-Er12)(n-y°r22)-

 

76

 

 

 

 

 

-ur122(1-2x€)(E-x-r22)(n-Y'r22)-8(ybn)(Xvé r12)(€-x-r22) +
[(1-r12)(1-r22)+ r122]3
+8r122y€(€-x-r22) - 24r122(x-€r12)(g-x-r22)2(n-y-r22)
[(1-r12)(1-r22)+ r122]“
2 , 2
a“G = _ a2[ 3 _ 12(z-n)2+ 8(xrnf‘_ 2Qr12 (Y'm’122m'y r2 )3
ay3an “"D r122 r12“ r126 [(1-r12)(1-r22)+ r1221“

-(1-r22)(1-2yn)+4yn(1-r22)-2(1-2yn)+2n(y-n)-3r22

 

- 2 - 2 2
(1 r1 )(1 r2 )+ r12

-4y(1-2yn)(1-r22)(n-y°r22)-0n(1-r12)(1-r22)(n-Y'r22)-6(n-y°r 2)°

+ 2

°(y-nr12)-8(y-n)(1-2yn)(n-Y'r22)+6(n-y-r22)-6r22(ybn)(n-y-r22)-

[(1-r12)(1-r22)+ r122]2

-r22(1-2yn)(1-r12)(1-r22)+6r22(y~n)(ybnr12)+2r1§ r22(1-2yn)+2r122(n-y'r22)

4(1-2yn)(1-r12)(1-r22)(n-y-r22)2-2“(ybn)(ybnr12)(n-y-r22)2+8(y-n)°
+

 

[(1-r12)(1-r22)+ r12213

°(n-y°r22)3+12r122r22(y~nr12)(n-y-r22)-8r122(1-2yn)(n-y-r22)

 

}

77

3"G = .. .2: { -1 + 8(x-€)2(y-n)2 + 2yn(1-r22)-(1-2yn)-r22

("TD 2 5
r12 r12

 

 

szayan

_ 2 _ 2 2
(1 r1 )(1 r2 )+ r12

4x2n(1-r22)(n-y-r22)-2n(1-r12)(1-r22)(n-y°r22)-2(ybnr12)(n-y°r22)+

 

+

+uxn(x-£)(n-y-r22)-2r22(ybn)(n-Y'r22)+4xyn(€-x'r22)(1-r22)-

 

[(1-r12)(1-r22)+ r122]2

-u(x-€)(E-x'r22)(1-2yn)+2(€-x'r22)2+2(y-n)r22(y-nr12)+

 

+r122r22(1-2yn)+4xn(y-n)(E-X'r22) + -8xn(€-X°r22)(1-r12)(1-r22)°

 

'(n-y-r22)-16(x-€)(y-nr12)(€-X°r22)(n-y'r22)+8(ybn)(E-X°r22)2°

 

[(1-r12)(1-r22)+ r122]3

, ,2 22_ 2 _,2 2 _,2 _,2_
(n-y r2 )+4r12 r2 (y nr1 )(n y r2 )+8r12 xn<€ x r2 )(n y r2 )

 

-8(ybn)(ybnr12)(E-X°r22)2- “r122(1-2yn)(€-x-r22)2

 

2 _ 2 _ , 2 2 _ , 2
- zur12 (y nr1 )(E x r2 ) (n y r2 )

 

[(1-r12)(1-r22)+ r1221“

78

- 2 - - _ 2
3‘06 = - .2: {- 1 + 8(x_€)2(Jy_n)2 + 2X€(1 r2 ) (1 2X5) r2
“ND
2 2
12

 

 

_ 2 _ 2 2
(1 r1 )(1 r2 )+ r12

“y2£(1-r 2)(€-x-r 2)-2£(1-r 2)(1-r 2)(€-X°r 2)-2(x-€r 2)(E-x-r 2)+
+ 2 2 1 2 2 1 2

 

+

uy€(y-n)(E-x'r22)-2r22(x-€)(E-x-r22)+ux'y€(1-r22)(5-y°r22)-

 

[(1-r12)(1-r22)+ r12212

«(y-n)<1—2ax)(n-y-r22)+2(n-y-r22)2+2(x-€)r22(x-ar12)+

 

+

r122r22(1-ZXE)+uy€(x-€)(n-y°r22) + -8y€(n-y°r22)(1-r12)(1-r22)°

 

 

(E-x'r22)-16(y-n)(xrir12)(n-y°r22)(E-x-r22)+8(x-£)(n-y-r22)2°

 

[(1-r12)(1-r22)+ r122]3

(E-X°r22)+“r122r22(X-Er12)(E-x-r22)+8y€r122(n-y°r22)(E-x-r22)-

 

- 8(X-E)(x-Er12)(n-y-r22)2-ur122(1-2x£)(n-y°r22)2

 

_ 24r122(x-§r12)(n-y°r22)2(€-x-r22)

 

[(1-r12)(1-r22)+ r122]“

79

- 2
__§16__ = - .23. {- 6(x-E) (y-n) + 8(x-n) 3(x—5) + “2‘“ 2"”2'2
axayzan ”"D r12“ r126 (1-r12)(1—r22)+ r122

 

 

+ -2y(1-2yn)(1-r22)(E-X°r22)-2n(1-r12)(1-r22)(E-X'r22)-

-2(y-nr12)(E-X'r22)-2(y-n)(1-2yn)(E-x'r22)+4(n-y°r22)(E-x'r22)-

 

2 2 2 2
[(1-r1 )(1-~r2 )+ r12 ]

-2r22(y-' (i-x'r22)-2x(1-2yn)(n-y°r22)(1-r22)+8(y-n)xn(n-y'r22)+

 

+2r22(x-€)(y-nr12)-2xnr122r22-2(x-E)(1-2yn)(n-Y'r22)

 

4(1-2yn)(n-y°r22)(1-r12)(1-r22)(E-x-r22)-16(y-n)(y~nr12)-
+

 

-(n-y°r22)(E-X°r22)+8(y-n)(n-y-r22)2(5-x-r22)+4r122r22(y-nr12)°

 

[(1-r12) (1-r22)+ r122]3

°(€-x-r22)-Ur122(1-2yn)(n-y-r22)(E-x-r22)-8(Xr§)(y-nr12)-

 

-(n-y-r22)2+8r122xn(n-y-r22) _ 24r122(y~nr12)(n-y-r22)2(€-x-r22)

 

 

}
[(1-r12) (1-_-r22)+ r122]“

APPENDIX B

COMPUTER PROGRAM FOR THE BOUNDARY INTEGRAL METHOD

APPENDIX B

COMPUTER PROGRAM FOR THE BOUNDARY INTEGRAL METHOD

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APPENDIX C

COMPUTER PROGRAM FOR THE POINT-FORCE METHOD

(writ!rwrwrmrarvrarrrarrr)rwrar)r1r)r)r)r1rvrararararwracar)

C
C

C
C
C

APPENDIX C

COMPUTER PROGRAM FOR THE POINT-FORCE METHOD

PROGRAM NENPLCLlINPUT.OUTPUT.TAPE531NPUT.TAPE6=OUTPUT)

000001
000002

l'l'l‘!OHI!Oi.'IUQGQCIOIHQIlflflifiilflfl§lflfllilflININCH.IGNIHIUIIGIIIUIlfifilo00O03

POINT FORCE HETHOD FOR ISOTROPIC PLATE BENDINS PROBLEHS *'**l

ARBITRARY PLAN FORH. TRANSVERSE LOAD. AND BOUNDARY CONDITIONS

REQUIRED INPUT VALUES ---

NEP =NUNEER OF BOUNDARY POINTS

NIP =NUHBER OF INTERNAL LOAD POINTS

NFP =NUHBER OF FIELD POINTS

XB.YB =POINTS ON 8 AT HHICH B.C. ARE SATISFIED.

BANX.BANY =COHPONENTS OF UNIT NORHAL TO B AT X89YB.

XXBoYYB =ENO POINTS OF HESHES AROUND B HHERE FICTITIOUS
FORCES ARE ASSIGNED.

XF9YF =FIELD POINTS

XIgYI =INTERNAL LOAD POINTS

PR =POISSON'S RATIO

EVALUE =YOUNG‘S HCDULUS

HVALUE =PLATE THICKNESS

RADIUS =RAOIUS OF THE FICTITIOUS CIRCULAR PLATE OF HHICH
THE DISPLACEHENT AT THE CIRCUHFERENTIAL BOUNDARY
IS SET TO ZERO.

NBTYPE =BOUNDARY CONDITION TYPE AT EACH BOUNDARY POINTS
NBTYPE = I --- CLAHFED
NBTYPE = 2 --- SIHPLY SUPPORTED
NBTYPE = 3 --- FREE

000004
000005
000006
000007
000008
000009
000010
000011
000012
000013
000014
000015
000016
000017
000018
000019
000020
000021
000022
000023
000024
000025
000026
000027
000028
000029
000030
000031

fi!!!‘!l‘ll§illl§ilfl§i§hifiHiilIlllfiili‘iifllliiilflfifiIfiillfil‘lfliilllilliloo0032

DIHENSION X8140).YB(40)cxxatel).YYB(81)pNBTYPE(40)
OIHENSION XI(100)9YI(100)oDEL12)

DIHENSION RB(60)oRN(80980)oPS(80).NKAREA(80).RL(80)
DIHENSION XF(181ngFl181).H(101):BHX(181)9BHY(1811
DIHENSION BANXl401pBANYl40)

100 FORHATIIOLOCTiabxoSXBI98X.‘YBF97X9‘ANX‘98XsFANY‘oQXoFNBTYPE‘.
O/‘OI/(1594F10.491511

110 FORMAT(ROLOCTI.IIXoFXXBP,I4X9FTYB!/¥01/(I4.8X.F9.2.8X.F9.2l)

200 FORMAT (IDLOCTi.11X.‘¥I‘TITXo’YIFpI4Xo/101/
OIIQvIIX9F6.2oIIXOF6.3))

300 FORMAT IPOLOCTPp19Xo’RLD‘.34X7‘RL3P/POF/fIQaOXvEZO.8u16X9
1E20.8))

400 FORMAT IPILOCT’919X9PPSP3/‘OP/lI4.8X.E20.O)1

500 FORHAT(FOLOCT’oIIXo‘XFIv17X’PYFI/3OF/T14911X9F6.3,11X.F6.3))

600 FORfiAT(IHIoGXo'NODE‘.IZXFXF‘916X9'YF’919X9PH‘.16XyPBHXFp
116X.iBHYI/(1H0.IID.2F20.1093E20.12))

700 FORMATllHloPINPUT VALUES .....‘u//1X.1NBP 3 ‘oI3oP NFP 3P:I3o
9‘ NIP 3 P9139‘ PR 3 ‘uF5.39
I/IHOoiYOUNGS HOOULUS 3 PoEI4.891 RADIUS OF THE PLATE 3 P:
6F7.1.¢ THICKNESS OF THE PLATE 839F6.3)

INPUT VALUES ..

READ(59')N3°oNIP.NFPoPR,EVALUEvRADIUSpHVALUE
NRITE(6.700)HOP.NFP.NIP.PRoEVALUEpRADIUSgHVALUE

90

000033
000034
000035
000036
000037
000038
000039
000040
000041
000042
000043
000044
000045
000046
000047
000048
000049
000050
000051
000052
000053
000054
000055
000056
000057
000058
000059

C

91

READIS.‘)(XB(I):I=19NBP)

READ‘STR1TYBTIToIglvN3P1
READlSoPllNBTYPETI1.1319N3P)
READ!5.')(BANX(11:1319NBP)
READ(59')‘BANT(I)91310NBPT
"RITETOuIOOTlIsXBTI’oTB1IToBANXlITTBANT(I)oNBTTPE(I)

191:19N5p1

C ASSIGN LOCATIONS OF FIELD POINTS. (XFoYF).

C

C

41

42

43

44

X0=5.08Y0=5.0
XFII)=10.-XOOYF(1)=-10.¢Y0
00 41 1:2.11
XF(I)=XF(I-1)-O.5
YFtI)=YF(I—1)

DO 42 J=1.10

K=J'17

DO 42 1:1.11
XFIIoKT=XF(I)
YFtIoK)=YF(I¢K-17)+1.0
XFIIZV=10.-XOQYF(12):-9.S¢YO
DO 43 1:13.17
XF(I)=XF(I-1)-1.0
YFII)=YF(I-1)

00 “Q J=199

K=JI17

00 44 1:12.17
XF(I+K)=XF(I)
YF(I+K)=YF(I*K-17)¢1.0
READ(5.*)IXI(1).I=1.NIP)
READ(5.*)IYI(I).I=1.NIP)

C ASSIGN LOCATIONS OF FICTITIOUS FORCES. (XX8.YY8).

C

25

26

27

NBP2=NBPl2$NBP2P1=NBP2*1
NBPP1=NEP+1

DIST1=4.0 0 DIST2=2.0
OEL11T=I10.02.lDIST1)/10.
DEL!2)=(10.42.'DIST162.*DIST21/10.
XXB(1)=5.¢DIST1-DEL(1)8YYB(1)=-5.-DIST1
XXBI41l=5.+DIST1+DIST2-DEL(2)0YYB(41)3-5.-DIST1-DIST2
DO 28 J=1.2
DELT=DEL(1)0IF(J.EO.2TDELT=OEL(2)
DO 25 I=2o10

K=I

IF!J.EO.2)K=KO40
XXBtK)=XXB(K-1l-DELT
YYBtK)=YYBtK-1)

DO 26 1:11.20

K=I

IFtJ.EG.2)K=K+40

XXBIK)=XX8(K-1)
YYE(K)=YY8(K-1)ODELT

DO 27 I=21.30

K=I

IF!J.EO.2)K=K§40
XXBIK|=XXBlK-1)4DELT
YY8(KT=YT8(K-1)

DO 28 1331940

K=I

000060
000061
000062
000063
000064
000065
000066
000067
000068
000069
000070
000071
000072
000073
000074
000075
000076
000077
000078
000079
000050
000081
000082
000083
000084
000085
000086
000087
000088
000089
000090
000091
000092
000093
000094
000095
000096
000097
000098
000099
000100
000101
000102
000103
000104
000105
000106
000107
000108
000109
000110
000111
000112
000113
000114
000115
000116
000117
000118
000119

28

C SET

C
C FOR
C

17

C
C FOR
C

92

IFIJ.EO.2)K=KO40

XXB!K)=XXB(K-1)

YYBTKD=YYB(K-11-DELT

XXBC8118XXB(41) 8 YYB(811=YY8(41)
NRITE(6v110)(IoXXB(I).YYB(I).I=19NSP2P1)
NRITE(6o500)IIoXF(IToYFlIioI=1.NFP)
NRITE!6:200!(I.XI(I),YITI).I=1oNIP)
DPLATE=EVALUEPHVALUE"3/(12.i(1.-PR"2)I
PI=4.'ATAN(1.)

COEF1=1./(16.'PI*DPLATE)

COEF2=COEF1'2 8 COEF4=COEF2"2 8 OLOAD=1.0 8 R2=RADIUS*'2

UP THE RL VECTOR AS SHOHN IN APPENDIX A.

DO 5 I=1oNBP

HL‘I1=D.O

HL(I9N?P1=O.O

N1X=XBII1

R1Y=YBIII

ANx=BANXII1

ANY‘BANYII)

00 6 leoNIP

R2X=XI(J)

R2Y=YIIJ1
Z13R1X-R2X8223R1T-R2Y8R1233Z19'2412992
,zs=ALostnle/n2)
1F(N3TYPE(IJ.EQ.3)GOT0 18
RL(I1=RL(I)-°LOAO'(R12$'ZS1
IF‘NBTYPEII1.EG.1)GOTO 17

SIMPLY SUPPORTED EDGE: ONLY ---

ZIS=ZIPI2 O ZZSIZZ'IZ 0 ANXS=LNXII2 s ANYS=ANYIfi2
PLII¢NBPJ=PL(I#NBP)-OLOAD!(l2.§215/P125¢1.025)IANXS
044.'21!22/R125*ANXGANY¢(2.4223/P12361.025)IANYS)
SOTO 6

CLAHPED EDGE: ONLY ---

RL!IQNBP)=RL(IONBP)-0L0AD'lZlGANXoZ2iANY)fil1.+25)
GDTD 6

FREE EDGES ONLY ---

18 215821992 8 228822992 8 ANXSSANX'IZ 8 ANYS=ANY992

6

R1288=R12$*!2
RLII1=RL(Il-OLOADRT(ANXSOPRRANYS)‘(2.'Z1S/R12541.625)
99(4.l(1.-PR)CANXPANY)lZl'ZZ/RIZS
ootANYSoPR'ANXSTIIZ.PZZS/R12801.925))
CISANXUT1.OANYS!I1.-PR))
C2=((2..PR-1.TOANXS612.-PR)GANYSIIANY
C3=((2.IPR-1.)GANYS4t2.-PR)PANXSIIANX
C4=ANYnI1.¢ANXSuI1.-PP))
RL‘10N8P33RLTI4NBPI-DLOADIICIPZIOI21543.5223)/R1255
OOCZ'ZZ"225-2131/R12350C3'21'I113-2231/R1255

0OC4'ZZ'(228921893.DIRIZSS)

CONTINUE

5 CONTINUE

C

C SET UP THE RH HATRIX AS SHONN IN APPENDIX 8, AND NOTE THAT

000120
000121
000122
000123
000124
000125
000126
000127
000128
000129
000130
000131
000132
000133
000134
000135
000136
000137
000138
000139
000140
000141
000142
000143
000144
000145
000146
000147
000148
000149
000150
000151
000152
000153
000154
000155
000156
000157
000158
000159
000160
000161
000162
000163
000164
000165
000166
000167
000168
000169
000170
000171
000172
000173
000174
000175
000176
000177
000178
000179

93

C FICTITIOUS FORCES ARE LOCATED AT THE CENTER OF THE ASSIGNED HESHES.

C

C

C SOLVE THE SET OF LINEAR EQUATIONS TO DETERHINE THE FICTITIOUS FORCES.

C

C
C COHPUTE DISPLACEHENTS AND SENDING

C

DO 8 I=1TNSP

R1X=XB(I)

R1Y3YBTI)

ANX=8ANX(I)

ANT=8ANY(I)

DO 7 J=1pNEP2
R2X=IXXBTJOIT¢XXB(J))/2.
R2Y=TYY8TJ¢11¢YYB(J))/2.
IF(NOPTION.EO.2.0R.J.NE.NBPDGOTO 33
R2X=(XXBI1)¢XXB(NBP))/2.
R2Y=(YY8(1)¢YYBIN8P))/2.

33 CONTINUE

11=R1X-R2X822=R1Y-R2Y8R123=Z1"2422'*2
25=ALOG(R12S/R2)

IFlNBTYPE(I).EO.3)GOTO 19

RHleJ1=R12S'ZS

IF(N8TYPE(I).EQ.1)GOTO 20

115321'l2 8 225322'82 8 ANXS=AN¥092 8 ANYS=ANY912
RNTIONBPpJT=t(2.”215/R12501.+25)'ANXS
664.‘Z1'ZZ/R125'ANX'ANT§(Z.PZZS/R12501.OZS)'ANYS1
SOTO 7

20 RH!IoNBPgJ)=(21*ANXOZZ'ANY)l(1.425)

SOTO 7

19 215321992 8 223322982 8 ANXS=ANX992 8 ANYS3ANYil2

PIZSS=P12$P§2
RH!I.J)=((ANXS+PRIANYS)'(2.9215/R12501.025)
06(4.'(1.-PR)GANX4ANY)*ZIPZZ/R125
00(ANYSOPRVANXSIfit2.'Z2S/R125*1.025)l
C1=INxvll.oaNY3Itl.-PR))
C2=((2.|PR-1.)lANXS¢(2.-PRlflANYS)lANY
C3=(12.-PR-1.liANYS#(2.-FR)IANXS)'ANX
C4=ANYit1.¢ANXSI(1.-FP))
PH(IoNBP.J)=(C1!21¥(21503.!225)/P1255
¢0C2522!(zzs-213)lPlzssoC3421fit213-225)/P1255
+0C4'zz'(225021$l3.l/PIZSS)

7 CONTINUE

O CONTINUE

CALL LEOTIFTRH.19NBP2oNBPZoRLTOTNKAREAoIER)
DO 29 1:1!NBP2

29 PSIIT=RLII1

HRITE (6.400) (I.PS(I).I=1.NBP2)
DO 9 1:1,15
HtI)=D.0
DHXTIT=0.D
DHY(I)=0.0
9 CONTINUE

DD 14 1:1.15
l1x=xrltl
IIY=YPTIT
DD 13 J=1.NIP
R2X=XI(J)
PZY=YIIJT

000180
000181
000182
000183
000184
000185
000186
000187
000188
000189
000190
000191
000192
000193
000194
000195
000196
000197
000198
000199
000200
000201
000202
000203
000204
000205
000206
000207
000208
000209
000210
000211
000212
000213
000214
000215
000216
000217
000218
000219
00022

000221
000222
000223
000224
000225
000226
000227
000228
000229
000230
000231

HOHENTS AT THE PRESCRIBED FIELD POIN000232

000233
000234
000235
000236
000237
000238
000239

94

Zl=RlX-R2X822=R1Y-R2Y8R125=21I!2022**2

215:21882 8 225=ZZPPZ

ZS=ALOG(RIZS/R2)

N11)=N(1)oCOEflltRIZS'ZSDlOLOAD

26:1.62.4215/R125*25 8 Z7=1.¢2.I225/R12$+ZS
Z8=DPLATEICOEP280LOAD

EHX(I)=BHXTI)-Z88(Z6OPR!Z7) 8 BHYtI)=8HY(I)-28*(Z74PR*ZGl
IFTI.GE.2)GOTO 13

13 CONTINUE

14 CONTINUE
00 16 1:1.15
R1X=XFTIT
R1Y:YF(I)

DO 15 J=1.NBP2
R2X=tXXBtJ¢1l4YXBtJl)/2.0
R2Y=1YYBTJ¢1)¢YYB(J))/2.D
IFTNOPTION.EG.2.0R.J.NE.NBP)GOTO 11
R2X=IXX8(1)¢XXB(NBP))/2.
R2Y=lYYBI1T+YYBtNBP))/2.

11 CONTINUE
21=R1X-PZX822=R1Y-R2Y8R125=Zl§82+22*I2
ZS=ALDGlR125/R2)

ZlS=Zl¥92 8 225:22ll2
H(1)=N(I)+F5(J)ICOEF18(R125525)
ZO=1.92.'Z15/R123415 8 Z8=1.¢2.4225/R125+25
211=PS(J)iCOEF2lZ6 8 212=PSIJTPCOEF2828
BHX!Il=8HXTI)-0PLATEl(2110PR8212)
8HY(IT=EHYTID-DPLATEG(212¢PR8211T

15 CONTINUE

16 CONTINUE
NRITE(6.6DO)(I.XF(I).YF(1).N(I).8HX(I).BHY(1).I=1.15)

END

4D.1DD.181.D.3.3D.E6.80..D.4
“.563.592.5’1.590.59’°.59'1.50'2.59'3.59'4.5610N'5.o
-4.5,-3.5:-2.59-1.So'0.590.591.5:2.593.5v4.5910*5.
109-5.9-4.59'3.59'2.59'1.59-0.590.591.5’2.593.594.50
10.5.94.593.5p2.591.59°.59’0.59'1.59'2.59'3.59'“.5
4092

10‘0.91°P'1.91°'0.91°'1.

10' 1 )10‘0..1°'1.61°PO.

O. 563.592.59 1 5.0.5.-0.5.-1.5.-2.5.-3.5.-4.5.
9.503.592.591. 590.59'°.59'1.59'z.59'3.59'4.59
4.503.582.591 59° 59-0.50“!.50’2.58‘3.59-“.59
4.593.592.591. 58° 5,-0.59-1.5.-2.5.-3.5.-4.5.
4.593.562.501 59°. 58-0.59'1.59'2.59'3.59'“o59
‘.5o3.502.591.59O.59‘°.53°1.50’2.56'3.59'4.59
4.5.3.5.2.5.1. 5 0. 5.-D.5.-1.5.-2.5.-3.5.-4.5o
4.5.3.5.2.5.1. 5 D. 5.-0.5.-1.5.-2.5.-3.5.-4.5,
‘.5p3.502.591 590. 59-0 59'1. 5,-2. 58-3. 59.6.5:
O.593. 582.561 590. 59’0. 59'1. 5": 56-3. 5"“.59
108-4. 5.108-3.5.10'-2. 5.108-1. s.10u-o.s.
1010.5.1081.5.10'2.5.1043.5.10'4.5

000240
000241
000242
000243
000244
000245
000246
000247
000248
000249
000250
000251
000252
000253
000254
000255
000256
000257
000258
000259
000260
000261
000262
000263
000264
000265
000266
000267
000268
000269
000270
000271

000273
000274
000275
000276
000277
000278
000279
000280
000281
000282
000283
000284
000285
000286
000287
000288
000289
000290
000291
000292

APPENDIX D

DERIVATIVES OF THE GREEN'S FUNCTION
OF AN INFINITE ORTHOTROPIC PLATE

APPENDIX D

DERIVATIVES OF THE GREEN'S FUNCTION
OF AN INFINITE ORTHOTROPIC PLATE

 

 

 

For ; > 1.0:
’ _ 32 _ 2 _ 2 _2 fl. 2
X uwDOmZ-AZ) a2 a2 .

+ 5{(x-£)f+>\2(y-n)2}_ mac—5) +8 j-TUZ} “ng y-..” arc t9 BEE/7:)
(x-i)‘+kz(y-n)2 (x-£)2+B2 (y-n)2

- arc tg M¥:%)]- 3(x-5.) (B-M}

ac;_ £2 (x-E)2+82(y-1'1)2 _ (x-€)2+A2(y-m2
a—y- (”Do 8(81- AT) {(y- n)[82n a2 Min a2

 

){(X‘E) 2+)? (Y'n) 2} +8{ (X'g) 2+82 (Y'n)2 } ] +2 (X’E) [arc tg_(.L._
(x-E) +Xz(y-n)2 (x-€)2+Bz(y-n)

- arc tg L331]- 3(y-n) (B-M}

95

96

x-E I B _ 1
9x3 21TDO(82-A2) (x-€)2+>\2(y-n)2 (x-€)2+B2(y-n)2

83

x3 ]
(x-E) 2+B2 (y-n) 2

(x-E) 2H2 (y-n) 2

3:9: «Zn-n)
3y2 2nDO(sz—A2)

 

 

 

 

 

 

 

a 3G -e2 (y-n) [ B .. 2‘

3x23y 21:90 (82-12) (x-g) 24.82 (y-n) 2 (x-g) 2+A2 (y-n) 2
a 3G -e2 (x-g) [ A _ B

8x8y2 ZwDo (82-12) (x-E) 2+A2 (y-n) 2 (x-n) 2+B2 (y-n)

For p < 1.0:

BG

3‘; = 1 {£an (x-E) “+20€2(x-€) 257-71) 2+6”(y-n)" _ 6]

I;
161rDO 111 a

+ 2 [(x-E) 2+£2(1-n) 2] [(x-E)3 +062 (x-E) (y-n)2] _ 2(x-€)arc
“1 [(x-E) “+2062 (x-C) 2 (y-n) 2+6“ (y-n) “ ] “2

_ _ 2
+ “(TOR SKY-D) 2 . “Uluzbt 5) (Y n)

 

 

u2 Kx-€)2+p€2(yhn)2]2+[2u1u2(y~n)2]2
_ 62(x-n) Rn u12(yhn)2+[(x-€)-u2(ybn)]: -
u u

1 2 1112 (y-n) 2+[ (x-ng (y-n)] 2

_ 2
2111112 (y n)

 

tg

(x-€)2+pe2 (y-n) 2

97

_ 52 (X-E) (y-n) 2{ (x-E) -u2 (y-n)} 2{ (x-€)+u2 (y-n) }

“1“: 1112 GMT) 2+{ (x-€)-u2 (y-n)]2 1112(y-n) 2+{ (x-«SHIJ2 (y-n)}2

 

 

1}

39.9. = 1 {_€2(y_-n).[£n <x—5)“+2oez(x-z)zg-n>2+€“(Y'm‘l 6]
3y 1671130 111 - a.2

 

+ 2{(x-£)2+€2(y-n)2}{0€2(x-€)2(x-n)+e“(y-n)2}+ 262(x-n) ,

u1{(x-€) “+2062 (x-E) 2 (y-n) 2+6“ (y-n) “} 112

- 2 - - 2
2u1u2(y n) “uluzw n) (x E)

_ (x-E) 2-e2fl-n)2,
(x-g) 2+p€2 (y-n) 2 112 [(x-§)2+oe2(y-n)2 J2? [Zulu2 (y-n)2]2

 

°arc

52(x-5),m “12(Y'”)2+ [(x-g)-u2(y-n)] 2_ e2(x-€)fl'n) .
111112 1112(y-n)2+[(X-€)+U2(Y’“)]2 “I“:

 

 

 

2U12(Y"T1) --2112{(x-€)-u2 (y-n)} 2u12(y-n) +2u2{ (x-E) +112 (y-n) }

 

 

1112 (y-n) 2+{ (3:49-112 (y-n) }2 1112 (17-71) 2+{ (X'§)+IJ2 (y-n) }2

336 = _1_ { 1_ (x—€)2+pe2(x-€) (y-n)2
3 ”ND Ll

3x 0 1 (x-g) “+2062 (x-E) 2 (y-n) 2+6“ (y-n) 2

 

2u1(x+€)(y-n)2

 

+
[(x-C) 2+oc2 (y-n) 2] 2+[2u1u2 (y-n) 2] 2

326 _ 1 1 062 (x-E) 2 (y-n) + e“ (y-n) 2
a"? " “ms 2 r 2’

Y o 1 (ac-2;) “+2er (ac-g) 2 (y-n) 2are“ (y-n) “

33c;
8x28y

For p

9.9:
8y

98

21: Wm) (x-E) 2
+ 1

 

[(x-é) 2+oe2 (y-n) 2] 2+[2u 1112 (y-n) 2] 2

 

 

 

 

 

- £2 { (X’€)"U2 (y-n)
8WDOu1u2 1J12(y_n)2+[(x.g)-p2(y’-n)]2
(x—EHu (y-n)
_ 2 }
ul2(y-n) 2+[ (X'E) +112 (y-n) 12
62(y-n)-u (X‘E)
= ‘ 52 { 2
8111301212212 1112 (y_n) 2+[ (x—E)-u2 (y-YTH
e2 (y-n)+u2 (X-E)
u12(y-n)2+[(X'€)+“2(Y'”)]
= 1.0:

(X-E) {2n (X-§)_2+€2(J-n) 2_ 2}

BNEDO a2

6 (x-n) 2n (x-E) 2+e2 (y-n) 2
811DO a2

1 (x-F.) 2-82 (y-n) (x-é)
”"5120 { (x-E) 2+e2 (y-n) }2

99

3 3G e (y-n) 3 (x-E) 2+rs2(y-T1)2

 

 

 

 

 

 

23y3 “mo (x-EJ) 2+€2 (rm) 2

826 = 1 {52(y-n)2-(x-€)2}8(y-n) 2

8x23y “Do 2 2 2
{(x-€)2+€ (y-n) }

826 = 1 (x-§)2-e2(j-n)2€(x-€)

3x8y2 “Do {(x-i) 2+€2 (y-n) 2 } 2

APPENDIX E

COMPUTER PROGRAM FOR AN ORTHOTROPIC PROBLEM

(TCTfoT(TCTCTCTCTCTFTCTFTCTCT(if)f1f1f9f3f3f9f9f8f8f8f8f‘f1

APPENDIX E
COMPUTER PROGRAM FOR AN ORTHOTROPIC PROBLEM

PROGRAM ORTPLCLIINPUT.OUTPUT.TAPE5=INPUT.TAPE6=OUTPUT)

000001
000002

NHHRHGINIIIINNINNIUNNNNNRRINUIUNMNNNUNRNNNUNRIN'UIUNNIIINNNRNNNNNUNNHNO00003

POINT FORCE METHOD FOR ORTHOTROPIC PLATE SENDING PROBLEMS.
ARBITRARY PLAN FWM. TRANSVERSE LOAD. AND BOUNDARY CONDITIONS '9'

SHONN HERE IS AN EXAMPLE FOR A SIMPLY SUPPORTED SQUARE PLATE.

REQUIRED INPUT VALUES ---

NBP =NUMBER OF BOUNDARY POINTS

NIP =NUMBER OF INTERNAL LOAD POINTS

NFP .=NUMEER OF FIELD POINTS

XB.YB =POINTS ON 8 AT NHICH B.C. ARE SATISFIED.

BANK.BANY =COMPONENTS OF UNIT NORMAL TO B AT XB.YB.

XXB.YYB =END POINTS OF MESHES AROUND B NHERE FICTITIDUS
FORCES ARE ASSIGNED.

XF.YF =FIELD POINTS

XI.YI =INTERNAL LOAD POINTS

VX =POISSON*S RATIO IN X DIRECTION.
DUE TO STRESS IN Y DIRECTION

EX.EY =YOLRTG'S MODULI IN X AND Y DIRECTIONS.
RESPECTIVELY

HVALUE 8PLATE THICKNESS

RADIUS =RADIUS OF THE FICTITIOUS CIRCULAR PLATE OF NHICH

THE DISPLACEMENT AT THE CIRCUMFERENTIAL BOUNDARY
IS SET TO ZERO.

000004
000005
000006
000007
000008
000009
000010
000011
000012
000013
000014
000015
000016
000017
000018
000019
000020
000021
000022
000023
000024
000025
000026
000027
000028
000029

.I.ININIUI'IINNUIUINUNIINIIINNIQIIINNIIUDHINNNINNINNNMN‘NINUNININNINIIOOOO30

DIMENSION XE(40).YB(40).XXBIE1).YYB(81).NBTYPE(81)
DIMENSION XI(100).YI(100|.DEL(2)

DIMENSION RLXl80).RB(80).RM180.80I.PS(80).NKAREA(80).RL(80l
DIMENSION XF(181).YF(181).H(181).EMX(181).BMY(181)
DIMENSION BANX(40).BANYI40).RMX(80.80l

REAL LUHDA.LUMDA2.MU1.MU2

100 FORMAT(‘0LOCT3.6X.‘XBF.8X.!TBF.7X.FANX‘.8X.FANY3.4X.8NBTYPE8.
O/POP/(1595F10.991511

110 FORMAT(10LOCT8.11X.‘XXB3.14X.FYTB8/101/(I4.8X.F9.2.8X.F9.211

200 FORMAT I80LOCT3.11X.‘XIF.17X.FYIP.14X./808/
O‘I“!11X1F6.2011X9F6.311

400 FORMAT (IDLOCTI.19X.$PSP¢/¢03/(I4.BX.E20.B)1

500 FORMAT!¢0LOCTF.11X.’XF8.17X.FYF$/¢08/II4.11X.F6.3.11X.F6.3)l

600 FORMATI1H1.BX.FNCDE1.12X1XFF.16X.FYF‘.19X.*H1.16X.FBMX1.
116X.!BMYl/(1H0.I10.2F20.10.3E20.1211

700 FORMATIIH1.’INPUT VALUES .....8.//1X.8NML 3 ‘.I3.8 NIP =F.I3.
1t NFP 3 8.13.8 PR 3.F5.3.8 DPSI 3 F.F5.3.P DETA 3.F5.3.
4/1H0.‘ EX 8 3.E10.4.F ET 3 ‘.E10.4.1 VX = '.F5.2.3 GXY F.E10.4
6/1H0.‘ RADIUS OF THE PLATE 3 I.
OF7.1.I THICKNESS OF THE PLATE =1.F6.3.1

1103 FORMATTI5.5X.E20.12.5X.E20.121

1014 FORMAT(1H1.PTHE FOLLONING IS A LIST OF DOUBLE CHECKING OF B.C.S‘.
1//1X.!NOTE ....
1 E-9 OR LESS. RESULTS HILL BE IN GOOD SHAPE.1.
1//.4X8NHL810X.8B.C. 1‘913X.PB.C. 2F.//1 .

701 FCRMATI1H1.FDX = P.E15.6.‘ DY = F.E15.6.F H = P.E15.6.1 G = F.
9E15.89P VT 3 P9F5.2.P RHO 3 ‘8'6-2,

100

DIST: !.F5.1)

000031
000032
000033
000034
000035
000036
000037
000038
000039
000040
000041
000042
000043
000044
000045
000046
000047
000048
000049
000050
000051
000052
000053
000054

IF ALL THE VALUES LISTED BELON ARE IN THE ORDER OF000055

000056
000057
000058
000059

702 FORMAT(1H0.FEPSLON = ’.E15.7.3
LUMDA 8 ’.E15.7)

C

C INPUT VALUES

C

(98'!

41

42

43

44

25

26

4‘

NEAD(5.'1N”L.NIP.NFP.PR.DPSI.DETA.EX.EY.VX.GXT.PADIUS.HVALUE.DIS
”RITE!6.700‘NUL.NIP.NFP.PR.UPSI.DETA12X.EY.VX.GXY.
4RADIUS.HVALUE.DIST

HEADLS.P1LXB‘IT.I=1.NHL1

NEADL5.'11TBTITTI=1.NMLT

NEADT59')(N3TTFETI).I=1.NHLT

READ!STPTIBANXLITOI=1.NHL1

NEAD(5.P)(BANY‘I1.I=1."UL1

NEADLS.‘1(XI(I1.I=1.NIP1

'IADLS.‘)1TI‘I1.I=I.NIPT
HHITEL6910011I.XB1119YB(IToBANXLI).BANT(I).NBTYPELI)

191:1.NHL1

X0=5.08Y0=5.0
XF(1)=10.-X08YF(1)=-10.+Y0

DO 41

XF(I)8XF(I-1)-0.5
YFtIT=YF(I-1)
DO 42 J31.10

=J'17

132911

00 “2 131911

XFTI.K):XF(I)
YFtIoKT=YFtI¢K-17)+l.o
XFT121:10.-x08YF(12)=-9.5¢Y0
DD 43 1:13.17
XF(I)=XP(I-1)-1.D
YFtI)=YP(1-1)

00 Q“ J:199

K=J917

DO 44 I=12.17
XFlI+K)=XF(I)
YF(I+KT=YF(IOK-17)41.0
ML2=NML928NML2P1=NML2+1

NMLP1=NHL¢1
NOPTION=1

DIST1=4.0 8 DIST2=2.0

DEL(1)=(10.42.*DIST1)/10.
DEL(2)=(10.¢2.'DIST102.'DIST21/10.
XX8(1)=5.0DIST1-DEL(1)8YY8(1)=-5.-DIST1

XXB(41)=5.0DIST1*DIST2-DEL(2)8YYB(41)=-5.-DIST1-DIST2

DO 28 J:182
UELT=OELI1I8IFIJ.EQ.ZIDELT=DEL(2)
DO 25 I32.10

K=I

IFTJ.EO.2)K=K¢4O
xxalKT=XXB(K-1)-DELT
YYB(K)=YYB(K-1T

00 26 1:11.20

K=I

IFTJ.EQ.2)K=K04D
XXthizxxatk-l)
YYBTK):YYBIK-1)+DELT
DD 27 1:21.30

K=I

101

DETA = I.E15.7.

000060
000061
000062
000063
000064
000065
000066
000067
000068
000069
000070
000071
000072
000073
000074
000075
000076
000077
000078
000079
000080
000081
000082
000083
000084
000085
000086
000087
000088
000089
000090
000091
000092
000093
000094
000095
000096
000097
000098
000099
000100
000101
000102
000103
000104
000105
000106
000107
000108
000109
000110
000111
000112
000113
000114
000115
000116
000117
000118
000119

27

28

38

102

IF‘J.EQ.2)K:KO4O

XXB1K1=XXBIK~IIOOELT

TYB(K1=YTO(K'I)

DO 28 I=3I.40

K=I

IFIJ.EG.21K=K04O

XXBIK)=XXB(K-1)

YYB1K1=YYB1K-11-OELT

XXB1811=XX51411 8 TT31811=YT31411
PI:‘8.PATAN11.T . Q=1.0

VT=ETPVX/EX 8 OX:EX'HVALUE.‘P3/(IE.‘(I.-VX'VT11
OY=EY9HVALUE“‘3/(12.'(1.-VX”VY11 8 Do=SQRT(OX‘DY)
NCOUNT=1

CONTINUE

DK=GXYPMVALUEHP3/12.

H=OXPVYOZ.'OK 8 RHO3H/OD
“ITE1691101119XXB1I19TTBLIT’I:19IHLZPI1
NHITE16.50011I.XF1I1.YF(I1.I:1.NFP)
WRITE16.20011I.XIII1.YI(I1.I=I.NIP1

E4=OX/OY 8 E23509T1E4) 8 EPSLON=SORT1E21
FACTCP=1.E-6 8 QLOAD=Q‘DETA*DPSI 8 R2=RAOIUSPPZ
KRITE16.701)DX.OT.H.GXT.VY.RHO

IF1RHO-1.012.1.3

FOR HMO .EQ. 1.0 'PP'PP‘H'P'
COEF1=1./116.8PI'EPSLONPOO) 8 COEF2=Z.RCOEF1
NTYPE=1

GOTO 4

FOR PHO .LT. 1.0 P‘”""'*'
COEF1=1./(32.*PIPDD) 8 COEF232.8COEF1
HU1=EPSLONRSQRT111.4RHOT/2.1
"U2:EPSLON‘SORT((1.-RHO)/2.1

NTYPE=Z

SOTO 4

FOR RHO .GT. 1.0 'P'P'PPP'P
EETAPEPSLCNRSOHTLRHOOSGRTLRHORPZ-I.11 8 BETA2=OETAPPZ
LUHOA=EPSLONPSQRT1RHO-SORTIRHO‘PZ-I.1) 8 LUHOAEPLUHOA'RZ
HRITEI6.7021EPSLON.OETA.LUHOA
COEF1=1./(O.'PIPOO'1OETA2°LUHDA211 8 COEFZ3COEF1'Z.
NTYPE=3

CONTINUE

00 5 1‘16”".

RL1I1=D.O

NL1I§NML1=D.D

R1X3X31I1

R1Y=YB1I1

ANx=BANN111

ANY=BANYlI1

DO 6 J=1oNIP

H2X3XI1J1

R2T=YITJI

ZI=R1X-R2X 8 zz=RIY-RZY 8 215321582 8 ZZSPZZ*'2
ANXS3ANXP'2 8 ANYSPANY'PZ

AAPOXHANXSOOT‘ANYS’VX

BBPOYPANTS¢OXPANXSPVT

CC32.'ANX'ANTROK

GOT0117.18.19).NTYFE

17 R125=ZIS+ZZSRE2 8 Z5=ALOG(R12S/R2)

C1=25-2.'E28225/R12S
C232502.9E2|ZZS/R125

000120
000121
000122
000123
000124
000125
000126
000127
000128
000129
000130
000131
000132
000133
000134
000135
000136
000137
000138
000139
000140
000141
000142
000143
000144
000145
000146
000147
000148
000149
000150
000151
000152
000153
000154
000155
000156
000157
000158
000159
000160
000161
000162
000163
000164
000165
000166
000167
000168
000169
000170
000171
000172
000173
000174
000175
000176
000177
000178
000179

an

18

10

c.
6.

45

6
5

33

20

21

103

C3=EPSLCNGZIRZZIR12S
RLTI)=RL1IT-OLOAD'(R12SPZS-13.'ZISOE2*ZZS)l

RLIIONMLT=RL1IONML)-QLOA08(AA*C1OBB'E2§C2¢2.REPSLONSCC8C3)iFACTOR

SOTO 6
Z7=ALOG1121599242.lRHO‘E2921552254E48ZZS9'21/R28921
Z8=ATAN12.'MUl‘MUZPZZS/lZISORHO'EZ‘ZZS)1
IFIZS.LT.0.)Z8=286PI

000180
000181
000182
000183
000184
000185
000186

Z9=ALOS11lMUl'ZZ148241Z1-MU2‘ZZ1'921/1(MU1822189201110MU2922198211000187

RLlI)=RL(I)-QLOAD*((ZISOEZ'ZZSl/MUIPZ7-2.91215-E2'ZZS1/MU2‘ZD
4-2.5E2821822/(MU1'MU21‘29-6.*12130E292251/MU11
C1=Z7/MU1-2.PZB/MU2 8 C2=Z7/MU102.PZ8/MU2 8 C3=-Z9/(MU1’MU2)
RL(I‘NWL)=RL1IONMLT-QLOAD‘lAA¥C14BBPE29C29CC9E29C31*FACTOR

SOTO 6

ZLC81=ALOGtIZISOLUVDA2'ZZST/R21 8 ZLOSZ=ALOS1(ZlSOBETAZ'ZZSl/RZT

IFtAESTZI).LE.1.E-6)SOTO 12
Z7=ATANfLUMDA‘ZZ/Z1) 8 ZS=ATAN1BETA822/21)
GOTO 45

Z7=PI/2. 8 28:27

CONTINUE

IF1Z7.LT.0.)Z7=Z76PI

IF128.LT.0.)Z8=ZSOPI

C1=BETA!ZLOS1-LUMDA'ZLOSZ 8 C23-LUMDA9ILOSIOBETA'ZLOS2

C3=~Z7028

000188
000189
000190
000191
000192
000193
000194
000195
000196
000197
000198
000199
000200
000201
000202

RL(I1=RL1I)-QLOAD9(8ETAI(ZIS-LUMDAZ'ZZS)PZLOSl-4.ILUMDA88ETA821'ZZ000203

0IIZ7-28l-LUMDA*(Z1S-DETA2'225T'ZLOSZ-3.'(BETA-LUMDA1
‘PTZISOLUMDA'EETA822511
RL(I§NML1=RL1IONML1-QLOADl1AA'CIOBB'EZICZOZ.PE28C315FACTOR

CONTINUE
CONTINUE

DO 8 I=I.NML

R1X=XB1I1

R1Y=YBTI1

ANX=BANX1I1

ANY=EANY1 I)

DO 7 J=1oNML2
R2X=1XX81J9110XXBIJ11/2.
R21=1YYB(J‘1)*YYB(JI)/2.
IF(NCPTIDN.EQ.2.0R.J.NE.NMLTSOTO 33
R2X=1XXBI1HXXBINML1V2 .
R2Y=(YYB(1)§YYB(NML))/2.

CONTINUE

21=R1X-R2X 8 12=R1Y-R2Y 8 ZIS=Z1882 8 225:22982
ANYS=ANX992 8 ANYS=ANY*‘2
AA=DX*ANXSOOY!ANYSlVX
EB=DYIANYSODXRANXSPVY
CC=2.!ANX'ANYPDK

SOTO (20.21.22).NTYPE

R125=ZIS¢ZZS9E2 8 ZS=ALOS1R125/R2)
C1=ZS-2.IE2'ZZS/R125

C2=Z502.NE2*ZZS/R125

C3=EPSLON'ZIPZZ/R125
RM!I.J)=R12S'ZS-13.IZ1S0E2'ZES)
RMII‘NML.J13(AA9C19888E28C292.REPSLONNCC'C319FACTOR
SOTO 7
Z7=ALOGI1ZIS'8242.'RHO'E2*ZISRZZS6E49225992T/RZRIZ)
Z8=ATAN(2.'MU1‘MU2*ZZS/(ZlSORHO‘EZ'ZZS1)
IF128.LT.0.’Z8=Z8§PI

000200
000205
000206
000207
000203
000209
000210
000211
000212
000213
000214
000215
000216
000217
000218
000219
000220
000221
000222
000223
000224
000225
000226
000227
000228
000229
000230
000231
000232
000233
000234
000235
000235
000237
000235
000239

22

23
66

7
8

10
1015

29

1102
1101

30

100

Z9=ALOSIl(HUI'Z21"20(ZI-HU2‘Z2)"2l/t(HUI'ZZ)*'2§(ZIOHU2'ZZ1'82))000240

RHII.J)=(l215*E2'ZZS)/flU1*Z7-2.'(ZIS-E2*ZZSi/HUZ'Z8
*-2.'E2*ZI'ZZ/(HU1*HU2)'Z9-6.*l21S‘E2‘ZZS)/HU11

C13Z7/HU1-2.928/HU2 8 C2=Z7/KU102.I28/HU2 8 C3=-Z9/(HU1'HU21

RHlI‘NHLoJ)=(AA‘CI‘BB'EZ'CZOCC‘EZ'C31'FACTOR
SOTO 7

ZLCSI=ALOSI(ZIS‘LUHOAZ'ZZSl/PZI 8 ZLOSZ=ALOSIlZ1S+BETA2'ZZSl/RZ)

IF(ABS(ZII.LE.1.E-6)SOTO 23
Z7=ATANtLUfiOA*ZZ/ZII 8 28=ATAN(BETA§ZZ/Zl)
SOTO 66

Z7=PI/2. 8 28:27

CONTINUE

IFlz7.LT.0.)Z7=Z7¢PI

IF(Z8.LT.0.)ZS=28+PI

C1=8ETI'ZL031-LUfiOA‘ZLOSZ 8 C2=-LUHOAlZLOSIOBETA‘ZLOSZ

C3=-Z7928

RH!I.J)=(BETA*(Z1S-LUflDA2'ZZSI'ZLOSI-é.‘LUHOA*8ETA‘ZI'ZZ

§I(Z7-ZOI-LUfiDA'IZIS'BETAZ'ZZS)“ZLOGZ-3.'(BETL'LUHOA)
¢I(Z156LUHOA‘BETA'ZZS))
RH(IONHLoJ)=(AA'CIOBB‘EZ*C2*2.‘EZ'C31'FACTOR
CONTINUE

CONTINUE

OO 10 I=1gNHL

OO 10 J=19NNL2

RflXlI.J)=PH(I.J)

RflxtloNHLoJ)=RN(IoNflL.J)

DO 1015 I=IgNflL2

PLX‘I)=RL(I)

CALL LEGTlFtFH.1.NflL2.NHL2.RL.O.HKAREA.1ER)
DO 29 1:1.NHL2

PSIII=PLtII

HRITE (6.600) (1.PS(1).I=1.NHL2)
HRIYEI6.1014)

00 1101 1:1.NHL2

SUfl=O.

00 1102 J=1.NHL2

SUH=SUH09HXII.J)IPS(J)

CONTINUE

RB(I)=SUH-RLXII)
NRITE(6.1103)(I.RB(I).RB(I¢NHL).I=1.NHL)
DO 9 1:10NFP

"(I)=0.0

BHX(1)=0.0

BHY(I)=0.0

CONTINUE

DO 16 1:1.NFP

R1X=XFlI)

R1Y=YF(I)

00 13 J=1.NIP

R2X=XI¢JD

RZY=YItJl

21:91X-R2X 8 22=01Y-R2Y 0 ZlS=ZlIiz 8 zzs=zz~~2

SOTO (30.31.32loNTYPE

91253213022S‘E2 8 ZS=ALOStR12$lR2)

Z6=2.lE2'225/R125
"(I)=H(I)OOLOAO'IRIZS'ZS-I3.'ZISOE2'ZZS))lCOEFI
BHXII)=BHX(I)-QLOAO*COEF2*OX*((ZS-Z6)+VY'E2‘(ZS+Z6))

000241
000242
000243
000246
000205
000246
000247
000248
000249
000250
000251
000252
000253
000254
000255
000256
000257
000253
000259
000260
000261
000262
000263
000264
000265
000266
000267
000268
000269
000270
000271
000272
000273
000274
000275
000276
000277
000278
000279
000289
000281
000282
000283
000280
000285
000286
000287
000258
000289
000290
000291
000292
000293
000294
000295
000296
000297
000298
000299

31

32

26
67

13
1:.

11

36

35

36

105

BHYII)=8HY(I)-OLOAD'COEF2‘OY* (EZ'IZSOZ6DOVX'(25-Z6))
SOTO 13
Z7=ALOSTl2159'262.*RH09E2*215'2256E6'225**21/929921
ZB=ATANI2.lHU1‘HU2'ZZS/(ZlSORHOlEZ'ZZS))
IF128.LT.0.)ZS=Z8OPI

000300
000301
000302
000303
000306

Z9=ALOS(I(HUI'ZZ1*fl2§(ZI-HU2'Z2)**2)/((HU18221"26(21*HU2*ZZ)"211000305

w(I)=H(I)¢0LOAD!((215+225225)/HUI*Z7-2.Itle-EZfiz2sI/Huzfiza
o-2.*EZ*21*22/(HUI!HUZIfiZO-6.I(2150E2IZZS)/MUI)iCOEFl
H2x=Z71HUI-Za'2./HUZ 0 u2v=zvxn01o2.uze/nuz
BHXII)=8HX(ID-OX'QLOAO'COEF2l(RZXOEZ'VYIHZYI
BurtI)=BHY(Ii-DYuOLOA0*COEF2«(H2Y¢EZ+VX§H2XD

GOTO 13

ZLCSl=£LOSt(leoLUHDA2IZZS)/RZD 0 ZLOGZ=ALOG((2150BETA2IZZSJIR2)
IF(ABS(ZII.LE.1.E-6)SOTO 26

Z7=ATANtLUfl0A*Z2/le 0 20=ATAN(BETA’ZZ/Z1)

GOTO 47

Z7=P1/2. 0 20:27

CONTINUE

1F(Z7.LT.0.)Z7=Z7+PI

1F(28.LT.0.)20=26+PI
HtI)=H(Il¢GLOA0!(BETAITZ1S-LUWDA2*ZZS)lZLOGl-6.fiLUVDAvBETA&ZI*ZZ
6n!27-281-LUHDA¥(le-EETA2'ZZS)GZL062-3.§(BETA-LUHDA)
0*(ZIS¢LUHOA*BETAIZZS)D'COEF1

Hzx=BETA!ZL061-LUHOAIZL002 0 H2Y=BETA*ZLOGZ-LUHDA&ZLOGI
BH¥(I1=BHX(Il-DX‘QLOAD'COEF2'(N2XOEZ‘VY‘H2Y1
BHY(I)=BHY(I)-OY'QLOAO*COEF2'(HZYIEZOVX'HZX)

CCNT IHUE

CONTINUE

DO 16 1:13NFP

RIX=XFtII

R1Y=YFIIJ

no 15 J=1.NHL2

RZX=(XXB(J+1)¢XXB(J))/2.0

921=(YYB(J01)OYYB(J))/2.0

IFtNOPTION.EG.2.0R.J.NE.NHL)GOTO 11

RZX=IXX8(1)OXX8(NHL))/2.
R2Y=(YYB(1)¢YYBINHL))/2.
CONTINUE

Zl=R1X~R2x 0 22:01Y-02Y 0 z1s=21fiiz 8 ZZS=ZZ**2

GOTO (36.35.36).NTYPE
0125:2150223l22 0 zs=ALOG(R125/R2)

26:2.GE2!225/R123
"(I)=u(I)oFS(J)fiCOEFlIlRlZS‘ZS-t3.02150E2I225))
BHX(I)=BHX(1)-PS(J1§COEF2*0X§((ZS-Z6l0VY«22I(ZS¢Z6))
BHYtII=BHY(1)-P$(J)!COEF2I0YG(EZIIZS¢Z6IOVX!(zs-Z6)l
GOTO 15
Z7=ALOSt(218*!2+2.GRHO*E2'Z1S‘ZZSOE6*ZZSG!Z)/R2*I2)
28=ATANI2.'HUI*HU2IZ25/(ZISORHO*E2*ZZS))
IF(28.LT.0.lza=zeoPI

000306
000307
000308
000309
000310
000311
000312
000313
000316
000315
000316
000317
000318
000319
000320
000321
000322
000323
000326
000325
000326
000327
000328
000329
000330
000331
000332
000333
000336
000335
000336
000337
000338
000339
000360
000361
000362
000363
000366
000365
000366
000367
000368
000369

Z9=ALOSI((HUllZ2)l'20(ZI-HU2*ZZD"2)/l(HUI‘ZZ)9i26(210HU2'22)*'2)1000350

“(I)3u1I)6PS(J)*(‘ZISOEZ'ZZS1/HU1.Z7-2.'IZIS'EZ'ZZS1/HU2'ZO
#-2.'E2*Zl'ZZ/(HU1'HU2)‘Z9-6.'(ZIS¢E292251/HU1)9COEF1
H2X=Z7/flUl-ZO'2./flU2 8 H2Y=Z7/HU102.*ZO/NU2
BHXll)=BHX(II-DxiPSIJ1§COEF2'(H2XOEZ*VY!H2Y1
BHY(ID=BHYIIl-OYGPSCJ)*COEF2*(H2Y'E29VXIH2X)
SOTO 15
ZLOSI=ALOStIZ1SOLUHOA2‘ZZSI/921 8 ZLOS2=ALOSt(Z1SOBETA20Z2$)/R2)
IF(ABS(ZII.LE.1.E-6)SOTO 37
Z7=ATANtLUHPAII2/21) 8 28=ATAN(BETA'22/21)

000351
000352
000353
000356
000355
000356
000357
000358
000359

106

SOTO 68

37 Z7=PI/2. 8 28:27

68 CONTINUE
IFTZ7.LT.0.)Z7=Z76PI
IF128.LT.0.128=ZS+PI
H(1)=H(11*PSIJ)'(BETA'(ZIS-LUNDA2*ZZS)§ZLOSl-6.iLUHOA'BETA*Z1*ZZ
OI!27-281-LUHOA'121S-SETA2*ZZS)‘ZLOSZ-3.*(BETA-LUHOA)
0512159LUHDA*BETA*ZZS)1'COEF1
H2X=8ETA*ZLOS1-LUfiDA‘ZLOSZ 8 H2Y=BETA*ZLOSZ-LUHDAlZLOSl
BHXII1=8HX1I1-OX'PSIJ)”COEF2'(H2X¢E2*VY*H2Y)
OHYII|=BHYII)-OY5PS(J)lCOEF2*(H2Y*E20VX*H2X)

15 CONTINUE

16 CONTINUE
HRITE(6.6001(1.XF(I).YF(I1.N(I).BHX(11.BHY(I1.1=1.NFP)
SOTO(53.53.51.52.53).NCOUHT

69 RNO=0.5 8 NCOUNT=2
SOTO 38

50 RHO=2.0 8 NCOUNT=3
SOTO 38

51 RHO=0.1 8 NCOUNT=6
SOTO 38

52 RHO=10. 8 NCOUNT=5
SOTO 38

53 STOP

END

609100.18190.3.1.091.Oo30.56.2.056.0.3.7.556o80..0.6.6.0
6.5.3.5.Z.5.1.5.0.59-0.5o'1.5.'2.59-3.5.-6.5.10'-5.u
'“.59'3.59'2.50’1.59’0.590.501.502.593.594.5910'5.
101-5..-6.5.-3.5.-2.5.-1.5.-0.5.0.5.1.5.2.S.3.5.6.5.
10*5..6.5.3.5.2.5.1.5.0.5.-0.5.-1.5.-2.5.-3.5.-6.5
6092

10'0..10'-1.910‘”..10'1.

10"1 91050 010.1 910.0.

6.5.3. 5 2 5.1.5 0. 59-0.5.-1.5.-2.5.-3.5.-6.5.

0 59‘°.50'1.50'2-5"3.59'4.59

0. 59-0.5o'1.5.-2.5.'3.5o-6.59

0. 59'°.50'1.50‘2.59'3.59'“.59

O. 5,-0.5.‘1.59'2.5.-3.5.-6.5.

0. 59'0.5.'1.5.’2.5.-3.5.'6.5.

0. 59'0.59'1.59’2.53'3.50'4.50

0.

0

0.

1

‘

tummemmmm

cat:
I
To

VI

50-0.59’1.59‘2.59’3.59'“.50
.5.-0.5.-1.5.-2.5.-3.5.-6.5.
5.-0. 5.-1. 5.- 2. 5.-3. 5.-6.5.
0"2.5’1°"1.591°.’°. 59

2. 5.1013. 5.10‘6. S

0 C ' . .C C O O

8
8
8
8
8
8
8
8
8

Hflbél‘EOaébb
"91d1fl1d1d1d1d1fl1d1fl
" “91181181111181“18113188;
H. v v o o o o o v o
CH“I»8V8“'UF°0°“8“ORD:

I‘D-
hit1fl1flUlUlUlUHm1fl1fl;

1.5.
1 5.
1.5.
1. 5.
1. 5.
1. 5.
1 5.
L 5.
1. 5.
-3. 5.
.5 10

000360
000361
000362
000363
000366
000365
000366
000367
000368
000369
000370
000371
000372
000373
000376
000375
000376
000377
000378
000379
000380
000381
000352
000383
000386

000386
000387
000388
000389
000390
000391
000392
000393
000396
000395
000396
000397
000398
000399
000600
000601
000602
000603
000606
000605

APPENDIX F

DERIVATIVES OF THE GREEN'S FUNCTION
OF AN INFINITE ANISOTROPIC PLATE

APPENDIX E

DERIVATIVES OF THE GREEN'S FUNCTION
OF AN INFINITE ANISOTROPIC PLATE

For simplicity, the derivatives are written in terms of the four
constants 01, 02, 03, and $0, and the eight functions L1, L3, R1, R3,

3_G - ___1.. 8R: 8R3 as_x ._ as;
3X - 8WD22¢1¢32 [(bBT +¢6"'— '1' ”((1’)( 3X ]
9.6. = _____13R1 8R3 - _3§_1_ _ 382
3y 8WD22¢1¢2 [$33Y +00? + “(a Y)(3 3y ]

_a_3G_ 1
3x 3 ”"022¢1¢2

8N1_8__N3
3x 8x

[033:1 +¢.§L3 + u<a~v><— J

a G_ 1 (02+82-20y)(a2+82)+(az-BZ)(Y2+XZ) 8L.

By3 “00220102 8 3y

 

 

 

+ (Y2+12-207)(Y2+12)+(1?-12)(az+82) 3L3
A By

-“[0(Y 22+A2)-y<a +52>1<§N1 - 3N3 )}

107

108

 

33G _ 1 3L1 3L3 3N1 8N3
axzay ' “RDzzRI¢-2‘¢3R‘i'*¢“57 * ““1”” ”837‘ " R;- ’1
33c = 1 [(02+82-2aY)(a2+82)+(a2-82)(Y2+A2) 9L1
Bxayz “WD22¢‘1¢2 L 3 3X

+ (Y2+k2-2ay)(12+A2);(y2-A2)(a2+82) 3L3

8x

§N_1-_3N_3)}

- u[a(v2+A2)—y<a2+ez>]( 3X 3X

where ,

¢1 (a-y)2+<e-x>2 ; ¢2 <a—y)2+(e+x)2

-2- 2_2 _2 2_2
¢3=(ay) B(E3 x>3¢u=(aY)+>\(B A)

L: = 2n L(X'€)+a(Y-n)lz+32(y-n)2

a2

in [(x-E) ”(y-n) 12H2 (y-n) 2

2
a

La:

R1

{[(x-€)+a(y-n) 12-82 (y-n) 2}- (L1- 3)

3"
u

{[(x-£)+y(y-n)] 2-A2 (y-n) 2 } - (La- 3)

N1

51

U)
(A)
I

.611
8x

3x

3L1 =

8y

9.12.
3y

811.

109

8(y-n)
5) +0 (y-n)

 

. _ “Yd”
' N3 ‘ arc t9 (X-EHYTy-n)

arc tg (X?

B(y~n)[(x-€)+a(y-n)](L1- 3)+{[(x-€)+a(y-n)]2-82(y-n)2}N1

- X(y-n)[(X-€)+Y(y-n)](L3- 3)+{[(x-€)+Y(y-n)]2-A2(y-n)2}N3

2{(x-€)+a§y-n)}
{(x-€)+a(y-n)}2+82(y-n)2

 

2{(x-€)+ij~n)}
{(x~€)+y(y'--n)}2+A2(y-n)2

2a(X-€)+2(02+82)jybn)
{(x-C)+a(y-n)}2+82(y-n)2

27(x-§)+2(I?+A2)jy-n)
{(x-£)+Y(y-n)}2+A2(y-n)2

-B(ybn)
{(x-£)+a(y-n)}2+82(y-n)2

-1(y-n)
{(x-€)+Y(y-n)}2+kz(y-n)2

8(x-E)
{(Xr€)+a(y-n)}2+82(y-n)2

8N3

Y

213.1.

3x

ES?"

110

A(x-€)
{(x-E)+Y(y-n)}2+kz(y’-n)2

= 2[(x-£)+a(y-n)](L1-3) )+ 3L‘{[(x-g )+a(y-n)]2-82(y-n)2}

-48(y-n)N1-48(y-n)[(x-€)+a(y-n)]§—N1

= 2[(X-€)+Y(y-n)](L3-3) )+ 3L3{[<x-a)+y<y-n)12-x2(y-n)2}

 

-ux(y-n)N3-ux(y-n)[(x-a)+y(y-n)]§§

2}§_L1_

= 2[a(x—i)+(a2-62)(y-n)](Ll-3)+{[(x-€)+a<y-n)]3-83(y-n)

'“Bl‘x‘€)+20<Y-R>JNl-u8(y-n)[(x-a)+a(y-n)]LN1

= 2[Y(x-€)+(Y2-A3)(y-n)](La-3)+{[(x-€)+Y(y-n)]2-X2(y-n)3}-§F-

'“X[(X-€)+2Y(y-n) )JNa-UA(y-n)[(x-€)+Y(y-n)]LN3

-——-= 8(y-n)(La-3)+B(y-n)[(X-€)+a(y-n)]gL1 + 2[(x-€)+a(y-nH'N1

+{[(x-€)+a(y-n)13-82(y-n)2}8N1

111

gig—3" My-n) (L3‘3)+X(Y('T1)[(X'E)+Y(Y’T1)]3L3 + 2[(X'€ “fly-”)1 °N3

+{[(x-£)+\(y-n112-12(Y“r) }§N3

.355]; = B[(X-C)+20.(Y-T1)](Ll-3)+B(y—n)[(x_5)+a n)] _a__L1
+ 2[O(X’€)+(32_B2) (y_n) ] .N1+{ [ (X'E) MY’U) ] 2’82 (y- n) 2}3N_1_
3—33— = A[ (X'€)+2Y(Y'T1)] (L3‘3)+)\ (y-n) [(X_€)+.Y (y_n)] L143

3_N_3

2[Y(x-E)+ < 2->2 )(y-n)] N3+{[(X‘5)+Y(y'n)]2-32 Y'”) 8y

APPENDIX G

COMPUTER PROGRAM FOR AN ANISOTROPIC PROBLEM

(TCTCTCO(TCTCTCTf1(TCTCUCT(if)(if?(TCTFT(TCTTTCTCTCTCTCTCTGMO

APPENDIX G

COMPUTER PROGRAM FOR AN ANISOTROPIC PROBLEM

PROGRAM ANIPLCLtINFUT.OUTPUT.TAPE5=INPUT.TAPE6=OUTPUT) 000100
000110
§¥III§¥§ICfiflIIOil.8".“lllliiliflifl'l‘filll5""!Nlfiflllflflllliflii’flflfilli'liflo 00 1 2 0
000130

POINT FORCE METNOD FOR ANISOTROPIC PLATE SENDING PROBLEMS. 000140
ARBITRARY PLAN FORM. TRANSVERSE LOAD. AND BOUNDARY CONDITIONS RP» 000150
000160

SNONN HERE IS AN EXAMPLE FOR A SIHPLY SUPPORTED SQUARE PLATE. 000170
000100

REOUIRED INPUT VALUES --- 000190
000200

NSP :NUMSER OF OOUNDARV POINTS 000210

NIP =NUMSER OF INTERNAL LOAD POINTS 000220

NFP =NUMSER OF FIELD POINTS 000230

xa.va =POINTS ON 8 AT NNICN B.C. ARE SATISFIED. 000240
BANX.BANY =COMPONENTS OF UNIT NORMAL To 0 AT XB.YB. 000250
XXB.YYB =END POINTS OF HESHES AROUND B NNERE FICTITIOUS 000260

FORCES ARE ASSIGNED. 000270

XF.TF :FIELD POINTS 000200

XI.TI =INTERNAL LOAD POINTS 000290

vx =POISSON-s RATIO IN x DIRECTION 000300

DUE TO STRESS IN T DIRECTION 000310

EX.EY tYOUNG'S MDDULI IN x AND Y DIRECTIONS. 000320
RESPECTIVELY 000330

sxv =8HEAP MODULUS 000340

NVALUE :PLATE THICKNESS 000350

RADIUS =RAOIUS OF THE FICTITIOUS CIRCULAR PLATE OF HHICH 000360

TNE DISPLACEMENT AT THE CIRCUMFERENTIAL DOUNDART 000370

IS SET TO ZERO. 000350

000390
.I.”.IlI”I.§II'IIIIII§C§liilliflifllfl‘lflifllll‘l...5"!“INKIUQI§§§5§I§IQ§IIOOOQOO
000410

DIMENSION XB(40).YB(40).X¥B(£1).YYB(81) 000420
DIMENSION XIIIOO).YIIIOO).DEL(Z) 000430
DIMENSION RLXTDO).RD(SD1.RM(SD.SD).PS(00).NKAREATDD).RL(SD) 000440
DIMENSION xF11011.TF(1011.N(101).3Mx(101).SMT(IDI) 000450
DIMENSION BANX(40).BANY¢60).RHN(80.80) 000450
REAL AVECTORT51.Mx.MT.HxT 000470
COMPLEX ERROR.ROOT(4) 000400
000490

100 FORMATtIDLOCTt.ex.¢xst.SX.¢TDF.7x.¢ANx1.Ox.¢ANvt.4x. 000500
o/aox/tIs.6F1D.411 000510
110 FORMATttDLOCT:.11x.txxat.14x.¢TvD:/:0¢/(I4.Ox.F9.2.Ox.F9.211 000520
200 FORMAT (COLOCTS.11X.¢XII.17x.1YI¢.14X./¢OI/ 000530
.(I4.11x.F6.3.11x.F6.3)1 000540
400 FORMAT tIILOCT!.19X.IPSPx/Iot/(14.8X.E20.61) 000550
500 FORHAT(30LOCT3.11X.£XF¢.17X.1YF¢/IOt/(16.11X.F6.3.11X.F6.3)) 000560
600 FORMAT:1N1.Sx.:NODEt.12x:xF¢.16X.:TF¢.19x.¢Nx.ISx.:SMx¢. 000570
116x.:DMv¢/t1N0.110.2F20.10.3Ezo.1211 000550
700 FDRMATTIN1.IINPUT VALUES .....a.//1x.:NOP : 1.13.: NIP =!.I3. 000590
4: NFP = P.13. 000600

9/1N091 EX ' '9E10.4.1 EY = 19E10.49‘ VX 3 '9F5.29' OX7 = $.E10.69 000610

S/lHOo‘ RADIUS OF THE PLATE = I. 000620
0F7.193 THICKNESS OF THE PLATE =‘OF6-3) 000630
703 FORHATllllylxoiTHE COEFFICIENTS OF THE CHARACTERISTIC i. 000660
OtPOLYNOHIAL ARE -----I./1X.5E12.5) 000650
704 EORHAT(1H091THE FOUR ROOTS OF THE CHARACTERISTIC EOUATIONtI. 000660
o(/1X92E12.5)1 000670
705 FORHAT11H0-‘ ’RROR FOR ROOT NO. 3.12.2X02E13.6.I IS I.2E10.6) 000680

112

 

C
C
C

113

707 FORHAT(1H0.1THE FOUR CONSTANTS ARE :1.
O/IXSFALPHA = ‘oE10.49F BETA = F.E10.0o
0‘ OAHNA 8 F.E10.4o‘ LUHBOA = 39E10.4)

1103 FO'HAT(IS.SX.E20.12.5X.E20.121
1016 FOQHAT11H190THE FOLLONIRG IS A LIST OF DOUBLE CHECKING OF B.C.Si.
I//.4X!NBP110X.FB.C. 1‘913XoFB.C. 2’1//)

708 FORMAT!1H0.’PAOIUS 3 19E10.3y‘ OISTI F ‘9F6.1v
OF OIST2 = l.F6.11
801 FORMAT(1H1.ITHE SENDING RIGIDITIES ARE ----- 3.

4/1x.¢011 : ¢.E10.4.R 012 = 1.E10.4.¢ 022 = x.E10.4.
o/Ix.tDeb = t.E10.4.x 016 = 1.510.4.: 026 = ¢.E10.4)

002 FOPHA711X.ISOHETHING Is NRONS NITN TNE INPUT MATERIAL CONSTANTSI.
41; THE DETERMINANT IS EITHER NEGATIVE OR zERo¢./1x.
.xCOMPUTATION Is TERMINATED. DET = c.515.71

005 FORMATTIN1.¢TNE AIJ ARE ---t./1x.tA11= R.E15.7.: A12: 1.
oEIS.7.v A22: :.E15.7.t A66: ¢.E15.7.t A16: :.E15.7.
.r‘ A26= 8.E15.7)

INPUT VALUES .........

'EADISo“1NBPDNIPRNFPDEXoEYoVXoGXYoRADIUSDHVALUE
READISS'IAIODAZO
"EA0150'1(XB(I)DI=10NBP1
READ159'11YB‘1191=10NEP1
"£00159'118ANX11101:19KBP)
'EAD‘ 59.318‘NY1 I 1 01:1 ,NBP)
lEADl5.l)(XI(I).I=1.NIP1
REAO‘50.1(YI(I)DI=10NIP1
NBP2:NBPl20N8P2P13N3P291
"OPTION:1

P13“..ATAN(1.1

A1131./EX 0 A12=~VX/EX 0 A2231./EY 0 A6631./GXY
HPITE1698051A110‘120A220A669A169A26
DET31A11'A22'AIZO'21.A6692.”A12*A16'A26'A11"26"2'AZZ'A16*'2
IFtDET.GT.O.1GOTO 00
"RITE169802105T
STOP

(to CONT INUE
ZZ=HVALUEi!3/(12.IDET)
0113(A22GA66-A26PPZIPZZ 0 022=(A11*A66-A16II2)‘ZZ
012=lA16'A26-A12!A66)'ZZ 0 066=lA11¥A22-A12*‘2)PZZ
016=lA12iA26-A22lA16)'ZZ 0 026=(A12*A16-A11PA26)RZZ
RADIUS380. 0 DISTI:Z.O 0 DIST2=2.
“R175160700INBPRNIPDNFPDEXDEYDVXDGXY0
ORADIUSRHVALUE
HRITE(6.708)RADIUS.DISTI.DISTZ
X035.°0Y0=5.°
XF111310.’X00TF‘113-10.9Y0
DO 41 132911
XF(I)8XFII-1)-O.5

‘1 TF1113YF1I'11
DO ‘2 J31910
K‘J'17
DO “2 131011
XE‘IOK13XF111

.2 T'IIOK13YFIIOK-171OI.O
XF112131°.'XOSYF(1213-9.59Y0
DO ‘3 I313917

000690
000700
000710
000720
000730
000740
000750
000760
000770
000780
000790
000791
000792
000793
000600
000810
000820
000830
000550
000850
000860
000870
000830
000890
000900
000910
000920
000930
000940
000950
000960
000970
000980
000990
001000
001010
001011
001012
001013
001014
001020
001030
001060
001050
001060
001070
001080
001090
001100
001110
001120
001130
001140
001150
001160
001170
001180
001190
001200
001210

43

44

25

26

27

28

C SET

706

114

XF(1)=XF(I~1)-1.0

YFII)=YF(I-1)

00 “a J=109

K=Jfll7

00 44 1:12.17

XFIIQK1=XF111

YF(14K|=VF(IoK-17)91.0
DEL(1)=(10.92.*DIST11/10.
DEL(2)=l10.92.l0157192.'01572)/10.
XX8(1)=5.901$T1-0EL(1)9YY8(1)=-5.-DIST1
XXB(41)=5.ODISTIOOISTZ-OEL(2)SYY8(4113-5.-OIST1-DISTZ
DO 28 J=1.2
0ELT=OELII)01FtJ.EQ.2)DELT=DEL(2)

00 25 1:2.10

K=I

IFIJ.EO.2)V=K940

XXBlK)=XXB(K-1)-DELT

YYBIK)=YY8(K-1)

DO 26 1:11.20

K=I

IF(J.EQ.2)K=K940

XXB!K)=XXB(K-1)

YYBKK)=YYB(K-1)OOELT

DO 27 1:21.30

K=I

IFtJ.E0.2)K=Ko40

XXBtK)=XXB(K~1)+DELT

YYBtK)=YY8(K-1)

DO 28 1:31.40

K=I

IFlJ.EG.2)K=K¢40

XXBIK)=XXB(K-1)

YYB(K)=YYB(K-1)-OELT

XX8(81)=XXB(41) 6 YY8(81)=YY8(411
HRITE(6.100)(I.XBII).YB(I).8ANX(I1.8ANY(I)pI=1:N8P)
HP!TE(6.110)(I.XXB(I).YY8(I).I=1.NBP2P1)
HRITE!6.500)(I.XF(I)»YF(I)’I=1.NFP)
HRITE(6.200)II.XI(I).YI(I).I=1.NIP)
HRITEI698011011.012.022.066.016.026

UP AND SOLVE THE FOURTH DEGREE CHARACTERISTEC POLYNOHIAL.

AVECTORII1=1.0 9 AVECTORt21=4.'O26/022
AVECTOR!31:2.il01292.'0661/022
AVECTOP(4)=4.'DI6/DZZ 0 AVECTORISJ=OIIIO22
HRITEI6-703)(AVECTOP(11.13195)

CALL ZFOLRlAVECTOPo4gflOOToIER1
"RITE(6.704)(ROOT(I1,131.41

DO 706 13194
ERROD=AVECTORII1§ROOTI11"49AVECTOQt21'ROOT11)FlIRAVECTORl3)*
SHOOTtI1"29AVECTORI41'ROOT(IIOAVECTOR(51
NQITE(6.705)IRROOT(I)oERROR

RI=REALIROOT1111 0 R2=AIHAGtROOTt111
R3=REAL(ROOT(3)) 9 R4=AIHAGtPOOT(3)1
H?ITE(6.7071R1.P29R3994

RIS=R1*'2 0 R2S832'F2 0 R3S=R3"2 0 H438R4'I2

CONSTG=IRI-P319.201929R41i'2 0 CONSTH=(R1-R31i'29(RZ-R41"2
COEF1=1./(8.5P19022'CONSTG'CONSTH1 0 COEF2=2.'COEF1
CONST1=((R1~R31952-(R25-R4S11/R2

001220
001230
001240
001250
001260
001270
001280
001290
001300
001310
001320
001330
001340
001350
001360
001370
001380
001390
001400
001410
001420
001430
001440
001450
001460
001470
001480
001490
001500
001510
001520
001530
001540
001550
001560
001570
001580
001590
001600
001610
001620
001630
001640
001650
001660
001670
001680
001690
001700
001710
001720
001730
001740
001750
001760
001770
001780
001790
001800
001810

60
61

62
63

6
5

115

CONSTzst(RI-R31-R241R2S-R4SI1/R4
CONST3=4.-(R1-R31
CONSTS=((RI-2.IR3)I(RIS¢RZS)9R11(R3S9R4SI)IRZ
CONSTesctR3-2.uR1191R3S+R4ST+R3utRISoR2511/R4
CONST7=2.ItRISost-R3S-R4S1
CON518=1(R189025-2.¢R1*R3)RLEIS9RZS19(318-RZSTGIRSSOR4S)1/02
CONST9=TtR3SoR4S-2.~RI-R31n1R3SoR4S141R3S-R4SIPLRISoR2511/R4
00N3T10=4.l(01*(R3S+R4S)-R3l(RISoRZS)1
0LOA0=1.0 S AszADIUSR'z

DO 5 1'10“?

IL(1)80.0

RLTI.NSP1=0.0

R1x:x0111

RIY=Y8(I)

ANx=DANXTIT

ANTsDANvtIT

00 6 J=1.NIP

RZX:XI(J)

RZY=YI(J)

zI=R1x-R2x o z2=R1v-R2v s 215=ZIEPZ S zzs=z2~~2
ANXS=ANXII2 S AN78=ANY‘!2 S ANxvaquANr
FUNCLI=ALOG¢IT214R1~z2IsuzoRESPZESI/Azr
FUNCL3=AL03tC(21903'221"2904S‘ZZS)/A2)
23=ZIRRIRZZ 0 z4=z1+R3-zz
IFTADS1231.ST.1.E-SICOTO 50

FUNCN1=PI/2.

GOTO 61

FUNCN1=ATAN(RZIZ2/Z3)

IF!A85(Z4).GT.1.E-6)GOTO 62

FUNCN3=PI/2.

SOTO 63

FUNCN3=ATANt R4'ZZ/Z4)

CONTINUE

IFIFUNCN1.LT.0.)FUNCN1=FUNQN19PI
IFTFUNCN3.LT.0.1FUNCN3=FUNCN3.PI
AA=011RANXSoDIERANv542.«DISSANxv

=ANXS'01260225ANY562.lANXY*026

CC'Z.‘016'INX592.‘026'ANY594.'066'ANXY
FUNCRIxL(ZIoRIRzz)Rcz-R2S92251R1FUNCL1-3.1-4.RRzuzzutz19R19z2)
«FUNCNI

FUNCRSst (219R3I22 )RIE-R4Suzzs m FUNCL3-3. 1-4.GR4*22*(21933'22)
«FUNCN3

001820
001830
001840
001850
001860
001870
001880
001890
001900
001910
001920
001930
001940
001950
001960
001970
001980
001990
002020
002010
002020
002030
002040
002050
002060
002070
002080
002090
002100
002110
002120
002130
002140
002150
002160
002170
002180
002190
002200
002210
002220
002230

FUNCSI=R2'22'(ZIORI'221*(FUNCL1-3.19((ZIOR191219'2-R25'22S)lFUNCN1002240
FUNCS3=R4l2291219R3922)I(FUNCL3-3.)9((ZIOR3'ZZ)F‘Z-R4S'2251‘FUNCN3002250

HXX=CONSTIIFUNCL1+CONST2IFUNCL3§COHST3fltFUNCNl-FUNCN3)
HXY=CONST5'FUNCL19CONST6'FUNCL3OCONST7'!FUHCNI-FUNCNS)
HYT=CONST8iFUNCL19CONST9‘FUHCL3-CONST1051FUNCNI-FUNCN3)
IL1113RLIIi-OLOAO'ICONSTI“FUNCRIOCONST29FUNCR3+CONST3¢
0(FUNCSI-FUNCS311
HLIIONBPJSRLIIOHOPl-OLOAO*(AAlHXX988§HYYOCC§HXY1
CONTINUE
CONTINUE

OO 8 1319N8P
RIXSXBTI)
I1Y=T81I1
ANX=8ANX(I)
AHY‘OANY111
DO 7 J313N892

002260
002270
002280
002290
002300
002310
002320
002330
002340
002350
002360
002370
002380
002390
002400
002410

33

70
71

72
73

10

1015

29

1102
1101

116

RZX=(XY8(J9119XX8(J)1/2.
RZY=(YY6(J91)OYTB(J)1/2.
IFINOPTION.EO.2.0R.J.NE.NBPIGOTO 33
R2X=IXXB(1)9XXB(NBP)1/2.
92Y=(YYB(116YYBINBP))/2.

CONTINUE

21:31X-R2X 0 22=R1Y-RZT 0 215321'92 0 225:229'2
ANXS=ANXP*2 8 ANYS=ANY§92 0 ANXTSAinANY
FUNCL1=ALOG(I(219R1'221"2632582251/A2)
FUNCL3=AL051((21993.221'929P4S'2251/A2)
Z3=ZI§FI*ZZ 0 24:21993922
IF(ABS(Z31.GT.1.E-6)GOTO 70
FUNCN1=PI/2.

GOTO 71

FUNCN1=ATAN1R2'22/Z3)
IF(A85124).OT.1.E-6)GOTO 72
FUNCN3=PI/2.

GOTO 73

FUNCN3=ATAN(R4'ZZ/Z4)

CONTINUE
IFIFUNCN1.LT.0.)FUNCN1=FUNCN19PI
IF‘EUNCN3.LTIO.1FUNCN3=FUNCN3991
AA=OII9AN¥59012§ANTSOZ.9016'ANXY
BB=ANXS'012*OZ2*ANYSOZ.iANXY5026
Cc=2.'016'£NX592.‘OZ6'ANYSO4.'O66*ANXY

FUNCRI=¢(214219221"2-RZSFZZS1FIFUNCL1-3.1-4.’RZ*ZZ*1210215221

oPFUNCNl

FUNCP3=1(219939221"2-R4892281ltFUNCL3-3.1-4.‘R4*ZZ'(ZIOR3*22)

0*FUNCN3

002420
002430
002440
002450
002460
002470
002480
002490
002500
002510
002520
002530
002540
002550
002560
002570
002580
002590
002600
002610
002620
002630
002640
002650
002660
002670
002680
002690
002700

FUNC51392'ZZ'(219N1‘ZZ)‘(FUNCL1'3.19‘121431.221"2'N2582231'FUNCN1002710
FUNCS3=P4'22*(ZIRR3*ZZ19(FUNCL3-3.)9!(21993'221"2-945'2251'FUNCN3002720

HXX=CONST1*FUNCL1OCONSTZFFUNCL30CONST3'(FUNCNI-FUNCN31
WY=CONST5*FWCL1OCONSTO'FFLMC L39CONST7N FUNCNl-FUNCN3)
HYT=CONST8FFUNCL1+CONST9‘FUNCL3-CONST10*(FUNCNI-FUNCN31
RH(IoJ)=(CONSTI‘FUNCPIOCONSTZ'FUNCR3OCONST3itFUNCSI-FUNCS3)1

RH!I+NBP9J)=(AA'HXX088'HYYOCC'HXY)
CONTINUE

CONTINUE

OO 10 131.1189

00 10 J=1yN5P2

PHXTI.J)=PN(I,J)
RHX(I¢NSP.J)=RN(I§NBP.J)

DO 1015 I=13N8P2

RLX(I1=RL(I)

CALL LEGTIF(RH.1.NBPZ.NBPZ.RL.0.HKAREA.IER)
DO 29 1:1.N8P2

PS(I)=RL(I)

HQITE (6.400) (I.PS(I).I=1.NBPZ)
HRITEL6.1014)

00 1101 1:1.NBP2

SUfl=o.

00 1102 J=1.NBP2

SUH=SUN4RMXII.J)IPS(J)

CONTINUE

RB(1)=SUH-RLXTI)
HRITET6.1103)(1.03!I).RB(1¢NBP).I=1.NBP)
00 9 I=1.NFP

Nt!)=o.o

002730
002740
002750
002760
002770
002780
002790
002800
002810
002820
002830
002840
002850
002860
002870
002880
002890
002900
002910
002920
002930
002940
002950
002960
002970
002980
002990
003000
003010

80
81

82
83

13

14

11

90

117

CHX111=°.0

BHT11130.°

CONTINUE

00 1“ IgloNFP

RIX=XFL I1

NIY:TE( I)

DO 13 J=10NIP

sz=x11 J)

NZT=T11J1

213N1X'Nzx 0 22=RIY-R2Y 0 215321"2 0 22$=ZZHZ
'Wng‘LOCI ( (21931‘22 1"29925‘z251/A2 1
FW'CL338L03111219N3'ZZ1"29R‘05'2231/A21
23321691'Z2 8 24:21993'22
IF‘ABS1Z3).GT.1 . E'6160T0 8°
FUNCN13p1/2.

SOTO O1

EUNCngATAN(PZ'ZZ/z31

1F( A851 Z4 1.8T . 1 . 5'6 TGOTO 82
FUNCN3=pI/2.

COTO O3

FUNCN33ATAN1R‘O‘ZZ/Z‘O 1

CONTINUE

IF! EWCNI . LT. 0 .1FWCN13FUNCN19PI
IF‘FUNCN3.LT.O.1FUNCN3=FUNCN39PI
FUTCP1311219R1‘221““2-925'2251'1FWCL1'3. 1'4 .‘NZ'ZZ‘1 219R1'z2)

O'FUNCNI

FUNCR3=I(21¢R3'221'82-R4592251'IFUNCL3-3.1-4.!R482291216R3'221

OiFUNCN3

003020
003030
003040
003050
003060
003070
003080
003090
003100
003110
003120
003130
003140
003150
003160
003170
003180
003190
003200
003210
003220
003230
003240
003250
003260
003270
003280
003290

FUNCSI=R2522N 219131.22 "(FUNCL1-3. “(1219919221992-2239225NFUTCNI003300
FUNCS3=R492251219R3922)‘(FUNCL3-3.)9((21933'221'92-R4592251fiFUNCN3003310

Hxx=CONST19FUNCL1 OCONSTZ'FUHC L3OCONST3NFUNCN1-FWCN3)
HXY=CONST59FUNCL1OCONST6'FUNCL39CONST7ilFUNCNI-FUNCN3)
NYY=CONST8'FUNC L1 OCONST9'FUNC L3-CONST10‘1FUNCNl-FUNCN3)
H(I)=H(I19(CONSTl‘FUNCRI9CONST2§FUNCR39CONST39(FUNCSI-FUNCS3)1

O'COEF1*OLO‘D

Hx=-COEFZP(OlliuxxoolzPHYvoz.POIéPHXY1RGLOA0
HY=-COEFZP(DIZGHXXROZZRHYYoz.'026¢HXY)POLOAO
HXY3-COEF2'1016*NXX00269HYYOZ.‘OOOFNXY1'GLOAD
OHX(I)=BHXTI)oMX

8NYII)=8HY(I)9NY

CONTINUE

CON'INUE

OO 16 I=1.NFP

R1X=XFtIl

RIT=YFKII

OO 15 J=I.NEP2

R2X=(XX8(J+1)OXXB(J11/2.0
R2Y=IYY8(J91)9YY8(J))/2.0
IF(N3PTION.EG.2.0R.J.NE.NBP)GOTO 11
R2X=IXX81119XX8lNBP))/2.
RZY=IYY8(1)+YYB(NBP))/2.

CONTINUE

218R1x-RZX 0 22=RIY-02Y 0 213:219'2 6 ZZS=ZZI92
FUJCL1=ALOGLt(21901.22)RRZoRZSPZZSI/Azl
FUNCL3=ALOSIt(Zlonsvzz)II24R4SRZZS)IA2)
23:21691'22 0 Z4=ZIoR3lZZ
IF(ABS(23).GT.1.E-6160TO 90

FUNCN1=PI/Z.

GOTO 91

FUNCNI=ATAN¢RZRZZ/ZS)

003320
003330
003340
003350
003360
003370
003380
003390
003400
003410
003420
003430
003440
003450
003460
003470
003480
003490
003500
003510
003520
003530
003540
003550
003560
003570
003580
003590
003600
003610

 

118

91 IFtABS(Z4).CT.I.E-6)OOTO 92 003620
FUNCN3=PI/2. 003630
COTO 93 003640

92 FUNCN3=ATAN(R4'ZZ/Z41 003650

93 CONTINUE 003660
IF(FUNCN1.LT.0.)FUNCN1=FUNCN19PI 003670
IFtFUNCN3.LT.0.)FUNCN3=FUNCN39PI 003680
FUNC91=((21991'221'92-PZS‘Z2S)*(FUNCL1-3.1-4.9R2*ZZ'(ZIRPIFZZ) 003690

ORFUNCNI 003700
FUNCP3=I(21+R3'221"2-R4S*ZZS)'(FUNCL3-3.)-4.'R4'ZE§(ZIOR3'ZZ) 003710
o'FUN2N3 003720

FUNC51=RZ'ZZ‘(ZIOR1lZZlltFUNCL1-3.)9((ZIORI'ZZ)fli2-RZS'ZZS1‘FUNCN1003730
FUNCS3=R4922*IZIOH3*22)'(FUNCL3-3.19((21623922)G'Z-R4S‘Z2SJ'FUNCN3003740

NXX=CONST1 'FLNC L1 RCONST 2&FUNC L3OCONST3NFUNCN1 ~FUNCN3) 003750
HXY=CONSTS'FUNCL1QCONST6RFUHCL39CONST7PIFUNCNl-FUNCNS) 003760
HYY=CONSTO'FUNCL1OCONST9§FUNCL3-CONST10I(FUNCNI-FUNCN3) 003770
"(I)=H(I)9(CONSTI'FUNCRIRCONSTleUNCR3RCONST3GIFUNCSI-FUNCS31) 003750
RCCOEFI'PSIJ) 003790
HX=-COEF2I(011*HXX9012*NYY42.5016'HXYIRPS(J) 003800
HT=~COEF2lt012'HXX90229HYY92.‘026'HXY)IPS(J) 003810
HXY=-COEF2I(016*HXX50269HYY92.‘066'HXY)RF$(J1 003820
BHXIII=BHXTIJOHX 003830
8HY(II=EHY(I)4HY 003540

15 CONTINUE 003850

16 CONTINUE 003860
HRITEI6.600)(I.XF(I).YF(I).H(I).8HX(I).8HY(I).I=1.NFP) 003870

END 003880
40.100.181.30.E692.0E6.0.3p8.8E4.80.90.4 003900
0..0. 003910
4.5.3.5.2.5.1.5.0.5.-0.5.-1.5o-2.5.-3.5.-4.5.10'-5.o 003920
-4.5.-3.5.-2.5.-1.5.-0.5.0.5.1.5.2.5.3.5.4.5.10'5. 003930
10"5.9’“.59’3.50'2.59'1.50'0.500.501.502-503.50“.59 003940
10'5-o“.503.592.591.590.59‘0.59‘1.59‘2.50‘3.50'“.5 003950
10'0..10*-1..10'0.310*1. 003960
10I-1.310'0..10*1..10'0. 003970
0.503.592.591.500-50’0.50'1.59‘2.59’3.59'“.50 003980
4.5.3.5.2.5.1.5.0.5.-0.5.-1.5.-2.5.-3.5.-4.5. 003990
0.503.502.501.500.59'0.50'1.50‘2.50'3.59'4.50 004000
4.5.3.5.2.5.1.5.0.53-0.5.-1.5.-2.5.-3.5.-4.5. 004010
4.5.3.5.2.5.1.5.0.5.-0.5.-1.5.-2.5.-3.5.-4.5. 004020
4.593.592.501.500.50'0.59'1.59'2-50'3.59‘“.59 003030
4.5.3.5.2.5.1.5.0.5.-0.5.-1.5.-2.5.-3.5.-4.5. 004040
4.5.3.5.2.5.1.5.0.5.-0.5.-1.5.-2.5.-3.5.-4.5. 004050
4.5.3.5.2.591.5.0.5.-0.5.-1.5.-2.5.-3.5.-4.5o 004060
0.503.502.591.500.50'0.50'1.50'2.59’3.59'“.50 004070
10"“.5010"3.5010"2.5010.'1.5010"0.59 ° 004000
10.0.5910'1.5010'2.§010‘3.5010‘4.5 004090

IFlOET.OT.0.)SOTO 40 011011

APPENDIX H

COMPUTER PROGRAM FOR THE VERIFICATION OF
EQUATIONS USED FOR ANISOTROPIC PROBLEMS

(TCIFT(TCTCTCICTCTCTCTCTCTCTf)(1C1f0€1f3f0f0€0€3f0f0f0f1(1

APPENDIX H

COMPUTER PROGRAM FOR THE VERIFICATION OF
EQUATIONS USED FOR ANISOTROPIC PROBLEMS

PROSRAH PLCLCHKfINPUT90UTFUT9TAPE5=INPUToTAPE6=OUTPUT)

THIS IS TO DOUBLE CHECK THE ANISOTROPIC PLATE PROCRAH.
USING A SIHPLY SUPPORTED SQUARE PLATE.

REQUIRED INPUT VALUES ---

NSP 8PRNBER OF BOUNDARY POINTS

NIP =HUHBER OF INTERNAL LOAD POINTS

NFP =NUHBER OF FIELD POINTS

X8oY8 =POINTS ON 8 AT HHICH B.C. ARE SATISFIED.

8ANX.8ANY =CONPONENTS OF UNIT NORNAL TO 8 AT XB’YB.

XX81YY8 gEND POINTS OF HESHES AROUND 8 HHERE FICTITIOUS
FORCES ARE ASSIGNED.

XF0YF SFIELD POINTS

XIpYI =INTERNAL LOAD POINTS

VX gPOISSONSS RATIO IN X DIRECTION
DUE TO STRESS IN T DIRECTION

EX5EY tTOUNG'S NODULI IN X AND T DIRECTIONS:
RESPECTIVELY

GXY =SHEAR HOOULUS

HVALUE *PLATE THICKNESS

RADIUS 8RADIUS OF THE FICTITIOUS CIRCULAR PLATE OF HHICH
THE DISPLACEHENT AT THE CIRCUHFERENTIAL BOUNDARY

IS SET TO ZERO.

000001
000002

MIRRRRRRRRRRRRIRRRRRRRRRRRRRRRRRRRRRRRRRRR!RRRRRRRRRRRRRRRRRRRRRRRRRRROoooo3

000004
000005
000006
000007
000008
000009
000010
000011
000012
000013
000014
000015
000016
000017
000018
000019
000020
000021
000022
000023
000024
000025
000026
000027
000028

lllil‘.INI'IHNRIIUU‘I'INI“IIIIGINHONIGIINNHUUNIUNHIII“IIIGOIIINHQINHRHOOOO29

DIMENSION X8(40)oYB(40)oXX8l81loYY8181)
DINENSION X1!1001.YI(1001.DEL(21

DINENSION RLX18019R818019RH(6098019PS(80)guKAREA(80).RL(801
DINENSION XF(181).YF(18119H(181108NX1181108NY(181I
DIMENSION 8ANXI401.8ANY(40)oRNX(80.80)

DINENSION BNXXtISl1oEFTY11811vBXY(181)

DINENSION X81t401.Y81t401:X¥81(811.TY81(81).XI1(100).YI1(1001
DIMENSION XF1(18119YF11181198ANX1(40108ANY1(40)

REAL AVECTOR(5).HX.HY.HXY
COHPLEX ERROR.ROOT(4)

100 FORNAT(FDLOCT196X9FX81.8X9‘YB‘.7X,3ANXI98X.IANY!.4X91N8TYPE1,

O/IOI/(I594F10.401511

110 FORHATlOOLOCT!.11XSFXXBi.14X0‘YT80/203/1I408X3F9.2.8X9F9.21)
200 FORNAT (IOLOCT8911X90XI¢.17X00YIFp14X9/201/

0‘1“OIIXOF603011XOF60333

300 FORHAT (00LOCT0019X00RLDFv34X.FRLS*/101/(I4.8X.E20.8o16X9

1E20.8))

400 FORHAT (FILOCT‘oI9Xo‘PSP1/200/TI4.8X:E20.811

500 FORNAT(IOLOCTthIX.IXF¢.17X.1YF!/!03/(I4911X0F6.3.11X.F6.3)1

600 FORHAT(1H1oOXo’NODEt.12X1XFI.16X.¢YFI.19X92H1916X.IBHX!.
116X.!8HYI/(1H0911092F20.1003E20.1211

700 FORNATIIHIoIINPUT VALUES .....89//1X90N8P 8 3,13.‘ NIP 32.13.

0: NFP 8 3,13,

9/1H00‘ EX 3 09:10.40‘ ET 3 09:10.00‘ VX 3 ‘9'5-20‘ GXT 3 00C10.09

o/IHO.’ RADIUS OF THE PLATE 3 ‘9

4F7.1,‘ THICKNESS OF THE PLATE =I.F6.3)

703 FORNATI///.1X.ITHE COEFFICIENTS OF THE CHARACTERISTIC 3»

OOPOLYNOHIAL ARE ----- to/1X95E12.5)

119

000030
000031
000032
000033
000034
000035
000035
000037
000033
000039
000040
000041
000042
000043
000044
000045
000040
000047
000040
000049
000050
000051
000052
000053
000054
000055
000055
000057
000050
000059

C

C
C

120

704 FORHAT(IHO.ETHE POUR ROOTS OF THE CHARACTERISTIC EQUATIONzP.
4(/1X.2E12.51)
705 FORHATIIHOo‘ ERROR FOR ROOT NO. PpIZoZX92E13.4o‘ IS ‘92E10.4)
707 FCRHAT(1H09ETHE FOUR CONSTANTS ARE :Eo
9/1xot‘Lpfl‘ 3 .0E10.40‘ BETA 3 ‘0!1°.“g
9‘ GAHHA 3 P9E10.“41 LUHEOA 3 30E10.41
1103 FORHATI1595X9E20.1275X9E20.121

1014 FC°HAT(1H1.‘THE FOLLOHINS IS A LIST OF DOUBLE CHECKING OF B.C.S‘,

1//.QX’NSP$10X9TB.C.
708 FOPHAT11H0gtRADIUS =

1’.13X.‘8.C.
‘9E10.3o‘

Zloll)
DISTI 3 P4F6.19

0’ DIST2 = ¢4F6.1)

801 FORHATIIHIo‘THE SENDING RIGIDITIES APE ----- 3:
0/1X4'011 = 34E10.4.l 012 = 3.E10.64i 022 = ‘oE10.44
O/1X43O66 = 14E10.4.l 016 8 PoE10.4.1 026 = ‘.E10.4)

805 FCRNATIIngiTHE AIJ ARE ---P./1X9‘A11= ioE15.7.P A12: 1.
0E15.79‘ A22= ‘oE15.7.‘ A66: 8.E15.7.‘ A16= ‘4E15.71
0! A26: 35E15.7)

INPUT VALUES .........

READ‘54'INBP9NIPyNvaEXoE79VXvGXY,RAOIU55HVALUE
RE‘O‘So'ILXB1117I=10NBP1
READ(59.’(YO‘I)0I=10NBP)
READ‘50'1(OANX(ITolzlvNBFJ
READ‘59P)‘EANT(I191:1.NBP)
REAO(59P)IXI(I791319NIP1
READ‘59')(TI(I’7I=19NIP)
NBP2=NBPPZSNBPEPI=NOFZOI

NOPTION=I

91:“.RAT‘N(1.,

NCOUNT=1

A1630. 0 A2630.

A11=I./Ex 0 A123'VX/EX 5 A2231./ET 0 A6631./5XT
"RITE(608051‘110A129A225A660A169A26
OET=LAIIPA22'A12'PZ)PA66‘2.PAIZ'AI0”A26'AIIHA26‘HZ'A22‘A16P'2
ZZ=HVALUE"3/(12.0DET)

0113(A22'A66-A260'21‘22 0 022=(A11'A66-016'&2)&ZZ
012=|A16'Az6-A12'A66)‘zz 0 O66=IAIIRA22-AIZ"2)‘ZZ
OI6=(A12'A26'AZZPA16)PZE 0 O26=(A12.A16-A11‘A26)‘zz
RADIUS‘OO. 0 DISTI=Z.D 0 OIST232.
OK=GXY‘HVALUE"3/12. 0 VT‘ET'VX/EX

O¥=O11 0 OYPOZZ 0 O330X‘VX62.'OK

":03 5 RNO‘U/SQ°T(OX’OY)
HRITE‘607OO)NBPvNIPoNvaEXoEToVX0sXTo
ORAOIU57HVALUE

"RITEI607OO)RAOIU59OISTloOIST2
"RITE(64709)OX9OY9RHO

EOPNATl/l/1X9’OX 3 '9E15.705X9‘OY :‘oE15.795X0‘RHO =P9FIO.5)
X035.00T0=5.0

XFII1:10.'XO0TF(1’='10.9TO

OO “1 132711

XEII|=XF(I'I)'0.5

TEtli=VE(I’I)

OO 42 J31710

K=J*17

OO “2 I‘loll

XEIIOKIPXF(I)

42 YFIIOKIPYE(IOK'I7)OI.O

709

41

000060
000061
000062
000063
000066
000065
000006
000067
000068
000069
000070
000071
000072
000073
000074
000075
000076
000077
000078
000079
000080
000081
000082
000083
000086
000085
000086
000087
000088
000089
000090
000091
000092
000093
000094
000095
000096
000097
000098
000099
000100
000101
000102
000103
000106
000105
000106
000107
000108
000109
000110
000111
000112
000113
000114
000115
000116
000117
000118
000119

43

44

25

26

27

28

804

807

301

302

303

304

121

XF(12)=10.-X0$YF(12)=-9.59Y0

00 43 1=13.17

XF(I)=XF(1-1)-1.0

YFtI)=YF(1-1)

DO 44 J=1.9

K=Jl17

00 44 1:12.17

XF(14K)=XF(1)

YF114K1=YF(1+K-17)91.0
0EL(1)=l10.92.'015T1)/10.
0EL(2l=(10.92.'015T142.'015T2)/10.
XX8(1)=5.9DIST1-DEL(1)0YY8(1)=-5.-DIST1
XXB(41)=5.¢DIST19015T2-DEL(2)5Y18t41i=~5.-DIST1-DIST2
00 28 J=1.2
0ELT=0ELI1)‘IFIJ.EQ.2)OELT=DELIZ)

DO 25 1:2.10

K=I

1F(J.EO.2)K=K440
XXB!K)=XXB(K-1)-DELT

YYBIK)=YYE'K-1)

00 26 1:11.20

K=1

1F(J.EG.2)K=K440

xxatxr=xxacx-11

YY8(K1=Y78tK-1)4DELT

00 27 1=21.30

K=1

IFCJ.E0.2)K=K+40

XX8(K)=XXB(K-1)+DELT

YYBtK):YYBtK-1)

00 28 1=31.40

K=I

1FlJ.EQ.2)k=K440

XX8(KI=XXB(K-1)

YYB(K)=YY8(K-1)-0ELT

xxa1011=xx01411 0 Y781811=YYB(41)
PH1=0.

HRITE!6.807)FH1

F0:HAT(1H0."II HHEN PHI IS 1.F7.3.5X.tDEGREES --¢)
FHI=PH10PIII80.

SINP=SIN£PH1) 0 COSP=COStPH1I

00 301 1:1.181
XF1l11=XF(I)‘COSP9YF(1)'SINP
YF1(1)=-XF(1)‘SINPoYF(I)'COSP

00 302 1=1.40
X81111=X811)“COSP9Y8(1)'SINP
Y81(1)=-X8(II¥SINP9Y8(1)*COSP
BANXII1)=8ANX(1)ICOSP+8ANY(I)‘SINP
8ANY11118-BANXI1)ISINP48ANY(I)ICOSP
00 303 I=1981
XXBIIII=XXB(I)!COSP9YY8tIDISINP
YYBltI)8-XX8(I)ISINP41Y8(IDICOSP

00 304 181.100
X11!1)=X1(1)ICOSP4Y1(1)*SINP
Y11t1)=-X1(11'SINP9Y1(1)¢COSP
HRITE(6.100)(14X8111)4Y81l1).BANX1(I).8ANY1(1).I=1.N8P)
HRITEI6.110)(1.XX81(1)oYY8111)oI=19N8P2P1)
HRITEI64500)II.XF1(I)9YF1(I).1=1.NFP)
HRITEI6.200)(19X11(114111(1191=1.N1P)
COSP2=COSPv"‘ 0 cosp4=cospzbuz

000120
000121
000122
000123
000124
000125
000126
000127
000128
000129
000130
000131
000132
000133
000134
000135
000136
000137
000138
000139
000140
000141
000142
000143
000144
000145
000146
000147
000148
000149
000150
000151
000152
000153
000154
000155
000156
000157
000158
000159
000160
000161
000162
000163
000164
000165
000166
000167
000168
000169
000170
000171
000172
000173
000174
000175
000176
000177
000178
000179

706

60
61

62
63

122

SINPZ:SINPPPZ 0 SINPQFSINRZPHZ
COSZP=COS(2.'PHI’ 0 OIN29331N12.'PHI1
011=DX¢COSF492.”03*SIHP2'COSP290YISINP4
022=OXPSINF492.'OS'SINPZ‘COSPZ‘OT‘COSPQ
066=OK61OX§OT'24.O3)PSINPZPCOSPE
Olz=OTPVY¢1OX§OY-2.'O3)lSINPZ'COSPZ
O16=°.5.1OT‘SINPZ'OX'COSPZ‘O3‘COSZP1'SIN2P
026:0.5'1OT'COSPZ'OXESINPZ'O3‘CCSZP)“SINZP
"RITE169OOI,O110012002200669O160026
AVECTOR‘I”I.O 0 AVECTOR121=4.'DZ6/022
‘VECT0913)=z.'(OIZ§Z.POOO1/O22
AVECTOR191=Q.POIO/O22 0 AVECTOR1513011/OZZ
"RITE16070311AVECTOR11191:1951

CALL ZPOLR1AVECTOQo‘0ROOT01ER)
“RITE16070311ROOT11101310“)

DO 706 I319“
ERROR=AVECTOR111‘RCOT1I’PPQOAVECTOR121'ROOT‘ITNHIOAVECTOR(31*
9ROOT11)“PZOAVECTOR141'POOT11TOAVECTOR151
”RITE169705110ROOT11ivERROR

RI‘REAL‘ROOT1111 0 R23AIHA51ROOT(I1)
R3=PEAL1ROOT1311 0 RQ=AIHAOIROOT1311
"RITE1647071R19R29R35R4

R15=Rl"2 0 R233RZPPZ 0 R333R3"2 0 RQS=R4PSZ
CONSTG=1RI'R3)'PZO(RZOR41"2 0 CONSTN=(RI’R3)P'ZO(R2‘R41":
COEF1=1./(O..RI'022'CONSTG'CONSTH1 0 COEF2=2.flCOEF1
CONST131‘RI‘RS’PRZ-‘RZS'RQS11/Rz
CONST2=I‘RI'R31'PZ01R23'R4511/R“
CONST3:Q.P(R1'R3)
CONST5=I(R1'2.'R31.(R139R2316R1.1R330R4511/R2
CONST631(R3'2.'R11’1R350R451*R3'1R130R23)1/R4
CON$T7=2.‘(RIS6R25‘R35'RQSI
CONSTO31(RIS‘RZS'E.NRINR3)'(R159RZS19(R15'RESTPIR339RNS11/R2
CONST9=1(R359R45'2.'R1'R31'1R3S9RQS19(R35'R431'1R150RZS)1/R4
CONST10=4.'(RI‘(R3S‘R451-R3'1R159R23)1
GLOA0=1.0 0 02=RA0103552

00 5 I=1oNSP

RLLI1=0.0

RL‘I‘NER):0.0

R1X=X81(1)

R1Y=Y81111

ANX=BANX1tIl

ANY=BANT11I1

OO 6 J=IoNIP

RZX=X111J1

R2Y=7111J1

ZI=R1X-R2X 0 ZZ‘RIY-RZY 0 ZISPZIPPZ 0 225322"2
ANXS3ANXIl2 0 ANT530NTP'2 0 INXT=ANXPANT
FUJCL1=AL051((ZIORIPZE1"2’R23PZZS1/A21
FUNCL3=AL051((216R3'ZZ1PPZOR9502251/‘21
Z3=ZI§R1‘22 0 z4=ZI9R3EZZ
IE1ABS1ZS’.GT.1.E'O’GOTO OO

FUNCN1=PI/2.

GOTO 61

FWCN13ATATHR2'ZZ/231

IE1‘BS1Z“).GT.1.E'OTGOTO 62

PUVCN33RI/2.

GOTO 63

FUNCN3=ATANIR4IZZ/Z4)

CONTINUE

IF‘EUNCN1.LT.O.TRUNCNIPFUNCNI9PI

000180
000181
000182
000183
000184
000185
000186
000187
000188
000189
000190
000191
000192
000193
000194
000195
000196
000197
000198
000199
000200
000201
000202
000203
000204
000205
000206
000207
000208
000209
000210
000211
000212
000213
000214
000215
000216
000217
000218
000219
000220
000221
000222
000223
000224
000225
000226
000227
000228
000229
000230
000231
000232
000233
000234
000235
000236
000237
000238
000239

6
5

33

70
71

72
73

123

1F1FUNCN3.LT.0.)FUNCN3=FUNCN3+PI
AA=011dANXS90125Anv542.5016CANXY
SBSANXS‘0120022'ANYSOZ.IANXT'026

€082 .0016uNx592 . .0269ANY594 . 5066“th

“H.201“ 111401-22 )«2-02352251-17154011-3. )-4.IR2!ZZ!(ZI+R1522 )

O‘FUNCNI

FUNCR3=t(ZIOR3‘ZZ1*‘2-R4S*2251'(FUNCL3-3.)-4.'R4‘ZZ'IZI¢R3*ZZ)

“FUNCNS

000240
000241
000242
000243
000244
000245
000246
000247

FUNCS1=R2RZZRCIIOR1'ZZ1'1FUNCL1-3.101(21¢R1'22)G'Z-RZS*ZZS)iFUNCN1000248
FUNCS3=R4'ZZ*IZIOR3'ZZ)fllFUNCL3-3.10¢(ZIOR3'221"2-R4S‘ZZSIRFUNCN3000249

NXX=CONST1IFUNCL1OCONSTZIFUNCL3OCONST3'1FUNCN1-FUNCN3)
NXY=CONST5*FUNCLIOCONST6IFUNCL39CONST7*(FUNCN1-FUNCN3)
NYY=CONST8*FUNCL1*CONST9'FUNCL3-CONST10'1FUNCNl-FUNCN3)
RLII)=RL(I)-QLOAD§(COHSTl'FUNCRI+CONST2'FUNCR3OCONST3*

4(EUNC31-EUNC5311

RLIIONBP1=RL1IONBP1-OLOA0'fAA‘NXX§BB*HYY+CC'HXY)
CONTINUE
CONTINUE

OO 8 I=19NBP

RIX:XOI(I)

R1T=T011I1

ANX‘OANXI‘I)

‘NT’BANYIII’

DO 7 J31 .NBPZ
RZX=IXXOIIJ011OXXO11J11/2.
RZT=(TTOI‘J6119YYBI(J)1/2.
IE1NOPTION.ER.2.0R.J.NE.NBP)GOTO 33
R2X=tXX8l11¢XX8tNBP))/2.
R2T317T81119TT31N3911/2.

CONTINUE

ZI=R1X-R2X S 22=R1Y-RZT 0 215=ZIPHZ 0 Z25=22**2
ANXSgANXP‘Z 0 ANY5=ANT"Z E ANXY3ANX'ANT
FUNCLI’ALOGI1(210RIPZZ1"20R2592251/A2)
FUNCL3=ALOSII(ZIOR3*zZ1"2‘R45‘ZZS1/A2)
Z3=ZI9R1822 0 Z4=ZI§R3'ZZ
IF11351231.GT.1.E°6)GOTO 7O

FUNCHl=PIl2 .

GOTO 71

FUNCN1=ATAN1R2'22/Z3l
IF‘ABS1ZQ1.GT.I.E-6)GOTO 72
FUNCN3=RI/2.

SOTO 73

FUNCN3=ATAN1RQPZZ/ZQ1

CONTINUE
IP1EUNCNI.LT.O.1EUNCN1=EUNCNI§PI
IFIFUNCN3.LT.D.TFUNCN3=FUNCN3OPI
AA=OIIPANXS4DIEPANYSOZ.“OIS'ANXY
OO=ANXSPOIZODZZPANYSOZ.‘ANXY'OZS
00:2.‘016'ANX862.‘026'ANY394.'066'ANXY
PUNCR1=C(219R1'221"2-R25'ZZS1'1FUNCLI-3.1-8.'R2*22*(ZIOR19121

0iFUNCN1

FUNCR3=1(21¢R3l221"2-R4S'ZZS)'(FUNCL3-3.1-4.RR4'22'(ZIOR3*221

OIFLMCN3

000250
000251
000252
000253
000254
000255
000256
000257
000258
000259
000260
000261
000262
000263
000264
000265
000266
000267
000268
000269
000270
000271
000272
000273
000274
000275
000276
000277
000278
000279
000280
000281
000282
000283
000284
000285
000286
000287
000288
000289
000290
000291
000292
000293
000294

FUNC51=RZRI2PIZIORIUZ2I'lFUNCLI-3.10((219R1'Z2li'Z-RZS‘ZZS)lFUNCN1000295
NMCS3=R4PZ2NZI¢R3RZZ )ll FLNCL3-3. ”(1210113922 M'Z-R‘oS'ZZS HFMN3000296

NXX=CONST1lFUNCLIOCONST2IFUNCL3OCONST3'IFUNCNl-FUNCN3)
WTSCWSTS'FUJCLI OCGCST6RFWC L3OCWST7I(FUNCN1-FLNCN3 1
NTY‘CONST8"”"CLIOCONST9IFUNCL3-CONST10'IFUNCNl-FUNCN3)

000297
000298
000299

10

1015

29

1102
1101

80
81

82
83

124

Rut1.J1:!CONSTIIFUNCRl+CONST2'FUNCR3OCONST3“(FUNCSI-FUNCS3))
Rflt1+NBP.J)=(AA'HXX¢BBIHYY4chuxy1

CONTINUE

CONTINUE

00 10 181.N8P

DO 10 J=I.NEPZ

RHXI1.J)=RH(1.J)

RNXtIoNBP.J)=RH(ION8P.J)

DO 1015 I=14NBP2

RLX(I)=RL(I)

CALL LEOTlFtRH.1.NBPZ.NBP2.RL.0.HKAREA.IER)
00 29 1:1.NBP2

PStI)=PL(1)

HRITE (6.400) (I.PS(I).I=1.NBP2)
HRITE(6.10141

DO 1101 1=1.N3P2

SUH=0.

00 1102 J=1.NBPZ

5UH=5UH40HX(1.J)&PS(J)

CONTINUE '

R8(I)=SUH-RLX(I)
NRITE(6.1103)(1.RB(IJ.RB(IoNBP).1=1.NBP)
00 9 1:1.NFP

H111=0.0

8HX(I1=0.0

BhYtI)=0.0

8XYlI)=0.

BHXX(II=0.

BHYYlII=0.

CONTINUE

OO 14 1:1.NFP

Rlx=XF1(1)

R1Y=YF111)

OO 13 J=1.NIP

R2X=X11(J)

RZY=YII(J)

21:91x-nzx 0 22:01v-02Y 0 le=Zli‘2 0 225:22942
FUNCL1=ALOGI ( (2191211422 15529029225 )MZ)
FUNCL3=ALOG(((2140352215'29R455225)/A2)
23:21901522 0 29=ZI+RSI22
IFIABSIZSD.GT.1.E-6)GOTO 60

FUNCN1=PI/2.

GOTO 01

FUNCN1=ATAN(R2'22/Z3)
IFIABS(Z4!.GT.1.E-6)GOTO 82

FUNCN3=P1/2.

GOTO 03

FUNCN3=ATANIR4*Z2/Z41

CONTINUE
IFIFUNCN1.LT.0.)FUNCN18FUNCN14PI
IFtFUNCN3.LT.0.)FUNCN3=FUNCN39PI
FUNCP1=I(Zloll‘zz)052-R2852253!(FUNCL1-3.)-4.!R2*22*(219R1522)
uFUNCNl

PUNCH!” (7.1493522 1442-0459zst-(rw013-3. )-4.IR4lZZI( 21911342)
4IFWCN3

000300
000301
000302
000303
000304
000305
000306
000307
000308
000309
000310
000311
000312
000313
000314
000315
000316
000317
000318
000319
000320
000321
000322
000323
000324
000325
000326
000327
000328
000329
000330
000331
000332
000333
000334
000335
000336
000337
000338
000339
000340
000341
000342
000343
000344
000345
000346
000347
000345
000349
000350
000351
000352
000353
000354
000355
000356

FUNCS1=R2'22'1210R1*ZZ11(FUNCL1-3.10((ZI‘R1'221*l2-RZS*ZZSTGFUNCN1000357
FUNCS3=R4'ZZI(ZIOR3*ZZ)‘(FUNCL3-3.)6!(21¢R3*22)GIZ-R4S'ZZS)iFUNCN3000358

NXX=CONST1*FUNCL19CDNST2‘FUNCL3OCONST3'(FUNCNI-FUNCN31

000359

13
14

11

90
91

92
93

15

125

NXYBCONSTS'FUNCLIOCONST6'FUNCL3¢CONST7'(FUNCNl-FUNCN31
HYY=CONST8EFIMCL1OCONST9lFLNCL3'CWST10'rf FUNCNI ~FLMCN31
“(I13H!I161CONSTI'FUNCRIOCONSTZ'FUNCR39CONST3'LFUNCSI-FUNCS311

O'COEFIROLOAD

HX=-COEF2§(DII'HXXOD12'NYY92.I016lNXY1lQLOA0
HY=-COEF2'(012'HXX9022§NYY¢2.‘DZ6'NXY1|0LOAD
HXY=-COEF2'(016*“XX4026'HYY62.5066*NXY1*QLOAD
8thI1=8HX(I19HX

8HT(I1=8HY(I14HY

8XY(I1=8XY(I14HXY
8HXX(I1=8HXII1'COSP298HYtI1ISINP2-2.i8XY(I1'COSP'SINP
8HYY11188HX(I1‘SINP298HYII1*COSP2-2.'8XY(I1'SINPICOSP
THE NEGATIVE SIGN IS FOR A REVERSED ANGLE.
CONTINUE

CONTINUE

DO 16 I=14NFP

R1X=XF11I1

R1Y=YF1(I1

DO 15 J=1.N8P2

R2X=lXY81tJ9114XX81(J1)/2.0
R2Y=1YT81(J§11+YYBI1J11/2 . 0
IFTNOPTION.EQ.2.0R.J.NE.N8P1GOTO 11
R2X=(XX81(119XX811N8P11/2.
R2Y=(YY81(119YY81(N8P11/2.

CONTINUE

ZI=R1X-R2X 9 22=R1Y-R2Y 0 218321'i2 0 ZZS=22*'2
FUNCL1=ALOG(t(ZIORI'ZZ1l*20R25522$1/A21
FUNCL3=ALOG(((214R3*221"29R4S'ZZS1/A21
l3=ZI¢R1'22 9 24=21+R3*22
IF(A88(Z31.GT.1.E-61GOT0 90

FUNCN1=PI/2.

GOTO 91

FUNCN1=ATAH(R2522/Z31

IF(A85(Z41.GT.1.E-61GOTO 92

FUNCN3=PI/2.

GOTO 93

FUNCN3=ATAN1R4'22/Z4 1

CONTINUE

IFtFUNCN1.LT.0.1FUNCN1=FUNCN19PI
IF!FUNCN3.LT.0.1FUNCN3=FUNCN3OPI
FUNCR1=((ZI+R1*ZZ1*‘2-R25*2251'(FUNCL1-3.1-4.*R2*22*(21#R15221

OGFUNCNI

FUNCR3=1(ZIOR38221"2-R4S'ZZS1‘1FUNCL3-3.1-4.'R4'22*1210R3*221

OlFUNCN3

000360
000361
000362
000363
000364
000365
000366
000367
000368
000369
000370
000371
000372
000373
000374
000375
000376
000377
000378
000379
000380
000381
000382
000383
000384
000385
000326
000387
000388
000389
000390
000391
000392
000393
000394
000395
000396
000397
000398
000399
000400
000401
000402
000403

FUNCSI=R2*ZZ‘1ZIORIPZZ1ltFUNCL1-3.1O((ZIOR1*221"2-R25'ZZS1'FUNCN1000404
FUNCS3=R4'22*(ZIOR3*ZZ1'(FUNCL3'3.101(ZIOR3'ZZ1*”2-R45'ZZS1'FUNCN3000405

NXX=CONST1*FUNCL19CDNST2'FUNCL3‘CONST3'1FUNCNI-FUNCN31
NXY=CONST5*FUNCL1OCONST6lFUNCL3OCONST7§(FUNCNI-FUNCN31
NTY=CONST85FUNCL1OCONST9'FUNCL3-CONST10'1FUNCNI-FUNCN31 ‘
N(I1=H(I1o(CONSTI'FUNCRI*CONSTziFUNCR3OCONST3‘(FUNCSI-FUNCS311

O'COEFI'PSTJ1

HX=~COEF2'(DII'HXXODIZ‘HYYOZ.‘016'NXY1lPS(J1
HY=-COEF2'(DIZ’HXXODZZ'HYY92."DZb‘NXY1'PS(J1
HXYS-COEF2'1016‘NXX+026‘HYY02.“066‘NXT1§PS(J1
BHX1I1=BHY(I1OHX

DHY(I1=8HT(I1OHY

OXY1I1=8XYtI1OHXY
DH¥X(I1=BNX(I1'COSP248HY(I1'SINP2-2.'8XY1I1'COSP'SINP
BHYT(I1=8HX(I1*SINPZOBHYII1'COSP2-2.IBXY(I1'SINPRCOSP
CONTINUE

000406
000407
000408
000409
000410
000411
000412
000413
000414
000415
000416
000417
000418
000419

 

4.593.542.591.5.0.59

...-0‘...

0 #Tfl‘flUIUIUIUHUIdId:

FIO'O‘S‘C’U'O‘OHO1’
“a

126

CONTINUE

"'ITE‘6060071I.XFII)0YF(IIOH(I7OBHXX(I'DBHTY11191310NFP,

SOTO!49.50.51.52.531NCOUNT
PHI=15. 0 NCOUNT=2
SOTO 804
PHI=30. 0 NCOUNT=3
SOTO 804
PHI=45. 0 NCOUNT=4
SOTO 804
PHI=60. 0 NCOUNT=5
GOTO 804
STOP

END

40,100,181.30.E692.0E6.0.398.8E4.80.40.4
-0.5.-1.5.-2.5.-3.5.-4.5.10*-5.9
-4.54-3.5.-2.59-1.5.-0.5.0.5,1.5.2.5.3.5.4.5.10'5.
ION'S.o'Q-50‘3.50’2.50’1.50’0.50O.501.502.503.50“.50
10'5.44.5.3.5.2.5.1.5.0.5.-0.5.-1.5.-2.5.-3.59-4.5
10.0.0100'1.010'O.510.1.

910.0. 910.1. .1050.

4. 5 3. 5 2.5 1. 5. 0. 5.-0.5.-1.5.-2.5.-3.5.-4.
5 0.50-0.50'1.50’2.50'3.59‘“.
5 0.54-0.5.-1.5o-2.5.-3.5.-4.
5 0.54-0.54-1.5o-2.5.-3.5.-4.
5 0.50’0450'1.50'2-50'3450'“.
5 0.54-0.59-1.5.-2.5.-3.5.-4.
.5.0.5.-0.5.-1.5.-2.5.-3.5.-4.
5 0
5 0
.5 0.
.5 1
1 !

0'-2 5910i- 1. 5 105-0. 5:
2. 5 10'3. 5.10.4. 5

.55'0-50'1.50'2.50‘3.50'“.
.50'0. 50-1 50-2 59-3 50'“.
50-0 50’1457’24 59-3-50'“.

59
So
5.
59
5.
5.
59
5.
5:
So

000420
000421
000422
000423
000424
000425
000426
000427
000428
000429
000430
000431
000432

000434
000435
000436
000437
000438
000439
000440
000441
000442
000443
000444
000445
000446
000447
000448
000449
000450
000451
000452

 

 

BIBLIOGRAPHY

10.

11.

12.

13.

BIBLIOGRAPHY

Timoshenko, S. and Woinowski-Krieger, 5., Theory of
Plates and Shells, McGraw-Hill Book Co., New York, (1959).

 

 

Muskhelishvili, N. I., Some Basic Problems of the Mathe-
matical Theory of Elasticity, Noordhoff, Holland, (1963).

 

Lourye, A., Application of Complex Variables Method to
Plate Problems, Journal of Applied Mechanics, USSR Aca-
demy of Science, 4, (1940).

 

 

Green A. E. and Zerna, W., Theoretical Elasticity, Oxford
University Press, New York, (1954).

Stevenson, A. C., Complex Potentials in Two-Dimensional
Elasticity, Proceedings, Royal Society of London, Series A,
184, (1954).

 

Morkovin, M. V., On the Deflection of Anisotropic Thin
Plates, Ph.D. Dissertation, University of Wisconsin, (1942).

Szilard, R., Theory and Analysis of Plates, Prentice-Hall,
Inc., (1974).

Zienkiewicz, O. C., The Finite Element Method, McGraw-Hill
Book Co., New York, (1979).

Altiero, N. J. and Sikarskie, D. L., A Boundary Integral
Method Applied to Plates of Arbitrary Plan Form, Computers
and Structures, (1979).

 

Michell, J. H., The Flexure of Circular Plates, Proceedings
of the Mathematical Society of London, (1902).

Suchar, M., On Singular Solutions in the Theory of Aniso-
tropic Plates, Bulletin de 1'Academie, Polonaise des
Sciences, Series des Sciences Techniques, 13, (1964).

Mossakowski, J., Singular Solutions in the Theory of Ortho-
trOpic Plates, Archives of Mechanics, (1954).

Mossakowski, J., Singular Solutions in the Theory of Aniso-
trOpic Plates, Archives of Mechanics, (1955).

127

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.
27.

128

Lekhnitskii, S. G.. Anisotropic Plates, Gordon and Breach
Science Publishers, (1968).

 

Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic
Elasticity Body, Holden-Day, Inc., (1963).

 

Ambartsumyan, S. A., Theory of AnisotrOpic Plates, Tech-
nomic Publishing Co., Inc., (1970).

 

Jaswon, M. A. and Maiti, M., An Integral Equation Formu-
lation of Plate Bending Problems, Journal of Engineering
Mechanics, II, (1968).

 

Maiti, M. and Chakrabarty, S. K., Integral Equations for
Simply Supported Polygonal Plates, International Journal
of Engineering Science, 12, (1974).

 

 

Jaswon, M. A., Maiti, M., and Symm, G. T., Numerical Bi-
harmonic Analysis and some Applications, International
Journal of Solids and Structures, 3, (1967).

Benjumea, R. and Sikarskie, D. L., On the Solution of
Plane Orthotropic Elastic Problems by an Integral Method,
Journallof Applied Mechanics, 32, Transaction of ASME,
23. (1972).

 

Altiero, N. J. and Sikarskie, D. L., An Integral Equation
Method Applied to Penetration Problems in Rock Mechanics,
Boundary Integral Equation Method: Computational Appli-
cations in Applied Mechanics, Edited by T. A. Cruse and
F. J. Rizzo, ASME, AMD-11, (1975).

 

 

 

Bares, R. and Massonet C., Analysis of Beam Grids and
Orthotropic Plates, Frederick Ungar Publishing Co., Inc.,
New York, (1968).

 

Vinson, J. R. and Brull, M. A., New Techniques of Solution
for Problems in the Theory of Orthotropic Plates, Pro-
ceedings 4th ULS. National Congress of Applied Mechanics,
University of California, Berkeley, June 18-21, (1962).

Rajappa, N. R. and Reddy, D. V., Analysis of an Orthotropic
Plate by Maclaurin's Series, Journal of Royal Aeronautical

Society, 61, (1963).

Mazumdar, J., A Method for Solving Problems of Elastic
Plates of Arbitrary Shape, University of Adelaide,
Adelaide, Australia, (1968).

 

Huber, M. T., Theory of Plates, L'vow, (1921).

 

Lekhnitzskii, S. G., Plane Statical Problem of the Theory
of Elasticity of Anisotropic Body, Prikladnaya Matematika
i Mekhanika, I, l, (1939).

 

28.

29.

30.

31.

129

Ashton, J. E., Halpin, J. C., and Petit, P. H., Primer
on Composite Materials: Analysis, Technomic Publishing
Co., Inc., (1969).

Edited by Weeton, J. W. and Scala, E., Composites: State
of Art, Proceedings and Sessions of 1971 fall meeting,

The Metallurgy Society of the American Institute of Mining,
Metallurgical and Petrolium Engineers, Inc., New York,
(1974).

Design with Composite Materials, The Institute of Mecha-
nical Engineers, Great BritEin, (1973).

 

Impact of Composite Materials on Aerospace Vehicles and
Prgpulsion Systems, NATO Advisory Group for Aerospace
Research and Development, Conference Proceedings, 112,
(1972).

 

 

"7114141174144“