A MATHEMATICAL MODEL FOR THE NONLINEAR ANALYSIS OF LAMINATED

GLASS PLATES SUBJECTED TO LATERAL PRESSURE

by

MAGDI EMILE MOHAREB, B. of C.E.

A THESIS

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

CIVIL ENGINEERING

Approved

August, 1990

COV'^ ACKNOWLEDGMENTS

The author wishes to express his deepest appreciation to his advisor

Dr. C , V. G. Vallabhan, for his vzduable encouragement and guidance in this

research. Sincere indebtedness is expressed to Dr. Y. C. Das, for taking the time

to revise the derivations presented in this document. In addition, the author

wishes to express his deepest gratitude to his family members for their patience,

support, and encouragement during the period of this research.

This work has been supported mainly by the National Science Foundation

under research grant number CES-8803146. Additional financial support &om

Monsanto Chemical Co., St. Louis, is also acknowledged.

u

TABLE OF CONTENTS

ACKNOWLEDGMENTS ii

LIST OF FIGURES iv

LIST OF SYMBOLS v

CHAPTER

1. INTRODUCTION 1

2. A MATHEMATICAL MODEL FOR LAMINATED GLASS UNITS 7

3. FINITE DIFFERENCE EXPRESSIONS FOR FIELD AND BOUNDARY EQUATIONS 21

4. SOLUTION ALGORITHM 39

5. EXAMPLE PROBLEMS 46

6. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 60

LIST OF REFERENCES 63

APPENDIX 65

m

LIST OF FIGURES

1.1 Layered Glass Units 6

1.2 Monolithic Glass Units 6

1.3 Sandwich Plate Units 6

1.4 Laminated Glass Units 6

2.1 Laminated Unit in the Deformed Shape 19

2.2 Axes of reference 20

3.1 Finite Difference Mesh for Lateral Deflection 37

3.2 Finite Difference Mesh for In-Plane Displacements 38

4.1 Interpolation Parameter a vs. Wmax/hav 45

5.1 Problem 1- Pressure vs. Maximum Deflection 52

5.2 Problem 1- Pressure vs. Stresses at the Center 53

5.3 Problem 1- Pressure vs. Maximum Principal Tensile Stress 54

5.4 Problem 1- Pressure vs. Maximum Principal Compressive Stress 55

5.5 Problem 2- Pressure vs. Maximum Deflection 56

5.6 Problem 2- Pressure vs. Stresses at the Center 57

5.7 Problem 2- Pressure vs. Maximum Principal Tensile

Stress 58

5.8 Problem 2- Pressure vs. Maximum Principal Compressive Stress 59

IV

LIST OF SYMBOLS

W = Matrix containing linear terms of the lateral displacement field equation

[B] = Matrix containing linear terms of the in-plane

displacement field equations

Di = Flexured Rigidity of the top plate

D2 = Flexural Rigidity of the bot tom plate

E = Glass Young's modulus

Ci,«7 Ci.y, Ci xy = Strain components of the top plate

^2,i^^2,y,€2,xy = Strain components of the bottom plate

G = Glass shear modulus

hi = Thickness of the top plate

/12 = Thickness of the bottom plate

hx = Finite diflference mesh subdivision length in the

X direction

hy = Finite difference mesh subdivision length in the

y direction

Ig = Length of the plates in the x direction

ly = Length of the plates in the y direction

nx,ny = Number of subdivisions in the x and

y directions, respectively

q = uniform applied lateral pressure

{RHS} vector containing the right hand side of the lateral displacement field equation

t = Thickness of the interlayer

= In plane displacement for the top plate in the X direction

U2 In plane displacement for the bottom plate in the X direction

F<;'

F<"

n'

^ xz 1^ yz

Vl

= Bending strain energy for the top plate

= Bending strain energy for the bottom plate

= Membrane strain energy for the top plate

= Membrane strain energy for the bottom plate

= Shear strain energy for the interlayer due to 7x2j7yz» respectively

= In plane displacement for the top plate in the y direction

V2

V

= In plane displacement for the bottom plate in the y direction

= Total potenti2Ll energy of the system

w — Lateral deflection of the top and bot tom plate

{w}

w max

= Lateral displacement vector

= Maximum lateral deflection

= Rectangular coordinates

= Shear strain components of the interlayer

VI

a,/3 = Interpolation parameters

^ = Poisson's ratio for the glass

Tj^ = Arbitrary displacement function Ui

0 = Angle undergone by an originally vertical fibre

of the interlayer

4> = Angle undergone by an originally horizontal fibre of the interlayer

n = Load potential energy function

v i i

CHAPTER 1

INTRODUCTION

Laminated Glass

Laminated glass consists of two monolithic glass plates glued together by an

elastomeric material such as the polyvinyl butyral to form one unit. Since it is a

combination of two or more plates, it is called laminated glass unit in literature

[1]. The elastomeric interlayer has a typical thickness varying from 1mm to

2mm. The material properties of the interlayer are completely different from

the properties of the glass. While the modulus of shear of the glass is about

4 X lO^pai, the corresponding modulus of the interlayer lies between lOOpsi and

300p5r at room temperature. This value is expected to drop further for higher

temperatures [1]. Laminated glass units have been extensively used in the manu­

facturing of aircraft and automobile windshields. A relatively recent application

is its use in the building industry as a cladding component. Laminated glass

units have been recently used in manufacturing insulating glass units, overhead

glaring, and safety glaring. They have the advantage of withstanding high wind

pressures and missile impacts without being susceptible to a total collapse or

dangerous shard formations.

In the building industry, it is commonly required to place laminated glass

units on very large openings. Laminated glass units are becoming popular for

use in buildings. Under extreme wind pressures, the out of plane deflections of

such plates are expected to exceed several times the nominal thickness of these

plates. Therefore, a flnite strain approach has to be used in deriving the fleld

equations and the boundary conditions governing the system. A small strain

approach, while simplifying the problem considerably, is too conservative to

follow, especially for high pressures and large spans.

Review of Previous Related Research

The nonlinear analysis of plates under bending action was investigated by

mzmy researchers. In 1910, Von Karman [17] developed a nonlinear plate bend­

ing theory for thin plates under lateral loads. The resulting field differential

equations being nonlinear, no close form solution could be obtained for most

practical problems. In 1936, Kaiser [8] solved the problem for a simply sup­

ported square plate. Later, Levy [10] gave a close form solution to the problem

for different edges condition using double Fourier series. In 1980, Tayyib [16]

developed a finite element.model for the nonlinear analysis of rectangular plates

under bending. Another finite element nonlinear analysis was used by Tsai

and Stewart [18] and by Moore [13] to solve the nonlinear glass plate bending

problem.

Later, Vallabhan and Wang [19,22] developed a more efficient finite difference

model to solve the Von Karman equations. Detailed experimental research on

glass plates subjected to lateral pressures conducted by Behr et al. [1] and

Vallabhan and Minor [20] confirmed the results obtained by the Vallabhan and

Wang model. In the same study, the behavior of laminated glass plates was

compared to those of layered plates (Figure 1.1) and a monolithic plate of the

same nominal thickness (Figure 1.2). It has to be mentioned that the analysis

of monolithic and layered plates could be handled theoretically by finite element

programs such as NASTRAN [14] and by the Vallabhan and Wang model as well.

The Vallabhan and Wang model is considerably faster than the finite element

model, and has obtained wide acceptance in the glass industry. However, to

the knowledge of the author, the nonlinear analysis of laminated plate units has

not been performed using a mathematical modelling. A brief review of the past

research related to sandwich plate units is given below.

Theoretical analysis on bending and buckling behavior of sandwich plates has

been investigated by several researchers. In an early paper, HofF and Mautner

[6] derived the differential equations of sandwich beams subjected to lateral

and in-plane loads using the principle of virtual displacements. Their results

have been shown to agree well with their own experimental results. Legget

and Hopkins [9] gave rigorous results and an approximate solution for simply

supported plates subjected to buckling. Van der Neut [21] analyzed the same

problem for various boundary conditions. Later, March and Smith [12] studied

the sanie problem for various other boundary conditions. In their research,

they presented approximate strain energy solutions. Bijlaard [2,3,4] proposed a

simple procedure giving rigorous solutions for simply supported plates as well

as approximate solutions for other boundary conditions.

In all of the mentioned work, plane section for the whole sandwich plate

system before bending was assumed to remain plane after bending (Figure 1.3).

Based on the same assumption, HofF [7] developed a linear plate bending and

buckling theory for rectangular sandwich plates using the principle of virtual

displacements. Experimental investigations confirmed the theoretical predic­

tions of the previous models for the case of units with a core shear modulus of

the order of one thousandth of that of the faces. A significant contribution to

the finite strain bending theory for sandwich plates was made by Reissner [15].

However, his theory is not applicable to the laminated glass problem since it

was derived for a thick core as compared to the two other plates (Figure 1.3).

Moreover, he assumed that the face plates were thin enough so that their bend­

ing resistance could be neglected. Later, a more refined nonlinear stress analysis

model for sandwich plates was developed by Das and Vallabhan [5], where the

transverse shear deformation as well as the compressibility of the core were taken

into consideration. Among the previous theories, only the later model is theo­

retically applicable to the nonlineax analysis of laminated glass units since the

assumptions used agree weU with the mechanics of the problem as observed in

experiments. However, the obtained field equations being too complex, have

not been solved numericzdly. In spite of the abundance of the above mentioned

theoretical work on the sandwich plate bending and buckling problem, the de­

velopment of a new theory specifically addressing the laminated glass units with

the assumption of nonlineax in-plane strains and conforming with the mechanics

of the problem as observed from experiments, is found to be necessary.

Scope of Research

For laminated glass units (Figure 1.4), the interlayer shear modulus being

very small as compared to that of the glass plates, the classic assumption that

plane section for the whole system before deformation remains plane after de­

formation, becomes non-realistic. Therefore, sandwich plate bending theories

already existing in the literature can not be satisfactorily employed for the

analysis of the laminated glass plate systems under bending. It is the scope of

this research to develop a nonlinear plate bending theory under the more real­

istic assumption that plane section before bending remains plane after bending

for each individued plate and the interlayer transmits a certain amount of shear

between the two glass plates. The derivation of the equations of this new math­

ematical model is presented in chapter 2. In chapter 3, the field differential

equations of the model are converted into nonlinear algebraic equations using

the central finite difference technique. The algorithm used for solving the ob­

tained algebraic nonlinear equations is described in chapter 4. Two laminated

glass unit problems are solved and compared with available experimental data,

in chapter 5 while conclusions and recommendations are given in chapter 6.

rm TTTt

Figure 1.1: Layered Glass Units

rm rm

TTTT

Figure 1.2: Monolithic Glass Units

/77T

Figure 1.3: Sandwich Plate Units (Reissner and Hoff Theories)

n77 rm

Figure 1.4: Laminated Glass Units

CHAPTER 2

A MATHEMATICAL MODEL FOR

LAMINATED GLASS UNITS

Introduction

In a laminated glass unit, the shear modulus of the interlayer is very small

as compared to that of the glass plates. Therefore, the classic assumption in

sandwich plate bending theories that plane section for the whole system be­

fore deformation remains plane after deformation is no longer valid. Therefore,

the existing theories can not be satisfactorily employed for the analysis of the

laminated glass plate units subjected to lateral pressure. It is the scope of this

research to develop a new nonlinear plate bending theory for the analysis of lam­

inated glass units taking into account the more realistic assumption that plane

section before bending remains plane after bending for each individual layer of

the system, i.e., for each of the glass plates and for the interlayer. The com­

monly used minimum potential energy theorem [11] is employed to obtain the

field equations and boundary conditions of the mathematical model governing

the laminated glass plate unit behavior.

Assumptions of Laminated Glass Units

The assumptions related to glass plates are:

1. The thickness of the plate is constant and very small compared to the

plate length and width, which justifies the neglection of the plate shear

deformation.

8

2. The plate material is completely elastic and obeys Hooke's law.

3. The material of the plate is homogeneous and isotropic.

4. Normals to the middle plane of the plate before deformation remain normal

to the middle plane after deformation.

5. The lateral deflections of the plate are of the same order as the plate

thickness but are still small in comparison to the other plate dimensions,

which causes the middle plane to be stretched under the effect of lateral

loads. Stresses are therefore induced in the middle plane and are referred

to as membrane stresses.

6. The in-plane displacement derivatives are small so that the higher powers

of their derivatives and values of their products are neglected in evaluating

the strain components.

The assumptions related to the interlayer are:

1. Plane section before deformation remains plane after deformation.

2. Material is homogeneous and isotropic.

3. Material is elastic and obeys Hooke's law, i.e., the interlayer shear modulus

is constant.

4. No slip occurs between adjacent faces of the plates and the interlayer.

5. The energy stored in the interlayer due to the normal stresses is negligible

as compared to the shear strain energy.

6. Linear shear strains are assumed instead of finite strains as a simplification

to the problem.

7. The interlayer thickness is very small so that its compressibility is negligi­

ble as compared to the lateral deflection of the whole system. Therefore,

the lateral deflection of the top plate is considered the same as that of the

bot tom plate.

Using these assumptions, the total potential energy V of the system can be

expressed as

V = F2' + F<'' + F« + F« + ir« + u[',' + n, (2.1)

where

Ujn — membrane strain energy for the ptate (i).

17^ = bending strain energy for the plate (i); t = 1,2 for the top and bot tom

plates, respectively,

iT^l^l — shear strain energy for the interlayer due to the shear strains 7^^

and 7yi, respectively, and

n = potential energy function due to applied loads.

Membrane strain energy functions can be expressed in terms of strains [11], as

(2.2)

10

where

E = Young's modulus of the glass plates,

fj. = Poisson's ratio of the plate,

hi = thickness of the plate,

Ix = length of the plate in the x direction,

ly = length of the plate in the y direction, and

Ci.xi Cj y, Ci^xy — nonlinear membrane strains, which are expressed in terms of the

displacements as

dui 1 , dw., ^- = ^ + o ( ^ ) ' ' (2-3) dx 2^dx

dvi l . ^ i u . ,

* - = a l ^ + 2(ai^) ' (2.4)

and

^ t dvi ,dw.,dw. . ^^

' ^ - = air + a^ + (a^^( sF^- ^^^ Here i = 1,2 denotes the top and bottom plates, respectively.

Similarly, the bending strain energy function is expressed as [11] follows:

v^'= fX)J_i)X'd^dy rl^/2 J./2 £?/!?

- • ' - / , /2 J-U/2 24(1-/i2)

[ (0) ' + {^r + M^){w) + 2(1 - M)(i )'l<i«<iy. (2.6)

11

For the interlayer (Figure 2.1), the average shear strains -^^z and 7y2 are given

as

fxz = (i> + e dw du

dx dz

+ [Ui -U2- -^{-T + -TTJl/t dx ' dx^ 2 2

. dw ,hi hi M , /« . \ = [ U 1 - U 2 - — ( ^ - f ^ + O l A , (2-7)

where t is the thickness of the interlayer.

Similarly,

Making use of the previous two equations, the interlayer shear strain energy

expressions axe written as

r',/2 /•i./2

^-I,/2 J-1,I2 .t .Z,/2 W«/2 1

= / / / O^^'^' Jo J-l^/2 J-lr/2 2

w,/2 rlm/2 Gi, dw.hi hi , , ,^j , ,^^. = -^[ui - U2 - ^ { ^ ^ ^-^ t)] dxdy (2.9)

./-Z,/2 J-lr/2 2t ox 2 Z

Uv

and

= / / / ^G^/7v. '^^ yo y-/,/2 J-1^12 2

J-i^/2J-ir/2 2t ay 2 2

where

G'j = interlayer shear modulus.

12

For the case of a uniformly distributed lateral pressure of intensity q acting the

laminated plate unit, the load potential function Cl is given by,

^ =S-C/2S-t2^dxdy

= f-ii%-tt/2-^^dxdy. (2.11)

From Equations 2.2 , 2.6 , 2.9 , 2.10 , and 2.11, substituting in Equation 2.1,

one gets the total potential energy V of the system as

r'»/2 flm/2

J-lt/2 J-lt/2

/ / Fdxdy, (2.12) J-1^/2 J-lr/2

where

Ph ^ ^ = 2(1 - u^)^^^''^ " ei.v^ + 2/xei.,ei.y + - ( 1 - /x)ei,,y^]

Ph 1 + 2(i-!t2)[^2. '^ + 2.v + 2/xc2.,e2.y + - ( 1 - M)e2,xy ]

2... 2 il2...2 ;j2... a2 . . . ;i2».. 2

+ T^TTT—l7[^-T + ^ - T + 2 / X - — — ^ - h 2 ( l - / x ) . •U;

24(1-/x2)Laa;2 ^ 2 "^dx^dy^ ' "^'dxdy'

Gi, dw,h\ /i2 . . '

(?/ , dw,h\ h2 . . '

— qw. (2.13)

The Principle of Minimum Potential Energy

The principle of minimum potential energy states that of all geometrically

possible configurations that a body can assume, the true one, corresponding to

13

the satisfaction of stable equilibrium, is identified by a minimum value for the

potential energy. It can be shown that the above statement can lead to the Euler

Equation [11], which is used in this derivation to obtain the field equations of

the system. Making use of Equations 2.3, 2.4, 2.5, 2.13, and applying the

Euler equation,

dF d dF d dF d^ dF d^ dF d^ dF

dui dx^dui^^ dy^dui,y^^ dx^^dui^x^ ^ dxdy^du^^^y^ "" dy^^du^^yy^ ~ '

(2.14)

where Ui denotes tt i ,^1,^2,^2, and lu, respectively,

Ui^x = the first derivative of Uj with respect to i ,

Ui^y = the first derivative of u, with respect to y,

^t,xx = the second derivative of Ui with respect to x,

Uiyy — the second derivative of Ui with respect to j / , and

Ui^xy — second order cross derivative of u^,

we get the five equations of equilibrium governing the laminated glass plate

system:

r_?L X l - / x ^ Gi(\-^i) ..\±JiJ!_. , rg/(l-/x). ^dx^ 2 dy^ 2Gh,t ^""'^ 2 dxdy^ ' ^ 2Ghit ^ '

dw.d^w l — ad^w, 1 + u d^w dw dx'dx^ 2 dy^' 2 dxdy dy

(^•('-'^\h + h+t)^, (2.15) 2Ghit ' 2 2 ^ dx

14

'a? + 1 - M a' G,ii - .)j^^ ^ (1+if »L]„, + [ ^ f e ^ ] . ,

2 5x2

dw ,d w dy dy^

+ 2G/iit ^ ' ^ 2 dxdy

1 — fjL d^w. 1 -\- fi d^w dw 2Gh^t

2 ax2 ] - 2 5x^2/ 5x

G f ( l - fi),hi ^2. X t ) ^

2Ghit ^2 2 hy' (2.16)

5x2 i-zxa^ GI(I-M) 1 + M ^ . , rg/(i-/x).

+ —::—':r-:: >.^. .—J^2 + [—;;—Q Q JT 2 + I O^L .. J^i 2 53/ 2Ghii 2 dxdy' 2Ghit dw ,d^w l — ud^w. 1-(-u d^w dw

dx dx^ 2 5y= 2 dxdy dy

2C? 2t ^ 2 2 ^ ax ' (2.17)

dy' + l - / x 5 2 C?j( l- /z) , ^ . 1 + M 52 , ,G/(l-/x),^^

2 5x2 dw rd w dy dy'

+ 2Ghit ''* ' ' 2 5x5j/-

1 — ^x5^1/;. 1 +/x 5^iy 5it7

2G/iit

5x' • 1 - 2 5x52/ 5x G/(1- /X) /ll ^ . s ^

2G/i2t ^ 2 2 ^ 53/' (2.18)

and

+

[(A + i^2)v^-Y(T + T + ) ' ^> = 9

+ Y^7^[(ei.. + /^ei.y)— -h (ei.y + /^ei..)— + (1

[(e2.x 4- /^e2.y)-^;:7 + ( 2.y + t^^2,x)-^-^ + (1 1-/X2 5x' 5y2

M)ei,

M)e2,xy

'"5x52/ 52iu .

dxdy'

Gjh^ h2 ,w^^i 5u2 ^^1 dv2. ' t^2'^ 2^^^^dx' dx'^ dy dy ^'

(2.19)

15

In the above equation,

d^ d^ d^

dx^ dx'dy' dy^

and

52 52 ^' = i-^^W (2.21)

In the equations obtained above, the left-hand side constitutes only of Hnear

terms. Nonlinear terms in the lateral deflection w are brought to the right-

hand side. This arrangement is essential for the iterative procedure discussed

in chapter 4, From the energy principle used here [11], the boundary conditions

axe obtained as.

At X = constant:

, dF d , dF , d , dF ,, ^ ' / u . h T - . ; - ^{-^ = 0. (2.22)

dui^x ox dui^„ dy Oui^^

Vu,x^ = 0. (2.23) OUi^xx

At y = constant :

.dF d . dF . d . dF dUi^y dx dUi^xy Oy OUi^yy

dF W ^ = 0. (2.25)

In Equations 2.22, 2.23, 2.24, and 2.25, u» represents Ui,ri,U2,V2, and w,

respectively. A notation r]^ is introduced to designate an arbitrary value of Ui.

Thus,

7y„. = the flrst derivative of T/t with respect to x and

"HiHy = the first derivative of 77^ with respect to y.

16

For the purpose of this research. Equations 2.22, 2.23, 2.24, and 2.25 are

applied for the case of a simply supported rectangular plate with no in-plane

restraints on the edges subjected to a uniform pressure. Only a quarter of the

plate is considered by virtue of symmetry. If the axes are taken as shown in

Figure 2.2, the obtained boundaxy conditions become

At X = 0 :

ui = 0. (2.26)

ei,xy = 0. (2.27)

U2=0. (2.28)

e2.xv = 0. (2.29)

^ = 0. (2.30)

A ( ^ + ^ ^ ) + | - [ 2 ( 1 - ^ . ) ^ ] = 0. (2.31) dx^ dx' ^ dy'^ dy^ ^ ^'dxdy^ ^ ^

At X = lx/2

At y = 0 :

c i^ + /iCi.y = 0. (2.32)

ei.xv = 0. (2.33)

e2^ + Me2.y = 0. (2.34)

e2,xv = 0. (2.35)

u; = 0. (2.36)

f ^ = 0. (2.37) ox*

vi = 0. (2.38)

17

(2.39)

(2.40)

(2.41)

(2.42)

5 ,d'w d'w, d , , . d^w , , ,

a;(a7 + ''a?^ + ai;>2(i-'')a^l = ''- (2-43) d'

^l.xy

V2

^2,xy

dw dy

" ^ ,

=

=

=

=

1,

0.

0.

0.

0.

?n

Aty = ly/2

ei,y-h^ei,. = 0. (2.44)

ei.:,y = 0. (2.45)

e2.y + Me2,x = 0. (2.46)

e2,xy = 0. (2.47)

w = 0. (2.48)

d'w ^ = 0. (2.49)

All the nonhnear field and boundary equations necessary for analyzing lami­

nated glass units under bending have been derived in this chapter. By omitting

the nonlinear terms in the field equations, and putting Ui = -Ui = U2 = ^2 = 0,

the field equation for small strains is obtained as

( A + D^)V* - ^ ( ^ + I + tfV')w = , . (2.50)

For a siniply supported unit allover the edges, the above equation has a Navier

type solution of the form

IQq °^ o" sm-J—sin-y"-

*" = (£>, + D,)^ „ S , - n = § , . . . mnlif + | ) > + £ i ( ^ + i l + ()2(=^ + | ) ] '

(2.51)

18

Expressions for stresses and strains can easily be obtained by taking the

appropriate derivatives of the above expression. This solution is valid for ^ ^ j ^

ratios of 0.5 or less.

It is the purpose of the next chapter to convert the obtained nonlinear differ­

ential equations in this chapter into algebraic equations using the central finite

difference technique.

19

Figure 2.1: Laminated Unit in the Deformed Shape

20

y-axis

x-axis

z-axis

Figure 2.2: Axes of Reference

CHAPTER 3

FINITE DIFFERENCE EXPRESSIONS FOR FIELD

AND BOUNDARY EQUATIONS

The governing equations for the mathematical model using the displacement

approach are given in chapter 2. The five field Equations 2.15 through 2.19 are

nonlinear with the function w, the lateral deflection. All the differential opera­

tors in Ui,Vi,U2, and V2 are Hnear. The domain of the problem is rectangular.

By virtue of symmetry, only one quarter of the plate is considered. Due to the

nonlinearity of the governing equations, an iterative numeric technique has to be

adopted for the solution. A close form solution for such a system of differential

equations is not known.

The left-hand side of equations 2.15 through 2.18 is Uneax in the in-plane

displacements ui,Vi,U2,V2^ Similarly, the left-hand side of Equation 2.19 consti­

tutes only of Uneax terms in w, the lateral deflection. All the nonlinear ternis in

Equations 2.15 through 2.19 were brought to the left-hand side of the equations.

The well known central flnite difference technique is used to transform the

continuous functions Ui,t;i, 1x2, 2? and w into discrete values at every point of

the finite difference mesh. The system of differential equations is transformed

therefore into a system of algebraic equations. The terms in the left-hand side

of the field and boundaxy equations, being linear, can be transformed into Hnear

differential operators, while all the nonlinear terms in the right-hand side are

condensed into a right-hand side vector. In a matrix form, the left-hand side of

the algebraic equations generated from field Equation 2.19 are stored in matrix

21

22

[A], while those equations generated from Equations 2.15 through 2.18 are kept

in matrix [B]. Therefore, the system of equations can be written as

[A]{u;}= {q + {fi{w,Ui,VuU2,V2)}, (3.1)

[B]{U}= {f2{w)}, (3.2)

where

{w} = the lateral displacement vector,

{U} = is the in-plane displacement vector, constituting of the values of iii,t;i,ii2,

and V2, respectively, at every finite difference mesh point, and

q = = appHed pressure magnitude.

The coefficients of matrices [A] and [B], and the corresponding right-hand

equation sides for grid points inside the domain as well as those at the boundaries

axe presented in this chapter. The details of the iterative technique used to reach

the final solution is given in the next chapter.

For the lateral deflection, the flnite difference mesh size is chosen to be n^ xny^

nx and ny being the number of subdivisions in the x and y directions, respectively

(Figure 3.1). The lateral deflection value at the simply supported edges being

zero, is not incorporated in the finite difference mesh in order to reduce the total

number of equations. For a point inside the domain, the finite difference form

of the lateral deflection field equation is

For 1 = 3, •• • n,. — 2; J = 3, • • Tiy — 2

23

+Jw;(ij+i) + Ju;(i,j-i) + Gw^ij+2) + Gw^ij.2)

{RHSy^ijy (3.3)

where

{i2^5}(i.,) = {q

, -E i r/ s5^iu . ^d'w . . d'w ,

+rT7:?i('>.' + '' >.v)-a^ + (^1.. + ''^i.') a ^ + (1 - ' ' )^' . '»a^l , Eh2 f, , .5^11? , .d'w , , , 5^iy .

+ r r 7 i ( ' » - + ' ' ^ ' - )a? + ( '.v + "«'•') aj;r + (i - '')^^.'»a^) Gi .h\ /i2 . x5ui 51X2 5^1 5^2 ii\ ri2 .. ,xrai UU2 OV-i C 7 V 2 N ,

t ( y + T +') (a7 - 1 7 + air - a^f'>''^" ('-'^

C = ( D , + A ) ( A + A + ^ ) + £ . ( ^ + i^ + i )2 (^ + ^ ) , (3.5)

B= ( Z 3 , + C , ) ( ^ + j j i , ) + £ i ( ^ + ^ + t ) = ( ^ ) , (3.6)

J f = ( A + A ) ; ^ , (3.7)

F= (D,+D,)-^, (3.8)

J = (D, + D,){=^^+^^) + ^(!f + !f+tY{:^), (3.9)

G = (Di + K j ) ^ . (3.10)

At the plate boundaries, Equation 3.3 has to be modijied to account for the

boundary conditions. The following equations are obtained:

For i = l ; j = 1

C B H 7^(i.i) + -j^ci+ij) + Y^(i+2.i)

+"2 (».i+i) + -J (».J+2) + i^^(t+i.i+i)

= ]{RHS}^ijy (3.11)

24

For i = I'J = 2

^ + G ^ rr J

= \{RES}^ijy (3.12)

For t = l ; j = 3 , . . . n y - 1

C ^ „ J

J G G

= \{RHS}^ijy (3.13) 1

2

For t = l ; j = Tiy

(C - G)w^ij) -h Bu;(i+ij) -f Hw^i+2j)

•WI2 -• nX -t- fi'-U}/.- -• *\ -I- Ptnir . , • ,

2 •^ ^ ^

= \{RHS}^ijy (3.14)

Fori = 2; i = 1

C -\- IJ B B TT —2—^(*'^) " y^(»+i.i) + ^ ^ ( i - i j ) + y ^(i+2.i)

-{-Fw^i+ij+i) -\- Fw^i_ij+i) + Cru;(ij+2)

= ^{RHSy^ijy (3.15)

F o r t 3 2 ; j = 2

(C -f- F + C?)u^(t.i) + Bw^i+ij) + BTi;(i_ij)iy(i+2.j) + ^^(i+2.i)

25

-^J'^{ij+i) + Jf^{ij-i) + Fw(^i+ij+i) + Fw(^i+ij+i)

= {RHSy^ijy (3.16)

For I = 2 ; j = 3 , . . . n y - 1

(C -H -5')u^(ij) + Bu;(,+ij) -h BTi?(i_i,j) + Hw(^i+2j)

+-^^(i+U+i) + ^^( i+Li- i ) + Fu;(i-i,i+i) + Fiy(i_i,j_i)

= {RHSy^ijy (3.17)

For t = 2, j = Tiy

(C - ^ + <^)^(i,i) + ^^ ( i+ i j ) + ^^( i - i . i ) + Bw^i^2,j)

= {RHSy^ij). (3.18)

For i = 3, • • • n , — 1; J = 1

C B B

H H

-\-Fw^i+ij+i) + Fu;(i_ij+i) + Gwi^ij+2)

= i{ i2^5}( i . , ) . (3.19)

For I = 3, • • • n , — 1; jf = Tiy

(C - C?)tz;(ij) -H 5u^(i+ij) + ^^( i - i . i ) + Hw^i+2,j)

+Hw^i_2j) + Fit;(i_ij_i) + Fw^i+ij.i) A-Jwf^ij.i) + Gw(^ij.2)

= { i l ^ 5 } ( i j ) . (3.20)

26

Fon' = nx;j = 1

/ ^ IT D TT

1

2

Fori = 71 ; ' = 2

= i{ i2^5}( i . , ) . (3.21)

(C - F + G)w^ij) + B^(i_i^) -H ^tx;(i-2.i) + i^^(t-i.i-i)

= {RHS}(ijy (3.22)

Fori = n,,; J = 3, • • • riy — 1

(C - H)w<^ij) + Bu;(i_i,j) -H Hw(^i_2j)

= {i i^5}(i . j ) . (3.23)

Fori = n , ; j = riy

(C - C? - -ff)uJ(ij) + 5tz;(i_ij) -h Hw^i.2j)

A-Jw(^ij.i) + Fu;(i_i,j_i) -f Gw(^ij.2)

= {RHS}^ijy (3.24)

For the in-plane displacements, the finite difference mesh size is ( n , -I- 2) x

(riy -I- 2). In addition to the edge displacements, fictitious points outside the

domain are considered in the proximity of the simply supported edges of the

27

plate (Figure 3.2). At every point of the finite difference mesh, there are four

unknowns Ui,t;i,Ti2, and V2, and four finite difference field equations are written

per point. Along the edges x = 0 and 2/ = 0, the field equations are modified to

account for the boundary conditions, while at the edges x = lx/2 and y = /y/2,

four additional boundary condition equations per point are appHed, so that the

total number of unknowns is equal to the number of equations. At the comer

point lying at the intersection of the simply supported edges, the field equations

axe modified and eight independent boundary conditions are written. The total

number of in-plane displacement unknowns is 4 x [{nx + 2) x (riy -I- 2) — 1], and

the corresponding finite difference equations are:

For t = 2, • • • n,.; j = 2, •. • n^

oitti(i,j) + hiu^i+ij) + 6iUi(i_ij) -I- CiUi(^ij+i) -f ciUi(i,j_i)

+<ivi(i+ij+i) - dv^i+ij.i) - dvK^i.ij^i) + <iri(i_ij_i) -f eiU2{ij)

dw . d^w 1 — A d^w. 1 + /x d'w dw

^~^^'d^ " 2 dy'^ T'dxdy'd^

G'(l - ' ' ) . (^ + % + , ) ^ ] , , , (3.25) 2Ghit ' 2 2 'dx

0'2Vl{iJ) + C2Vi(i,_,+i) -I- C2Vi(iJ-i) + 62Vl(i+l,i) + hzV^.u)

+(iui(i+i,j+i) - (fui(i_ij>i) - duK^i+ij-i) -H dui^i-ij-i) + eit>2(i.i)

dw d^w 1-fid^ l-\-fi d^w dw

^ ' ~ 5 ^ ^ 5 ^ " ^ 2 5 x 2 ^ " 2 dxdydx

G / ( l - M ) A + ^ + , )^ l , .^ . , . (3.26) 2C?/iit ' 2 2 ' 5y

"3ti2(t.j) + biU2{i+l,j) + 6iTX2(i_i,j) + CiTX2(t,j+l) + CiTXi(i,j_i)

28

-^dv2(i+ij+i) - dv2{i+i,j-i) - di;2(i-i.j+i) + dv2{i-ij-i) + e^u^ij)

. dw, d'w 1 — /x d'w, 1 -h /x 5 11? 5iy

5 x ' 5x2 ' 2 5y2 ' 2 5x51/ 52/ ^7/(1 —/x)^/ii /i2 .v5u;. / U - / x ; ^ a i , ^2 C7u; (3.27)

a4V2(i,j) + C2U2(i,i+l) + C2V2(i.i-l) + ^2^2(^+1, ) + t2V2(,-l,i)

-(-dii2(i+i,i+i) - citi2(t-i,i+i) - <itt2(i+i,j-i) + du2{i-i,j-i) + e3t;i(ij)

. dw.d^w 1 — fid^w. 1 +/x 5^iy 5iu ^~ 5y ^ 5y2" " 2 5x2^ 2^dxdyJ^

Gi{l-fi) hi h2 dw

In t h e above equat ions .

a i =

02 =

as =

a4 =

6i =

62 =

ci =

C2 =

d =

Cl =

- 2 1 - / X G j ( l - / x )

2Ghit

Gi{l-fi) 2Ghit

Gi(l-fi) 2Gh2t

Gi{l-fi) 2G/i2t

hi - 2

- 2

/•i - 2

hj 1 - M

hi 1 - M

/•J 1 - M

''J hi

1 - / X

2 / i2

1 - / X .

1.

2hl

pi-V

1+/X Shyhx

(1 - M)g/ 2Ghit

(3.29)

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

(3.36)

(3.37)

(3.38)

29

'^ = f^GXT-l- (3.39)

For t = 1; J = 1

^i(i.j) = 0. (3.40)

^HiJ) = 0. (3.41)

^2(i.i) = 0. (3.42)

V2iij) = 0. (3.43)

For 1 = 2 , . . -Tia.; j = 1

+2(fvi(i+ij+i) - 2(iri(i_ij+i) + eiU2(i,j)

. 5ii; 5^iy 1 — /x 52^; 1 -I- /x 5^iy 5iu

^~ 5 ^ ^ 5 ^ ^ 2 5y2 ) 2 ~ 5 ^ 5 ^

G / ( l — / x ) , / i i h2 .dw. , ^

t'i(M) = 0. (3.45)

a3li2(tj) + ^1^2(i+lJ) + hy'2{i-lj) + 2CiTX2(i,j+l)

+2dv2(i+ij+i) - 2du2(i_ij>i) + e3txi(ij)

. dw 5^it; 1 — M ^ "" \ 1 + M ^ ^ ^ ^'"'

^~ 5 ^ ^ 5 ^ "^ 2 5y2^ ~dxdy'dy

. Gi{l-fi) hi /l2 , .^^T^T ,« .^v

2G/i2t ' 2 2 ' 5x

^2(ij) = 0. (3.47)

For z = n , - f l ; j = 1

30

Also,

aiTii(i,i) 4- biuii^i+ij^ -f 6iWi(i_ij) + 2ciu^ij+i)

+2<fri(i+ij+i) - 2ciTJi(,-_ij+i) + eiiX2(ij)

<^j(l - ^i) ,hi h2. ^ i\( — \ 2Ghit ^2 " 2 ^ ' ^^5xV, ) - (3.48)

^i(i.i) = 0. (3.49)

0'2'^2{i,j) + 6lU2(i+ij) -I- 6lU2(i_i.j) -H 2Citt2(i,j+i)

+2dr2( i+i j+i) - 2<ft;2(i_ij+i) -h Cau^ij)

Gi(\ - ti) ,hi h2 X i\( — \

2GM ^2 " 2 ^ ^^dx\ij) (3.50)

''2(iJ) = 0. (3.51)

2 ^ ^ i ( i + i j ) - ^ ^ i ( » - i . i ) + x : ^ ^ ( ^ - ^ ' + ^ )

2/1

1_ 2/ i .

1 1 2^^ i ( .> i J ) - 2^^i(^-i . i)

1 1 /x ti2(.+l.i) - :77-^2(t-l.j) + T-V2(i.i+1)

2^dx\i,3)

= 0.

2;ix

1 /ly

1 2^dx\ij)-

2K V2(i+l.i) - 2^^2 ( i - l . i ) = 0.

(3.52)

(3.53)

(3.54)

(3.55)

For I = l ; j = 2 , . . . n .

^i(i.i) = 0. (3.56)

31

<^2Vl{i,j) + C2Vi(ij+i) + C27;i(,j_i) + 262Vi( i+iJ)

-\-2dui(^i+ij+i) - 2dui(^i+ij_i) + eiV2{ij)

. dw d'w 1 — M ^^^ 1 + /x 5^it; 5ti; ^'l^^l^ " 2 5x2^ Y~dxdy'di

Gi{l-fi) hi /i2 , ..5u;

2G/iit ' 2 2 ' dy

y'2{i.j) = 0. (3.58)

0'4V2{i,j) + C2r2(i,j+i) + C2r2( t j - i ) + 262^2(1-1-1 J)

-\-2du2{i+ij+i) - 2du2{i+ij-i) -\- eiT;i(ij)

. dw 5^it; 1 — /i 5^iu. 1 + /x 5^u; dw ^''dy^'dy' " 2 5x2^ 2 dxdy'dx

, g / ( l - ^ ) > i /l2 , . X ^ ^ l / « .QN

For 1 = 1; J = TXy -I- 1

ui(i.,) = 0. (3.60)

0'2Vi{ij) + C2r i ( i j+ i ) -I- C 2 r i ( t j - i ) + 262^1(^.^1^)

+2<iui(i+ij+i) - 2(iui(i+ij_i) -I- eix>2(tj)

,Gi{l-fi),hi . h . d w

-^-2GhJ-^^^'2^'W^''^^' ^ ^

ti2(i.i) = 0. (3.62)

°'AV2{i,j) + C2V2{iJ+i) + C2r2(ij-1) + 262^2(^+1^)

Also,

-\-2du2(i+ij+i) - 2(iu2(i+i,j-i) -f eiri(,j)

- I jGijl - M ) > I /i2 57x;

" ^ 2GM ^T + T^^^5;"i('-^)-

32

(3.63)

^i(i.i+i) = 0. (3.64)

2r^i(»^>i) 1 /X ir—^2 (3.65)

^2(i.i+l) = 0. (3.66)

1 1 ^2(iJ-H) - 7

"y *"«'y

For I = n, + 1; J = 2 , . . • riy

1 .5 w 2h,.'''^''^^'^ ~ 2h:.'''^''^-'^ ^ Vx""'^'^''^^ = 2 5j )( >>)-

(3.67)

aiui(ij) -H 6iUi(i^.ij) -f 6iui(._ij) -f ciui(i,j+i) -h ciTii(i,j_i

+<'ui(t+i.i+i) - dv^i+ij^i) - rfvi(i-ij+i) + <iui(,_i.j_i) -\- eiU2{ij)

Gi{l-ji) hi h2 .,dw

- ~ 2Ghit ( T ^ T ^ ' ) ( 5 ^ ) ( ^ - > (3.68)

a2Vi(»j) + C2Vi(ij+i) + C2ri(i,j_i) -h 62 1( +1 ) + 62Vi(i_i.y)

-^dum+ij+i) - rfui(i_ij+i) - ciui(i+i,j_i) -I- (itxi(i_i,j_i) -h eit;2(i,_,)

1 -\- u, d'w dw. • ( ^ r ^ —)(ij)- (3.69) 2 5x52/ dx

< 3 2(t,i) + 6iU2(t+ij) + 6i1X2(i_i,_;) + CiU2(t,j>i) + CiUi(ij_i)

+<^V2(t+l,i+l) - fiv2(t+l,i-l) - dV2(t-l,i-|-l) + dv2{^i-i,j-i) + e3Ui(i,_y)

33

_ Gj{l-fi)h, h dw 2Gh.t ^ 2 ( T + T + 0 ( ^ ) ( . . dx (3.70)

0'4V2{iJ) + C2r2(t.i+1) + C2V2(ij-i) + 62r2(.+ij) + 62i;2(i_ij i)

+<^^2(i+i.i+i) - du2(i.ij+i) - du2(i.^ij.i) + du2{i-ij.i) + esvi^ij)

1 + /x 52it; dw

2 dxdy dx (3.71)

Also,

1 1 / /x 1 dw

(3.72)

1 1 1 1 2;,^^i(».i+i)-2r^i(M--i) + 2r''i(»+W)-2^t;a(,_i.^^^ = 0-

(3.73) 1 1

2x;^2(i+i^) - 2X:^2(.-ij) + 2r;^2(i..>^) - 2^r2( , , . i ) = -:,{^)lijy A* . M 1 / ^ ^ N 2

1 1 1 1 2x:^2(i.i+i) - 2r''2(ij-i) + ^r^2(i+i.i) - :nr^2(i-i.i) = o. "y *-..y

For t = 2, • •. n , ; J = Tiy -h 1

2/i, 2hx

(3.74)

(3.75)

ai^i(i.i) + ^i"i(i+i.i) + t i^ i ( i - i j ) + ciixi(ij+i) -h ciui(ij_i

+<iT'i(i-(-ij-n) - <ifi(i+ij-i) - <ivi(t-i,i-|.i) + <ivi(i-ij_i) + eiti2(t.i)

1 ->r fi. d^w dw

2~^5l5^5^^ '' ' ' (3.76)

^2Vi(i.i) + C2t;i(ij+i) -f C2t;i(ij_i) -h 62ri(i+i,j) + 62ri( i_ i j )

+<^^i(t+i,i+i) - rfui(i_ij+i) - <iui(i^.i,i_i) + dui^i.ij.i) + eit;2(i.j)

G / ( l - / x ) . / i i h2 .,dw^

= --2GM ^T + T +')(ar)(-> (3-")

34

^3ti2(i.i) + hy'2{i+l,j) + biU2{i-ij) + CiU2(i,j+i) + CiUi(ij_i)

•^dv2{i-^.ij+i) - dv2(i-^.ij.i) - dv2(^i-ij+i) + dv2{i-ij-i) + e3Ui(i,j)

1 -I- /x d'w dw. <-^Z^.-^hj)- (3.78) 2 dxdy dy

^4V2(i.j) + C2r2(i.j+1) + C2r2(i.j_i) -H 62^2(^+1,^) + 62^2(^-1^)

+dU2( i+ i j+ i ) - <iti2(t-l,i+l) - dU2{i+ij-i) -I- du2( i_ i j_ i ) + e3Vi(i.j ^i)

G/(l—/x),/ii /12 .,dw^

Also,

(3.80)

1 1 , M M 1/^^x2 2^^i(i.i-.i) - ^X^ 'iCM-i) + 2^^i(.^i.i) - 2^^2(.-i.i) = - 2 ^ ^ \ i , y

(3.81)

2x;'^2(i..>i) - 2j:;;^2(.i-i) + ^V2^i^^,i) - 2C'^'-''^ = °'

(3.82)

2^t'2(M>i) - 2^^2(M-i) + 25^^2(i+iJ) - 2j^^2(i-i^ = -2(a^)(^J)-

(3.83)

At the comer point (i = n,. + l ; i = riy 4- 1), making use of the boundary

conditions 2.33 through 2.36 and 2.45 through 2.48, it can be shown that the

field equations reduce to

35

(1 - ^)-^ - ^;7n-M^ - ^2) = 0, (3.85) 5x2 2Ghit

d'u2 Gi

dx' ^ 2Gh2t

I, ,d'u2 GI , (1 - / ^ ) ^ i r + ^;Hr-M^ - ^2) = 0, (3.86)

and

(1 - ' ^ ) | i r + ^ ( ^ 1 - ^2) = 0. (3.87) 5x2 2G;i2t

At the same point, the boundary conditions may be expressed as

;3.88)

;3.89)

;3.90)

;3.91)

;3.92)

;3.93)

[3.94)

[3.95)

For t = n,. -I-1; J = Tiy -I-1

d'vi

dx'

d'v2

dx'

52ui

^ '

5^1X2

52/2

5x 52t;i

^ dy' du2

~d^ d'v2

^ dy'

d'ui

^ dx' dvi ^—^— dy

d'u2

^ dx' dv2

dy

=

^

^z

—

0.

0.

u.

0.

0.

0.

0.

0.

36

Also,

2h:^i(.+i.i) - 2fc^i(»-i.i) = 0-

[-27T + 2^]t;i(,i) = 0.

27i;^2(i+l.i) - 2fc^2(i-l.i) = 0.

7^^2(t-H.i) + 4 ^ 2 ( i - l j ) - )SrU2(tj+l) - -^Vi^iJ-i)

[ -24 -f 2^]t;2(,,) = 0.

(3.100)

(3.101)

(3.102)

(3.103)

4^ i ( i J+ i ) + 7^^i(i.i-i) - )|ui(i-n.i) - /l'^i(i-i.i)

[ _ 2 i + 2 ^ ] n i ( . ^ ) = 0.

^ ^ 2 ( i j + l ) + 5^^2(i.i-l) - 7i|^2(»+ij) - )irii2(t-i.i)

[ _ 2 ^ + 2^]tx2(i,-) = 0.

2^Ti2(ij>l) - 2 tu2(M-l ) = 0.

(3.104)

(3.105)

(3.106)

(3.107)

37

ly/2

y-axis

]=ny+1 j=ny

Simply Supported

j=2

j=1 i=1 i=2 i=nx

Simply Supported

i=nx+1 x-axis

lx/2

Figure 3.1: Finite Difference Mesh for Lateral Deflection

38

y-axis

J=

ly/2

ny+2

ny+1 J= j=ny

Simply Supported

j=

2

1

Simply \ Supported

1=1 i=2 i=nx l=nx+1 i=nx+2 x-axis

lx/2

Figure 3.2: Finite Difference Mesh for in-Plane Displacements

CHAPTER 4

SOLUTION ALGORITHM

Field Equations 2.15 through 2.19 have been converted to algebraic equations

at every point of the finite difference mesh. The resulting equations may be

expressed in a matrix form as

MW(o'= {q-^-{Mw,uuv^,^„v,}}'-;i (4.1)

m{uY'^= {/2(«')}S-/ (4.2)

where,

{w} = the lateral displacement vector.

{U} = the in-plane displacement vector, constituting of the values of iti,Ui,iX2,

and r2, respectively, at every finite difference mesh point, and

q — the appHed pressure magnitude.

Subscripts (i) designate the ith iteration, while superscripts (A;) denote the kth

increment.

Matrices \A\ and \B\ are Hnear, while / i and J2 a^e nonhnear functions of

the lateral displacement function w. The original system of nonhnear algebraic

equations is transformed into a set of quasi-Hnear equations in the following

manner: Values for it;,TXi,Vi,U2, and V2 from the (i — l)t / i iteration are used to

form the right-hand side of system 4.1 of equations of the ith iteration. Equation

4.1 is solved for {ix?}. The new value of {ii;} is used to calculate the right-hand

39

40

side of 4.2 and Equation 4.2 is solved for {?/}, i.e., ui,Vi,U2, and V2. The

procedure is repeated until a selected convergence criterion is achieved. However,

it is found that the above iterative scheme will converge only in the case of

small deflections. For a similar type of problem, Vallabhan [19] has developed

an efficient iterative procedure. A similar approach is adopted in this research,

with some modiflcations. Four different algorithms are adopted for the cases

(i = l,fc = 1), (i = l,fc 7 1), (i ^ l,fc = 1), and (i t l,fc ^ 1), respectively,

i being the increment number and k the iteration number. Following is the

detailed description for each individual edgorithm.

For the first load increment {k = 1), and the first iteration (i = 1):

1. Assume {ty}(J),{txi}/o), {i'i}(o)5{^2}(o)i<^'^<^{^2}(o) to be zeros to calculate

{g + /i(^u,i*i,Vi,U2,r2)}}oj.

' • • \ y 2. Solve Equation 4.1 to get {'u;}Lx,

(1)

3. Calculate the value of a corresponding to '^''^^, a being an interpolation

parameter determined from numerical experimentation, tu)„aa.(i) the maxi­

mum deflection, and hav the average thickness of the glass plates (Figure

4.1).

4. Using {iy}(}j, calculate {/2(ty)}(l)-

5. Solve Equation 4.2 to obtain {Tii}(}j,{vi}(lj,{w2}(}j, and {v2}(}j-

6. Go to next iteration.

41

For the first load increment (fc = 1), and the iterations

i(i = 2, • . ' maximum number of iterations ):

1. Use {tx?}jjli),{tii}jj!i),{t;i}[,^li),{u2}|jli),an(f{v2}jj!i) to calculate {g +

/i(i(;,Ui,ri,U2,V2)}(J/.

2. Solve the Equation 4.1 to get {ii;}/^.

3. Check convergence. If satisfied, exit the loop and start computations for

the next increment; otherwise, go to the next step.

4. Obtain the interpolated values {lF}| .x, using the interpolation parameter

a:

{W}<.')' = (1 - «){«,}<;) + a{lF}<'l,,

5. Using {tZ?}(J), calculate {f2i'w)}[]y

6. Solve 4.2 to get {ui}jj)\{t;i}g,{u2}jj)^ and {v2}\}l

7. Go to step 1.

For the load increment k{k = 2 , . • • maximum number of increments ),

and the first iteration i{i = 1):

1. Calculate estimated values {t^y}(ij,{Tiia}(i),{viff}(ij, {it2p}(t), and {v2fl}(i),

for the present increment by Hnearly extrapolating the displacement con­

verged values of the previous two increments:

42

{«„}<*) = 2{u,}( ' -" - {mY'-'l

{^..}W = 2{r,}<'-" - {^:}'*-"

2. Calculate the value of a corresponding to ""j^", to be used for the present

increment iterations.

3. Use {w,y''\{u[%{vi,y''),{u2gy''\ and {t;2,}W to calculate

{g-f/i(u;,Ui,ri,TX2,r2)}[i?

4. Solve Eiquation 4.1 to get {ii'}!!').

5. Obtain the interpolated values {ii7}L|:

Mjj; = (1 - a){,.}<JJ + «{„,,}(')

6. Using {u;}Lj, calculate {/2('"')}(i)-

7. Solve 4.2 to get {ui}[]l {vi}[]l {u2}\^l and K l J J j .

8. Go to next iteration.

For the load increment k(k = 2, • • • maximum number of increments ),

and iterations i{i = 2,. •. maximum number of iterations ):

1. Obtain the interpolated values of the in-plane displacements {^}(j\, {^}|j\ ,

{^}(,V ^ d {v^}[i^:

{u-i}[^^= ( l -W^i}S*! , )+^{ i i r} j f l , )

43

{^}Sf)' = (1 - /3}{«2}Sf2:, + ^{^}|*!.,

/3 is another interpolation parameter, it is taken to be equal to one for the

purpose of this research.

2. Use {w}\^l,y {u^}\}l K}[J)\ {u-2}\^l and {rJl}|J) to calculate

(0 {g-H/i(u;,ui,vi,u2,V2)}{*?.

3. Solve Equation 4.1 to get {u?}!*).

4. Check convergence: if satisfied, exit the loop and start computations for

the next increment, otherwise go to next step.

5. Obtain the interpolated values {u;}/^^, using the interpolation parameter

a :

{u?}W = (1 - a){^}S5 + a{12?}W„.

6. Using {ic;}/ x, calculate {/2(^)}(i) •

7. Solve Equation 4.2 to get {iii}(i), {vi}(i), {ti2}(,)» and {u2}(t)-

8. Go to step 1.

The iterative scheme described above has been implemented by developing

a FORTRAN computer program. A Hsting of the program is given as an ap­

pendix. The expledned procedure has been successful in solving the problems

demonstrated in the next chapter. However, it is subject to further research in

44

order to minimize the number of iterations per increment and the computing

time accordingly.

45

GO •

O

LJ

LJ

CC Ctl GZ CL.

^ ' -o o

CM •

0.0

0

" " \

0.0 1.0 2.0 3.0 4.0 5.0

WMRX/RVERRGE PLRTE THICKNESS 6.0

Figure 4.1: Interpolation Parameter a vs.Wmax/hav

CHAPTER 5

RESULTS

It is the purpose of this chapter to present solutions obtained from the math­

ematical model developed in this research. The solutions are compared with

those obtained from experiments conducted at the Glass Research and Testing

Laboratory, Texas Tech University, The laminated glass imit sample tested is a

simply supported squaxe unit with an interlayer thickness of 0.06zn. The detailed

dimensions of the units are given later. Experiments conducted by researchers

at Texas Tech University on special 2 x 2m. blocks of laminated glass, indicate

tha t the shear modulus C?/, of the interlayer is nonhnear and varies with the

average shear strain in the interlayer. The value of the shear modulus of the

interlayer varies from 50 to 400p5i. The mathematical solution is obtained for

various values of Gj such as 0,100,200, and 400p5i. The influence of Gj on the

behavior of the laminated glass unit is thus illustrated.

Example 1. The first sample problem is a simply supported uniformly loaded

squaxe plate. Edges are movable in the in-plane directions. The dimensions and

properties of the plate are given as follows:

l^ = ly = 60zn.

hi= /i2 = 0.1875in.

t = 0.06in.

^ = l O V i

/x = 0.22.

46

47

The plate is subjected to a uniformly increasing static lateral pressure up to

a maximum pressure in the proximity of 1 psi. Because of the double symmetry

of the laminated plate and the appHed load, the theoretical model performs the

analysis for one-quarter of the plate in order to minimize computer storage and

to increase the computational efficiency.

A 10 X 10 finite difference mesh is selected to analyse the problem. Fur­

ther studies might be needed to investigate the mesh size effect on the solution

convergence. However, the selected mesh is found to give acceptable results as

compared to the experimental data. A tolerance of 10~^ is chosen as a conver­

gence criterion. The plate is loaded incrementally until a peak value near Ipsi

is reached. Forty load increments are taken for this purpose.

Five cases are investigated: Gj is assigned the values 0 (to simulate the

layered case), 100,200, and 400p5i . The monoHthic case is successfully simulated

by putting Gj = 0 and doubHng each plate thickness as weU as the appHed load

q. Each individual plate hence carries a uniformly distributed load q.

Four series of plots are generated. The first series (Figure 5.1) gives the

relationship between the appHed lateral pressure and the maximum deflection

at the center of the plate. It clearly shows that the degree of nonHnearity in

the pressure-displacement relationship increases with the ^ ^ " ratio, w^ax being

the maximum deflection of the plate and hav the average thickness of the two

glass plates. This effect is due to the increased stiffness of the plate caused

by membrane action. For a given load, the value of the maximuni deflection

decreases as the interlayer shear modulus increases.

48

The second series of plots (Figure 5.2), gives the plate center maximum

compressive stress at the the top face of the top plate as well as the plate center

maximum tensile stress at the I oltom face of the bot tom plate. These stresses

axe compsu-ed to those obtained experimentally. For a value of Gi of lOOpsi,

theoreticzd and experimental results are found to be in good agreement.

Since membrane stresses in the center of the plate are of tensile nature,

the magnitude of the principal tensile stress is found to be higher than the

compressive stresses at the same point. This phenomenon is interpreted by the

fact that membrane and bending stresses have the same signs at the bot tom

face of the plates while they have opposite signs at the top face.

For high values of the appHed pressure, the location of the maximum principal

stresses in the plate is found to shift from the center of the plate towards the

edges. While the maximum bending stresses in the plate occur at the center of

the plate, the maximum membrane stresses are located at the plate edges. The

resulting principal stress is found to be at the center of the plate only if the

bending stresses are more predominant than membrane stresses, i.e., for small

values of the appHed pressure. The third and fourth series of curves (Figures

5.3 and 5.4) give the maximum tensile and compressive principal stresses in the

plate. The magnitude of the maximum principal compressive stresses is larger

than that of the majcimum principal tensile stresses in the plate. However, the

maximum principal tensile stresses values are of practical importance since they

represent the failure criterion for glass units.

Example 2. The second example constitutes of a uniformly loaded rectan­

gular plate, simply supported on all edges. Edges are movable in the in-plane

49

directions. The dimensions and properties of the plate are given as follows:

Ix = 60in.

/y = 96in.

hi = /i2 = 0.125m.

t = 0.03in.

E = lOV*

/x = 0.22.

The plate is subjected to a uniformly increasing static lateral pressure until

divergence occurs. It has to be reported that after ^^°^ reaches a certain value,

convergence could not be obtained for a wide range of interpolation parameters

a and 13. A stress function approach in the analysis of geometrically nonhnear

single plates has been successfully performed by Vallabhan [19] for higher degrees

of nonHnearity (i.e., ^^")• However, a similar degree of nonlinearity could not

be obtained using the displacement approach employed in this research. Further

research is needed to investigate the convergence characteristics of the problem

for highly nonhnear behavior.

Here also, a 10 x 10 finite difference mesh is selected to analyze the problem.

A tolerance of 10"'* is chosen as a convergence criterion.

Five cases are investigated: Gj is assigned the values 0,100,200, and 400p5i,

respectively. The fifth case is that of a monolithic plate of the same nominal

thickness.

50

Similarly to problem 1, four series of plots are generated. The first series

(Figure 5.5), gives the relation between the applied pressure and the maximum

deflection at the center of the plate. For a given load, the maximum deflection

value decreases for a higher interlayer rigidity modulus. The lateral deflection

is found to be more sensitive to the Gj value as compared to the first problem

since the interlayer thickness is smaller than for the former case. For a value

for Gj = AOOpsi, the obtained maximum deflection is sHghtly smaller than that

of a monoHthic system of the same nominal thickness subjected to the same

appHed pressure. A system with small interlayer thickness and high interlayer

shear modulus might therefore be stiffer than a monoHthic system with the same

nominal thickness. This result is of practical importance for the laminated glass

industry and needs to be confirmed experimentally.

The second series of plots (Figure 5.6) gives the plate center minimum and

maximum principal stresses at the top face of the top plate and the bot tom

face of the bot tom plate, respectively. It is observed that for all the examined

values of Gj, the stresses at the center of the plate behave more closely to the

monoHthic system than to a layered system.

The third and fourth series of curves (Figures 5.7 and 5.8) give the maximum

tensile and compressive principal stresses in the plate. The magnitude of the

maximum principal compressive stresses is larger than that of the maximum

principal tensile stresses in the plate. However, the maximima principal tensile

stresses values are of practical importance since glass failure is primarily due to

tension, rather than compression.

51

It has to be mentioned that for high Wmax/hav ratios (i.e., for w^ax/hav in

the proximity of 5), the solution obtained by the model seems to converge to an

exaggerated value of the lateral displacement and the stresses. This phenomenon

may be at tr ibuted to the fact that the finite difference mesh used in this solution

is not fine enough to represent the continuum. Another explanation would be

that for high lateral pressure, the system of equations becomes ill-conditioned,

so that the solution converges to an exaggerated value of the displacement. The

previous two hypotheses have to be tested in order to determine whether the

first one, the second one, or a combination of both is responsible for the error.

In figures 5.1 through 5.8, these limiting points axe represented by a -f symbol.

Results seem to be unreHable for displacement and stress values beyond these

points.

52

rs)

o •

^ ^ CD •

M (I

NC

HE

S

.6

C

1

O ^ •—^

CJ LJ _ ] L- ^

Q °

rg • o ~

o o _

1 ///

/ o /

/4> / yy

y/y

.Si

_£. '

^ ^ ^

/

CXPCRIfCNlflL RESULTS

^ ^ ^ < ^ < 3 8 t >

60 IN. X 60 IN.

HI-H2-0.1875,7-0.06 IN.

E-10,000,000 PSI,M-0.22

10 X 10 nCSH

0.0 0.2 0.4 0.6 0.8 PRESSURE (PSI)

1.0 1.2

Figure 5.1: Problem 1- Pressure v&Maximimi Defiection

53

0.4 0.6 0.8 PRESSURE (PSI)

Figure 5.2: Problem 1- Pressure vs. Stresses at the Center

54

0.4 0.6 PRESSURE (PSI)

Figure 5.3: Problem 1- Pressure vs. Maximum Principal Tensile Stress

55

0.4 0.6 PRESSURE (PSI)

Figure 5.4: Problem 1- Pressure vs. Maximum Principal Compressive Stress

56

0.2 PRESSURE (PSI)

Figure 5.5: Problem 2- Pressure vs. Maximum Deflection

57

0.2 PRESSURE (PSI)

Figure 5.6: Problem 2- Pressure vs. Stresses at the Cent er

58

0.2 PRESSURE (PSI)

Figure 5.7: Problem 2- Pressure vs. Maximum Principal Tensile Stress

59

0.2 PRESSURE (PSI)

0.4

Figure 5.8: Problem 2- Pressure vs. Maximum Principal Compressive St ress

CHAPTER 6

SUMMARY, CONCLUSIONS

AND RECOMMENDATIONS

Summary

A nonhnear plate bending theory for laminated glass plates is developed with

the assumption that plane section remains plane for each individual layer for the

laminated units, rather than for the whole laminated unit. The principle of min­

imum potential energy has been used to obtain field and boundary equations of

the mathematical model. Using the central finite difference technique, the ob­

tained field equations are converted into nonlinear algebraic system of equations.

These equations are solved iteratively using interpolation parameters.

The model handles laminated plates of different thicknesses. Different mesh

sizes along the x and y directions are incorporated in order to handle any rectan­

gular size of the unit. The model is capable of simulating layered and monoHthic

units as well as laminated units.

Conclusions

1. Results from the developed model reasonably agree with experimental data

for a value of the interlayer shear modulus in the proximity of lOOpsi. Pre-

Hminary experimental investigations confirmed this value for small shear

strains.

2. For small interlayer thicknesses and high values of the interlayer shear

modulus, a laminated unit may be stiffer than a monoHthic plate of the

60

61

same nominal thickness.

3. The solution of a given load increment does not depend on the solution

obtained for the previous increments. Hence, the accumulation of errors

is avoided in the used procedure.

4. For high values for the interlayer shear modulus, the system behaves closely

to a monolithic plate of the same nominal thickness under the same applied

load, and can be approximated to act as monoHthic for practical purposes.

5. The model successfuUy simulates the layered and the monoHthic cases by

equating the shear modulus to zero. For the monolithic case, both the

plate thicknesses and the applied pressure are doubled.

Recommendations

Further research is needed to investigate the foUowing topics:

1. Development of the optimum values for interpolation parameters to mini­

mize the number of iterations needed for convergence.

2. Investigation of the mesh size effect on the solution accuracy.

3. Studying the magnitude of the appHed load increment versus the number

of iterations needed for convergence in order to minimize the number of

computations for a given problem.

4. Non-dimensionalizing field Equations 2.15 through 2.19 in order to develop

a parametric study for laminated glass units.

62

5. Put t ing Matrix [B] in a banded form to minimize the storage and number

of computations.

6. Further experimental investigation of the interlayer shear modulus value

under various temperatures. The developed model shows a high sensitivity

of the behavior of laminated units to the interlayer shear modulus value.

7. Further studies on different boundary conditions using the same approach

adopted in this research (i.e., fixed end conditions, simply supported with

immovable edges, ends on elastic supports).

8. The use of same formulation type to study the post-buckHng behavior of

laminated systems. A different load potential energy Q, is employed in this

case.

9. Modifying the field equations in order to account for the material nonHn­

earity of the interlayer, PreHminary investigations of the interlayer shear

modulus show a nonhnear behavior with shear strain.

10. The use of the same approach to model different plate shapes (i.e., circular,

plates with incHned boundaries).

11. The development of a finite element solution and a comparison of the

results of the two models.

12. Further laminated unit bending experimental investigation in order to as­

sess the niodel assumption validity.

LIST OF REFERENCES

1. Behr, R. A., Minor, J. E., Linden, M. P., and Vahabhan, C. V. G", "Lami­nated Glass Units under Uniform Lateral Pressure," Journal of Structural Engineering, ASCE, 111(5): 1037-1050, 1985,

2. Bijlaard, P. P., "On the Elastic StabiHty of Thin Plates Supported by a Continuous Medium," Proc. Roy, Netherlands Acad. Sci., No, 10, 1946.

3. Bijlaard, P. P., "On the Elastic StabiHty of Thin Plates Supported by a Continuous Medium," I. Proc. Roy. Netherlands Acad. Sci., No. 1, 1947,

4. Bijlaard, P. P., "On the Elastic StabiHty of Thin Plates Supported by a Continuous Medium," II, Proc. Roy, Netherlands Acad. Sci., No. 2, 1947,

5. Das, Y. C , and Vallabhan, C. V. Girija, "A Mathematical Model for Non­hnear Stress Analysis of Sandwich Plate Units," Mathematical Comput. ModeUing, Vol. 11, pp. 713-719, 1988.

6. Hoff, N, J., and Mautner, S. E., "Bending and Buckling of Sandwich Beams," Journal of Aeronautic Sciences, Vol. 15, No. 12, pp. 707-720, December, 1984.

7. Hoff, N. J., "Bending and BuckHng of Sandwich Beams," National Advi­sory Committee for Aeronautics, Technical note 2225, November, 1950.

8. Kaiser, R., "Rechnerische und ExperimenteUe Ermittlung der Durchbiegun-gen und Spannungen von Quadratischen Flatten bei freier Auflagerung an den Randem gleichmassig verteilter Last und grosser Ausbiegungen," Z. F. A. M. M., Bd, 16, Heft 2, pp. 73-98, April, 1936.

9. Legget, D. M. A., and Hopkins, H. G., "Sandwich Panels and Cylinders under Compressive End Loads," R, k M. No. 2262, British A. R. C , 1942,

10. Levy, S., "Bending of Rectangular Plates with Large Deflections, " NACA, TN No. 846, 1942.

11. Langhaar, H. L., Energy Methods in AppHed Mechanics, John Wiley and Sons, Inc., New York, 1962.

12. March, H. W., and Smith, C, B,, "Buckling Loads of Flat Sandwich Panels in Compression, Various Types of Boundary Conditions," Mimeo. No. 1525, Forest Products Lab., U. S. Dept. Agriculture, March, 1945.

13. Moore, D. M., "Proposed Method for Determining Glass Thickness of Rectangular Glass Solar Collector Panels Subjected to Uniform Normal Pressure Loads," JPL PubHcation 80-34, Jet Propulsion Laboratory, Pasadena, California, October, 1980,

63

64

14. "MSC NASTRAN AppHcation Manual," Macneal Schwendler Corpora­tion, Los Angeles, California, 1981,

15. Reissner, E,, "Finite Deflections of Sandwich Plates," Journal of Aeronau­tic Science, Vol. 15, No. 7, pp. 435-440, July, 1948.

16. Tayyib, A, H., "Geometrically Nonhnear Analysis of Rectangular Glass Panels by Finite Element Method," Ph, D, Dissertation, Texas Tech Uni­versity, 1980.

17. Timoshenko, S., and Woinowsky-Krieger S,, Theory of Plates and Shells, McGraw-HiU Company, I n c , New York, 1965.

18. Tsai, C. R., and Stewart, R. A., "Stress Analysis of Large Deflection of Glass Plates by Finite Element Method," Journal of Ceramic Society, Vol. 59, Nos. 9-10, pp. 445-448, 1976.

19. Vallabhan, C. V. G., "Iterative Analysis of Nonhnear Glass Plates," Jour­nal of Stmctural Engineering, ASCE, 109(2): 2416-2426, 1983.

20. VaUabhan, C. V. G., and Minor, J. E., "Experimentally Verified Theo­retical Analysis of Thin Glass Plates," Proceedings of the 2nd Interna­tional Conference, Computational Methods and Experimental Measure­ments, Southampton, July, 1984.

21. Van der Neut, A., "Die Stabilitaet Geschichteter Flatten," Rapport S, 286, Nationaal Luchtvaartlaboratorium, September, 1943.

22. Wang, B, Y., "Nonhnear Analysis of Rectangular Glass Plates by Finite Difference Method," M, S, Thesis, Texas Tech University, 1981.

APPENDIX

PROGRAM LISTING

65

66

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

0011

0012

0013

0014

0015

0016

0017

0018

0019

0020

0021

0022

0023

0024

0025

0026

0027

0028

0029

0030

0031

c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c

PROGRAM LAMPLATE « * « « « * < » « « i « * 4 i « *

FORTRAN PROGRAM TO PERFORM THE NONLINEAR ANALYSIS

OF LAMINATED GLASS UNITS SUBJECTED TO

UNIFORM LATERAL PRESSURE

»»**«»***«at*****4i*««*

CODED BY: MAGDI MOHAREB, RESEARCH ASSISTANT

CIVIL ENGINEERING DEPARTMENT. TEXAS TECH UNIVERSITY

LUBBOCK. TEXAS 79409

MAY 1990

67

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026

C C C C C C C C C C C C C C C C C

c c c c c c

*mm*m*miti****m*mmmmm*m mm*m*m*

* INPUT DATA DEFFINITIONS *

**•• CARD «1 (SUBROUTINE DATINPUT LINE 0019) *•** WOVERTO A 15 ELELMENT ARRAY CONTAINING NONDIMENSIONAL

MAXIMUM UTERAL DISPLACEMENT (WMAX/HAV) ALPHA 0 A 15 ELEMENT ARRAY CONTAINING THE INTERPOLATION

PARAMETER ALPHA VALUE CORRESPONDING TO (WMAX/HAV)

***• CARD «2 (SUBROUTINE DATINPUT LINE 0025) **** NX NUMBER OF SUBDIVISIONS IN X DIRECTION NY NUMBER OF SUBDIVISIONS IN Y DIRECTION XL PUTE HALF LENGTH IN THE X DIRECTION(IN.) YL PLATE HALF LENGTH IN THE Y DIRECTION(IN.) HI UPPER PLATE THICKNESS (IN.) H2 LOWER PLATE THICKNESS (IN.) T INTERLAYER THICKNESS (IN.) ELAS GLASS YOUNG S MODULUS (PSI) PR GLASS PQISSON S RATIO GI INTERLAYER RIGIDITY MODULUS (PSI) Q APPUED PRESSURE VALUE (PSI) NINC NUMBER OF LOAD INCREMENTS ERR PERMISSIBLE ERROR MAXIT MAXIMUM N. OF ITTERATIONS

68

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

0011

0012

0013

0014

0015

0016

0017

0018

0019

0020

0021

0022

0023

0024

0025

0026

0027

0028

0029

0030

0031

0032

0033

0034

0035

0036

0037

0038

0039

0040

0041

0042

0043

0044

0045

0046

0047

0048

0049

0050

0051

0052

0053

0054

C

C

C

C

c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c

ROUTINE DESCRIPTIONS

1 LAMPLATE: MAIN PROGRAM

HANDLES THE ITERATIVE PROCEDURE DESCRIBED IN

CHAPTER 4

2 DATINPUT: READ AND WRITE INPUT DATA

3 AMATRIX: CONSTRUCT [A] MATRIX, CORRESPONDING TO THE LEFT

HAND SIDE OF THE LATERAL DISPLACEMENT FIELD EQUATION

3,1 ALINE: USED IN SUBROUTINE AMATRIX

PUT [A] MATRIX ENTRIES CORRESPONDING TO A GIVEN

FINITE DIFFERENCE MESH POINT (I.J)

4 ADECOMP: PERFORM A UTDU DECOMPOSITION OF MATRIX A

5 ASOLVE: PERFORMS FORWARD AND BACKWARD SUBSTITUTION

TO SOLVE [A]-CW} » {R>

6 BMATRIX: CONSTRUCT [B] MATRIX, CORRESPONDING TO THE LEFT

HAND SIDE OF THE FOUR IN-PUNE FIELD EQUATIONS

FOR A GIVEN POINT (I.J). FOUR EQUATIONS ARE GIVEN

BY THE APPROPRIATE SUBROUTINE:

FOR CORNER POINTS

FOR FIRST ROW

FOR LAST ROW

FOR FIRST COLUMN

FOR LAST COLUMN

FOR POINTS INSIDE THE DOMAIN

THROUGH 6.6 CALL THE FOLLOWING SUBROUTINES

6.1 BCORNERS:

6.2 BFTRSTROW

6.3 BTOPROW:

6.4 BFIRSTCOL

6.5 BUSTCOL:

6.6 BCORE:

SUBROUTINES 6

6.A ASUB:

6,B BCSUB

6.C DSUB

6.D PSUB

6.E KSUB

FORM THE 4X4 SUBMATRICES ON THE DIAGONAL OF MATRIX B

FORM MATRIX [B] OFF DIAGONAL 4X4 SUBMATRICES

DETERMINE THE LINE NUMBERS IN [B] CORRESPONDING

TO A GIVEN FINITE DIFFERENCE POINT (I.J)

7 BDECOMP: PERFORM THE L-U DECOMPOSITION OF MATRIX [B]

8 BSOLVE: PERFORM FORWARD AND BACKWARD SUBSTITUTION TO SOLVE

[B]{R>-CR}

THE SOLUTION VECTOR IS STORED IN THE RHS VECTOR {R}

9 SELECT: SEPARATES THE IN-PLANE DISPUCEMENT VECTORS

•CU1>. {Vl>, {U2}. {V2} GIVEN THE VECTOR {RHS2}

10 RHSIDEA: TO FORM THE RIGHT HAND SIDE OF MATRIX A

IT CALLS THE FOLLOWING SUBROUTINES:

10.1 FIRSTDERU: COMPUTE FIRST DERIVATIVES OF U DISPLACEMENTS

10.2 FIRSTDERV: COMPUTE FIRST DERIVATIVES OF V DISPLACEMENTS

10.3 FIRSTDERW: COMPUTE FIRST DERIVATIVES OF W DISPLACEMENT

10.4 SECDERW: COMPUTE SECOND DERIVATIVES OF W DISPLACEMENT

11 RHSIDEB: TO FORM THE RIGHT HAND SIDE OF MATRIX B

IT CALLS THE FOLLOWING SUBROUTINES:

11.1 DERWTWO: COMPUTE THE DERIVATIVES OF FOR (NX+1)*(NY+1)

POINTS. GIVEN THEIR VALUES AT THE INNER NX*NY

MESH POINTS

11.2 R2C0RE:

11.3 R2B0T:

11.4 R2LEFT:

11.5 R2T2:

11.6 R2T1:

11.7 R2R2:

COMPUTE RHS OF [B] FOR MESH CORE POINTS

COMPUTE RHS OF [B] FOR BOTTOM EDGE POINTS

[B] FOP I:FT EDGE POINTS

[B] FOR POINTS RIGHT BEFORE TOP EDGE

COMPUTE RHS OF [B] FOR TOP EDGE POINTS

COMPUTE RHS OF [B] FOR POINTS RIGHT BEFORE RIGHT EDGE

COMPUTE RHS OF

COMPUTE RHS OF

69

0055

0060

0061

0062

0063

0064

0065

0066

0067

0068

0069

0070

0071

0072

0073

0074

0075

0076

0077

0078

0079

0080

0081

0082

0083

0084

C

C

c c c c c c c c c c c c c c c c c c c c c c c c

11.8 R2R1

12 COPY:

13 CHECK:

14 RBINTERP:

COMPUTE RHS OF [B] FOR RIGHT EDGE POINTS

SET VECTOR {A} OF DIMENSION N EQUAL TO VECTOR {B}

CHECK IF CONVERGENCE IS SATISFIED

CALCULATE INTERPOLATED VALUES OF {W} USING THE

INTERPOLATION PARAMETER ALPHA

CALCULATE INTERPOLATED IN-PUNE DISPLACEMENTS

USING THE BETA PARAMETER («1 FOR THIS CASE)

PRINT OUT DISPLACEMENTS IN CASE CONVERGENCE IS

ACHIEVED

PRINT OUT A MESSAGE IN CASE DIVERGENCE OCCURS

CALCULATE GUESS DISPLACEMENT VECTORS FOR THE

NEXT INCERMENT BY LINEAR EXTRAPOLATION OF THE

PREVIOUS TWO INCREMENTS

DETERMINE THE VALUE OF ALPHA CORRESPONDING

TO MAXIMUM DEFLECTION

CALCULATE AND PRINT OUT PRINCIPAL STRESSES

IT CALLS THE FOLLOWING SUBROUTINES:

STRAINM: CALCULATE MEMBRANE STRESSES GIVEN DISPLACEMENTS

STRESSM: CALCULATE MEMBRANE STRESSES GIVEN MEMBRANE STRAINS

BENDSTRESS: CALCULATE BENDING STRESSES GIVEN DISPLACEMENTS

PRINCIP: COMBINE BENDING AND MEMBRANE STRESSES. CALCULATE

PRINCIPAL STRESSES. AND DETERMINE MAGNITUDE AND

AND LOCATION OF MAXIMUM PRINCIPAL STRESSES

20.5 PRINTSTRESS: PRINT OUT PRINCIPAL STRESSES

21 OPENFILE: OPEN FILES FOR OUTPUT

22 COMMENT: PRINT OUT COMMENTS AT THE BOTTOM OF OUTPUT FILES

15 RAINTERP:

16 PRINTRES:

17 PRINTDIV:

18 GUESS:

19 XALPHA:

20 STRESS:

20,

20,

20,

20

70

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

C C C C

C c C C

PROGRAM LAMPLATE COMMON/Bl/ A(100,21) C0MM0N/B2/ NX.NY.NUM.NUMl C0MM0N/B3/ W(IOO) C0MM0N/BMATX/B(576,576) COMMON/LOAD/Q.QINC.NINC,MAXir,ERR,AL C0MM0N/DISP/U1(121).Vl(121).U2(121).V2(121) C0MM0N/DISP0LD/U10LD(121).V10LD(121),U20LD(121),V20LD(121).WOLD(IOO) C0MM0N/DISPNEW/U1NEW(121),V1NEW(121).U2NEW(121),V2NEW(121).WNEW(IOO) C0MM0N/DISPR/U1PR(121),V1PR(121).U2PR(121).V2PR(121).WPR(100) C0MM0N/DISPP/U1PP(121),V1PP(121).U2PP(121).V2PP(121).WPP(100) C0MM0N/DISPG/U1G(121).V1G(121).U2G(121),V2G(121).WG(IOO) C0MM0N/DER/DU1X(100),DU1Y(100),DV1X(100).DVIY(IOO). 1 DU2X(100),DU2Y(100),DV2X(100),DV2Y(100)

COMMON/RHSW/RHSl(100) C0MM0N/RHSIDE2/RHS2(576)

CALL OPENFILE CALL DATINPUT CALL AMATRIX CALL BMATRIX

FIRST INCREMENT FIRST CYCLE:

INC-1 QINC»1./REAL(NINC)*Q WRITE (*,*) 'INCREMENT f.INC,' FIRST CYCLE' CALL RHSIDEA(U1G.V1G.U2G.V2G.WG.RHS1) CALL ASOLVE(A,RHS1) CALL COPY(WOLD.RHSl.NUM) CALL RHSIDEB(WOLD.RHS2) CALL BS0LV£(B,RHS2) CALL SELECT(RHS2,U10LD.V10LD,U20LD,V20LD)

FIRST INCREMENT NEXT CYCLES

HCONV-0 DO 15 ITER-2,MAXIT IF (HCOHV.EQ.O) THEN CALL RHSIDEA(U10LD,V10LD,U20LD.V20LD,W0LD,RHS1)

CALL ASOLVE(A,RHS1) CALL CHECK(RHS1,WOLD,NUM.NCONV)

IF (NCONV.EQ.O) THEN CALL XALPHA(RHSKD.AL) CALL RBINTERP(WOLD,RHS1,WNEW,AL) CALL RHSIDEB(WNEW,RHS2) CALL BSOLVE(B.RHS2) CALL SELECT(RHS2,U1NEW.V1NEW,U2NEW,V2NEW)

CALL COPY(UIOLD.UINEW.NUMI) CALL C0PY(V10LD,V1NEW,NUM1) CALL C0PY(U20LD,U2NEW,NUM1) CALL C0PY(V20LD.V2NEW,NUM1)

71

0055 0056 0057 0058 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081 0082 0083 0084 0085 0086 0087 0088 0089 0090 0091 0092 0093 0094 0095 0096 0097 0098 0099 0100 0101 0102 0103 0104 0105 0106 0107 0108

C-C C C

C C C C

C c c c

CALL COPY(WOLD,WNEW,NUM) ELSE

CALL PRINTRES(RHSl,U10LD,VIOLD,U20LD,V20LD,INC,ITER) CALL STRESS(UIOLD.VIOLD,U20LD,V20LD,RHSl) CALL COPY(UlPR.UlOLD,NUMl) CALL C0PY(V1PR,VIOLD,NUMl) CALL COPY(U2PR,U20LD,NUMl) CALL COPY(V2PR,U20LD,NUMl) CALL COPY(WPR,RHS1,NUM)

END IF END IF

IF (ITER.EQ.MAXIT.AND.NCONV.EQ.O) CALL PRINTDIV(INC) 15 CONTINUE

INCREMENTS 2 TO NINC

IF (NCONV.EQ.l) THEN DO 35 INC>2.NINC

IF (NCONV.EQ.l) THEN

INCREMENT « INC FIRST ITERATION:

WRITE(*.*) 'INCREMENT t'.INC,' FIRST CYCLE' QINC»REAL(INC)/REAL(NINC)*Q CALL GUESS(U1G,V1G.U2G.V2G.WG.

1 U1PR,V1PR.U2PR.V2PR,WPR. 1 U1PP.V1PP,U2PP,V2PP.WPP)

CALL XALPHA(WG(l).AL) NDIV-1 DOWHILE (NDIV.EQ.1.AND.AL.GT.0.05) IF (NCONV.EQ.O .AND. AL.GT.0.05) AL«AL-0.005 WRITE (•.•) 'ALPHA-'.AL CALL RHSIDEA(U1G.V1G.U2G,V2G.WG,RHS1) CALL AS0LVE(A,RHS1) CALL RBINTERP(WG,RHS1,WNEW,AL) CALL COPY(WOLD,WNEW.NUM) CALL RHSIDEB(WOLD.RHS2) CALL BSOLVE(B,RHS2) CALL SELECT(RHS2,UIOLD,VIOLD,U20LD,V20LD)

NCONV-O

INCREMENT « INC NEXT ITERATIONS:

DO 25 ITER=2,MAXIT

IF (NCONV.EQ.O.AND.ABS(RHS1(1)),LT,50,) THEN

IF (ITER.EQ.2) THEN CALL RAINTERP(U1,V1,U2,V2.

1 UIOLD,VIOLD,U20LD,V20LD, 1 U1G.V1G,U2G.V2G)

ELSE CALL RAINTERP(U1,V1,U2,V2.

72

0109 1 U10LD,V10LD,U20LD,V20LD, 0110 1 U1NEW.V1NEW,U2NEW,V2NEW) 0111 END IF 0112 CALL RHSIDEA(U1,VI,U2,V2,WOLD,RHSl) 0113 CALL ASOLVE(A,RHSl) 01-4 CALL CHECK(RHS1,WOLD,NUM.NCONV) 0115 IF (ABS(RHS1(1)).LT.50.) THEN 0116 IF (NCONV.EQ.O) THEN 0117 CALL RBINTERP(WOLD,RHS1,WNEW,AL) 0118 CALL RHSIDEB(WNEW.RHS2) 0119 CALL BSOLVE(B.RHS2) 0120 CALL SELECT(RHS2,U1NEW,V1NEW,U2NEW.V2NEW) 0121 CALL COPY(UIOLD.UI.NUMl) 0122 CALL C0PY(V10LD.V1,NUM1) 0123 CALL COPY(U20LD.U2.NUMl) 0124 CALL C0PY(V20LD.V2.NUM1) 0125 CALL COPY(WOLD,WNEW,NUM) 0126 ELSE 0127 NDIV-0 0128 CALL PRINTRES 0129 1 (RHS1,U1,V1,U2.V2.INC,ITER) 0130 CALL STRESS(U1,V1,U2,V2.RHS1) 0131 IF(INC.NE.NINC) THEN 0132 CALL COPY(U1PP,U1PR,NUMl) 0133 CALL C0PY(V1PP,V1PR,NUM1) 0134 CALL COPY(U2PP,U2PR,NUMl) 0135 CALL COPY(V2PP,U2PR,NUMl) 0136 CALL COPY(WPP.WPR.NUM)

0137 C 0138 CALL COPY(UlPR.Ul.NUMl) 0139 CALL COPY(VlPR.Vl.NUMl) 0140 CALL C0PY(U2PR.U2.NUM1) 0141 CALL COPY(V2PR,U2,NUMl) 0142 CALL COPY(WPR.RHSl.NUM) 0143 END IF 0144 END IF 0145 END IF 0146 END IF

0147 IF (ITER.EQ.MAXIT.AND.NCONV,EQ,0) THEN

0148 CALL PRINTDIV(IHC)

0149 NDIV«1

0150 ENDIF 0151 25 CONTINUE 0152 END DO 0153 END IF 0154 35 CONTINUE 0155 END IF 0156 CALL COMMENT 0157 END

73

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

C

c c c c c

c

c c c

c

c

c

c

SUBROUTINE DATINPUT

READ k PRINT INPUT DATA CALCULATE SOME OF THE CONSTANTS COMMONLY USED IN THE PROGRAM

C0MM0N/B2/ NX.NY.NUM.NUMl

COMMON/BINPUT/ GI.ELAS.PR.H1.H2,T,HX,HY,G COMMON/LOAD/Q,QINC,NINC,MAXIT,ERR,AL COMMON/CE/CCl,CC2,CC3 C0MM0N/CC/C2X,C2Y.XY COMMON/BD/CA,CB,CD.CE C0MM0N/INTERP/ALPHA(15).W0VERT(15),HAV

WRITE(6,5)

5 FORMATC WOVERT k CORRESPONDING ALPHA VALUES') DO 10 I«l,15

READ(4.«) WOVERT(I).ALPHA(I) WRITE(6,*) WOVERT(I),ALPHA(I)

10 CONTINUE

TO READ AND WRITE INPUT

READ (*,*) NX,NY,XL,YL,H1,H2,T.ELAS, 1 PR.GI.Q.NINC.ERR.MAXIT

WRITE (6.6) NX.NY.XL.YL.H1.H2.T.ELAS, 1 PR.GI.Q.NINC,ERR,MAXIT

6 FORMATC NUMBER OF SUBDIVISIONS IN X DIRECTION ',15 / 1 ' NUMBER OF SUBDIVISIONS IN Y DIRECTION '.15 / 2 ' PLATE HALF LENGTH IN THE X DIRECTION(IN.)..'.E12.5 / 3 ' PUTE HALF LENGTH IN THE Y DIRECTION(IN.) .. ' .E12.5 / 4 ' UPPER PLATE THICKNESS (IN.) ' .E12.5 / 5 ' LOWER PLATE THICKNESS (IN,) ' .E12,5 / 6 ' INTERLAYER THICKNESS (IN,) '.E12.5 / 7 ' GUSS YOUNG S MODULUS (PSI) •.E12,5 / 8 ' GUSS POISSON S RATIO ' .E12,5 / 9 ' INTERLAYER RIGIDITY MODULUS (PSI) '.E12.5 / A ' LOAD VALUE (PSI) ',E12.5 / B ' NUMBER OF LOAD INCREMENTS '. 15 / C ' PERMISSIBLE ERROR ' .E12.5 / D ' MAXIMUM N. OF ITTERATIONS ' .15 ///)

NUM-NX*HY NUM1-(NX+1)*(NY+1) HX-XL/REAL(NX) HY-YL/REAL(HY)

G-0.5*EUS/(1.+PR)

CONST-ELAS/(1.-PR*PR) CC1-C0NST«H1

74

0055 0056 0057 0058 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070

C

C

C

C

CC2=C0NST*H2 CC3-GI*(0.5*Hl+0.5*H2+T)/T

C2X-0.5/HX C2Y-0.5/HY XY=C2X*C2Y

CA«0.5*(1.+PR) CB=0.5*(1.-PR)

CD»GI*CB/(G*H1*T)*(0,5*H1+0,5*H2+T) CE=CD*H1/H2

HAV-0,5*(H1+H2)

RETURN END

75

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

C C C

c c

c

c

c

c C

C

C

c

c c

SUBROUTINE AMATRIX CALCUUTE THE A (NX, NY) MATRIX FOR A MAXIMUM MESH SIZE OF 10X10 SUBDIVISIONS

COMMDN/Bl/ A(100,21) C0MM0N/B2/ NX,NY,NUM,NUM1 COMMON/BINPUT/ GI,ELAS,PR,H1,H2.T.HX.HY,G

CALCUUTE CONSTANTS D1«ELAS*H1**3/(12*(1-PR*PR)) D2«ELAS*H2**3/(12*(1-PR*PR)) D=D1+D2 AA-GI*(0,5*Hl+0.5*H2+T)*»2/T

CXY-1./(HX*HY) CX2-1./(HX*HX) CY2»1./(HY*HY) CX4«CX2*CX2*D CY4»CY2*CY2«D CXY2-CX2*CY2*D

CX2-CX2*AA CY2»CY2*AA

C«6*CX4 + 6*CY4 + 8*CXY2 + 2*CX2 + 2*CY2 6- -4*CX4 -4*CXY2 -CX2 H> CX4 F- 2*CXY2 E- -4*CY4-4*CXY2-CY2 C» CY4

MESH CORNER POINTS: CALL ALINE(1.1,0.25*C.0.5*B.O,5*H,0,,0,5*E.F,0.5*G) CALL ALINE(2.1.(C+H)*,5.0,5*B.0.5*H.F.E.F,G) CALL ALINE(1.2.(C+G)*.5,B,H,0.0.5*E,F,0.5*G) CALL ALINE(2,2,C+H+G,B.H.F.E,F.G)

CALL ALINE(HX-1,1,.5*C,,5»B,0.,F,E,F,G) CALL ALINE(NX.1,0.5*(C-H).0,.0.,F,E.O,.G) CALL ALINE(NX-1,2.C+G,B.0,,F,E.F,G) CALL ALINE(NX,2,C-H+G,0,,0.,F,E,0.,G)

CALL ALINE(l,NY-l,.5*C.B.H.0...5*E.F.O.) CALL ALINE(1.NY,.5*(C-C),B,H.0..0..0..0.) CALL ALINE(2.NY-l,C+H,B,H.F,E,F.O.) CALL ALINE(2,NY,(C-G+H).B,H,0.,0.,0.,0.)

CALL ALINE(HX-1,NY-1,C,B.0.,F,E.F.0.) CALL ALINE(NX,HY-1,C-H.0.,0,,F,E.0.,0.) CALL ALINE(NX-1,NY,C-G,B,0.,0,,0..(".,0,) CALL ALINE(NX.NY.C-G-H,0,.0,,0..0..0,.0,)

MESH ROWS «1,2.NY-1. k NY

76

0055 0056 0057 0058 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081

C C

C C

C C C

10

DO 10 I«3,NX-2 CALL ALINE(I,1,.5*C,.5»B,.5*H,F,E,F.G) CALL ALINE(I,2,C+G.B.H.F.E.F.G) CALL ALINE(I,NY-l,C,B.H,F,E,F.O.) CALL ALINE(I.NY.C-G,B,H,0..0.,0.,0.)

CONTINUE

MESH COLUMNS «l,2,NX-l.k NX: DO 20 J»3.NY-2

CALL ALINE(1.J,.5*C.B.H.0,..5*E.F.0,5*G) CALL ALINE(2,J.C+H.B.H.F,E.F,G) CALL ALINE(NX-1,J,C,B,0..F,E.F,G) CALL ALINE(NX,J,C-H,0.,O..F,E,0.,G)

20 CONTINUE

MESH CORE POINTS: DO 30 >3,NX-2 DO 30 J-3,NY-2

CALL ALINE(I,J.C,B,H,F,E,F,G)

30 CONTINUE

DECOMPOSE THE A MATRIX

CALL ADECOMP(A)

RETURN END

77

0001 C-0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014

SUBROUTINE ALINE(I.J.C.B,H,Fl,E,F2,G) COMMON/Bl/ A(100,21) C0MM0N/B2/ NX,NY,NUM.NUMl K»(J-1)*NX+I A(K,1)«C IF (B .NE.O.) A(K,2)-B IF (H .NE.O.) A(K,3)»H IF (F1.HE,0.) A(K,NX)=F1 IF (E .NE.O.) A(K,NX+1)=E IF (F2.NE.0.) A(K.NX+2)»F2 IF (G .NE.O.) A(K,2*NX+1)»G RETURN END

78

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025

C

30

20

10 C

SUBROUTINE ADECOMP(S) COMMON /B2/ NX,NY,NUM,NUM1 DIMENSION S(100,21)

MBAND=2*NX+i DO 10 N-l.NUM IF (S(N,1).NE,0.0) THEN DO 20 L>2.MBAND

IF(S(N,L).NE.O.O) THEN C=S(N,L)/S(N.l) I=N+L-1 J«0 DO 30 K«L,MBAND

J-J+1 S(I,J)=S(I.J)-C*S(N.K)

CONTINUE S(N.L)-C

ENDIF CONTINUE ENDIF CONTINUE

RETURN END

79

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042

C c c c

c c c

c c c

SUBROUTINE ASOLVE(A.R) PERFORMS FORWARD AND BACKWARD SUBSTITUTIONS TO SOLVE CA]{W>-[R} THE SOLUTION VECTOR IS STORED IN THE R VECTOR

C0MM0N/B2/NX.NY,NUM.NUMl DIMENSION A(100.21).R(100)

FORWARD REDUCTION

MBAND«2*NX+1 DO 10 N«1,NUM IF (A(N.1).NE.0,0) THEN

DO 20 L»2,MBAND IF (A(N,L).NE.O.O) THEN

I-N+L-1 R(I)»R(I)-A(N,L)*R(N)

ENDIF 20 CONTINUE

R(N)«R(N)/A(N,1) ENDIF

10 CONTINUE

BACKSUBSTITUTION

DO 30 M-l.NUM N-NUM+1-M

IF (A(N.l).NE.O.O) THEN DO 40 L-2,MBAND IF (A(H,L).NE.O.O) THEN

K-N+L-1 R(N)-R(N)-A(N,L)*R(K)

ENDIF CONTINUE 40

ELSE R(N)-0.0

ENDIF

30 CONTINUE

RETURN END

80

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

C C C c

c c c

C c C c

SUBROUTINE BMATRIX

TO FORM THE B MATRIX B CAN BE FORMED FOR A MAXIMUM MESH SIZE OF 10X10 SUBDIVISIONS

C0MM0N/BC0NST/A1,A2,A3,A4,B1,B2,C1,C2,D,E1,E3 COMMON/BK /K1,K2,K3,K4 C0MM0N/BP0INT/MH,N1,N2,NNX1,NNX2.NNX3.L1,L2,LNX1,LNX2,LNX3

COMMON/BMATX /B(576,576) C0MM0N/BINPUT/GI,EUS,PR,H1,H2,T,HX,HY,G C0MM0N/B2/NX,NY,NUM.NUMl COMMON/BC/CX.CY,CX2,CY2.CXY,CXP.CYP,XOVY.PXOVY,YOVX.PYOVX

CONSTANTS TO BE USED IN THE B MATRIX

CX-l./HX CY-1./HY CX2«CX*CX CY2-CY*CY CXY-CX*CY

CXP-CX*PR CYP-CY*PR YOVX-CX*HY PYOVX-PR*YOVX XOVY-CY*HX PXOVY-PR*XOVY

HXl-HX+1 NYl-NY+1 HX2-NX+2 HY2-NY+2 MB -8*HX2+11 MH "4*HX2+6

Nl-4 N2-8 NNX1-4*NX1 NNX2-4*NX2 HNX3-4*NX2+4

LI—4 L2—8 LNXl—4*NX1 LNX2—4*NX2 LNX3—4*NX2-4

CALCUUTE MATRIX CONSTANTS

RESET THE VALUE OF C PREVIOUSLY ALTERED IN AM MATRIX SUBROUTINE

C-0.5*ELAS/(1.+PR) B1-CX2 C1-0.5*(1.-PR)*CY2

81

0055 0056 0057 0058 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081 0082 0083 0084 0085 0086 0087 0088 0089 0090 0091 0092 0093 0094 0095 0096 0097 0098 0099 0100 0101 0102

C

C

C C

c c

c c

c c

c c

c c

c c

D «0.125*(1.+PR)*CXY E1»0.5*GI*(1.-PR)/(G*H1*T) Al—2*CX2- (1. -PR) •CY2-E1

B2-0.5*(1.-PR)*CX2 C2-CY2

A2—2. *CY2- (1. -PR) *CX2-E1

E3=E1*H1/H2 A3—2. *CX2- (1. -PR) *CY2-E3

A4«-2.*CY2-(1.-PR)«CX2-E3

MESH CORNER POINTS CALL BCORNERS

MESH FIRST ROW DO 10 I-2,NX+1 CALL BFIRSTROW(I)

10 CONTINUE

MESH TOP ROW DO 20 1-2,NX CALL BTOPROW(I)

20 CONTINUE

MESH FIRST COLUMN DO 30 J-2,NY+1 CALL BFIRSTCOL(J)

30 CONTINUE

MESH UST COLUMN DO 35 J-2,NY CALL BUSTCOL (J)

35 CONTINUE

MESH CORE POINTS DO 40 I-2,NX+1 DO 40 J-2.NY+1

IF((I.NE,NX+1).0R,(J.NE,NY+1)) CALL BCORE(I,J) 40 CONTINUE

L-U DECOMPOSITION OF THE B MATRIX CALL BDECOMP

RETURN END

82

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

C C C

C C C

C C

c

C C C

SUBROUTINE BCORNERS COMMON/BK /K1,K2,K3,K4 C0MM0N/BP0INT/MH,N1,N2,NNX1,NNX2,NNX3,L1,L2,LNX1,LNX2,LNX3

COMMON/BMATX /B(576.576) C0MM0N/BINPUT/GI.EUS,PR,H1.H2.T,HX.HY.G C0MM0N/B2/NX,NY,NUM,NUMl COMMON/BC/CX.CY,CX2,CY2,CXY,CXP,CYP,XOVY,PXOVY,YOVX.PYOVX C0MM0N/BC0NST/A1,A2,A3,A4,B1.B2,C1,C2.D,E1,E3

MESH BOTTOM LEFT CORNER

B(l,l)-1, B(2,2)-l, B(3,3)-l, B(4.4)-l,

MESH BOTTOM RIGHT CORNER

CALL KSUB(NX+2,1) CALL BCSUB(0,5*CX,0,5*CX,0) CALL BCSUB(-0,5*CX,-0,5*CX,L2) CALL PSUB(CYP,0.0,NNX1)

MESH TOP RIGHT LEFT CORNER

CALL KSUB(1,NY+2) CALL BCSUB(0,5*CY,0.5*CY,0) CALL PSUB(0,0,CXP,LNXl) CALL BCSUB(-0.5*CY,-0.5*CY.2*LNX2)

MESH TOP RIGHT CORNER

PP—2. • (1. -PR*PR) *CX2 PQ—2. * (1. -PR*PR) *CY2

Pl-PP-El P2-PQ-E1 P3-PP-E3 P4-PQ-E3

CALL KSUB(NX+1,NY+1) CALL ASUB(P1.P2.P3.P4,E1,E3) CALL BCSUB(-PP,0.0,LI) CALL BCSUB(0.0,-PQ,LNX2)

CALL KSUB(NX+2,NY+1) CALL BCSUB(0.0,-PR*CY2,NNX1) CALL BCSUB(0.5*CX,CX2,0) CALL BCSUB(0.,-2.*CX2+2.*PR*CY2,L1) CALL BCSUB(-0.5»CX,CX2,L2: CALL BCSUB(0.,-PR'»CY2,LNX3)

CALL KSUB(NX+1,NY+2)

83

0055 0056 0057 0058 0059 0060 0061 0062

CALL BCSUB(CY2.0.5*CY.O) CALL BCSUB(-PR*CX2.0.0.LNXl)

CALL BCSUB(-2.*CY2+2.*PR*CX2.0.0,LNX2) CALL BCSUB(-PR*CX2.0.0.LNX3) CALL BCSUB(CY2.-0.5*CY.2*LNX2)

RETURN END

84

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023

SUBROUTINE BFIRSTROW(I) C0MM0N/BC0NST/A1,A2.A3,A4,B1.B2,C1,C2,D.E1,E3 C0MM0N/BK/K1,K2,K3,K4 C0MM0N/BP0INT/MH,N1,N2,NNX1,NNX2,NNX3,L1,L2,LNX1,LNX2.LNX3 C0MM0N/BMATX/B(576,576)

CALL KSUB(1,1) CALL ASUB(A1.1..A3.1,.E1.E3) B(K2.K2+2)-0,0 B(K4,K4-2)-0.0 CALL BCSUB(B1,0.0.N1) CALL BCSUB(B1.0.0,L1) CALL BCSUB(2.*C1.0.0.NNX2) CALL DSUB(2.*D,NNX3) CALL DSUB(-2.*D,NNX1) B(K2,K2+NNX3-l)-0.0 B(K4.K4+NNX3-l)-0.0 B(K2.K2+NNXl-l)-0.0 B(K4,K4+NNX1-1)=0.0

RETURN END

85

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016

SUBROUTINE BTOPROW(I) COMMON/BK /K1.K2,K3.K4

C0MM0N/BP0INT/MH.N1.N2,NNX1,NNX2.NNX3.L1,L2,LNX1.LNX2.LNX3 COMMON/BMATX /B(576,576) C0MM0N/B2/NX.NY.NUM,NUMl C0MM0N/BC0NST/A1,A2,A3,A4,B1,B2.C1,C2,D.E1.E3 COMMON/BC/a. CY. CX2, CY2, CXY, CXP, CYP, XOVY .PXOVY, YOVX,PYOVX

CALL KSUB(I.HY+2) CALL BCSUB(0.5*CY,0.5*CY,0) CALL PSUB(0.5*CX,0.5*CXP,LNXl) CALL PSUB(-0.5*CX,-0.5*CXP.LNX3) CALL BCSUB(-0.5*CY.-0.5*CY.2*LNX2) RETURN END

86

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023

SUBROUTINE BFIRSTCOL(J) C0MM0N/BC0NST/A1,A2.A3,A4.B1.B2,C1.C2.D.E1.E3 C0MM0N/BK/K1.K2.K3.K4 COMMON/BPOINT/MH.Nl.N2.NNX1.NNX2.NNX3,L1,L2,LNX1,LNX2,LNX3 COMMON/BMATX/B(576,576)

CALL KSUB(1,J) CALL ASUBd. ,A2,1.,A4.E1.E3) B(Kl.Kl+2)-0.0 B(K3.K3-2)-0.0 CALL BCSUB(0.0.2.*B2.N1) CALL BCSUB(0.0,C2,NNX2) CALL BCSUB(0.0.C2.LNX2) CALL DSUB(2.*D.NNX3) CALL DSUB(-2.*D,LNXl) B(K1.K1+NNX3+1)«0.0 B(K3.K3+NNX3+1)=0.0 B(Kl,Kl+LNXl+l)-0.0 B(K3.K3+LNXl+l)-0,0

RETURN END

87

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016

SUBROUTINE BLASTCOL(J) COMMON/BK /K1.K2.K3.K4 COMMON/BPOINT/MH.N1.N2,NNX1.NNX2.NNX3,L1.L2,LNX1,LNX2,LNX3 COMMON/BMATX /B(576.576) C0MM0N/B2/NX,NY,NUM,NUMl COMMON/BC/CX.CY.CX2,CY2,CXY,CXP,CYP,XOVY,PXOVY,YOVX.PYOVX

CALL KSUB(NX+2,J) CALL BCSUB(0,5*CX,0,5*CX,0) CALL BCSUB(-0.5*CX,-0.5*CX,L2) CALL PSUB(0.5*CYP,0.5*CY,NNX1) CALL PSUB(-0,5*CYP,-0.5*CY,LNX3)

RETURN END

88

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020

SUBROUTINE BCORE(I,J) C0MM0N/BC0NST/A1,A2,A3,A4,B1,B2,C1,C2.D.E1.E3 C0MM0N/BK/K1.K2,K3.K4 C0MM0N/BP0INT/MH,N1.N2,NNX1.NNX2,NNX3,L1.L2.LNX1,LNX2.LNX3

COMMON/BMATX/B(576.576)

CALL KSUB(I.J) CALL ASUB(A1.A2.A3.A4.E1,E3) CALL BCSUB(B1.B2,N1) CALL BCSUB(B1,B2,LI) CALL BCSUB(C1,C2.NNX2) CALL BCSUB(C1.C2.LNX2) CALL DSUB(D.NNX3) CALL DSUB(-D,NNX1) CALL DSUB(D,LNX3) CALL DSUB(-D.LNXl)

RETURN END

89

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018

SUBROUTINE ASUB(A1.A2.A3.A4.E1.E3) C0MM0N/BK/K1,K2.K3,K4

COMMON/BPOINT/MH.Nl.N2.NNX1.NNX2.NNX3.LI.L2,LNX1,LNX2.LNX3 COMMON/BMATX/B(576,576)

B(K1,K1)-A1 B(K2,K2)-A2 B(K3,K3)-A3 B(K4,K4)-A4

B(K1.K1+2)=E1 B(K2,K2+2)-El B(K3.K3-2)=E3 B(K4,K4-2)-E3

RETURN END

9

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012

SUBROUTINE BCSUB(B1,B2,NN) C0MM0N/BK/K1,K2,K3,K4 COMMON/BMATX/B(576.576)

B(K1.K1+NN)=B1 B(K2.K2+NN)»B2 B(K3.K3+NN)-B1 B(K4.K4+NN)-B2

RETURN END

91

0001 0002 0003 0004 0005, 0006 0007 0008 0009 0010 0011 0012

SUBROUTINE DSUB(D,NN) COMMON/BK/Kl.K2.K3.K4 C0MM0N/BMATX/B(576,576)

B(K1,K1+NN+1)-D B(K2.K2+NN-1)-D B(K3.K3+NN+1)-D B(K4.K4+NN-1)»D

RETURN END

92

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012

SUBROUTINE PSUB(Pl.P2.NN) COMMON/BK/Kl.K2.K3.K4 COMMON/BMATX/B(576,576)

B(K1.K1+NN+1)»P1 B(K2.K2+NN-1)-P2 B(K3,K3+NN+1)-P1 B(K4,K4+NN-1)»P2

RETURN END

93

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012

SUBROUTINE KSUB(I,J) COMMON/BK/Kl.K2,K3,K4 COMMON/B2/NX.NY.NUM.NUM1

K-(J-l)*(NX+2)+I K4-4*K K3-K4-1 K2-K4-2 I1-K4-3 RETURN END

94

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041

C C C

C C C

C c c

SUBROUTINE BDECOMP COMMON/BMATX /B(576.576) C0MM0N/B2/NX,NY,NUM,NUMl

N«4*((NX+2)*(NY+2)-1) DO 10 J»2,N B(J,1)-B(J.1)/B(1.1)

10 CONTINUE

DO 20 J-2.N

CALCUUTE U TERMS OF COLUMN J

DO 30 K=2,J SUM-0,0

DO 40 I-1,K-1 SUM-SUM+B(K,I)*B(I.J)

40 CONTINUE B(K.J)-B(K,J)-SUM

30 CONTINUE

CALCULATE P VECTOR OF COLUMN J

DO 50 K-J+1,N SUM-O.0

DO 60 I»1,J-1 SUM-SUM+B(K,I)*B(I,J)

60 CONTINUE B(K.J)-B(K,J)-SUM

50 CONTINUE

CALCUUTE L TERMS OF COLUMN J

DO 70 I»J+1,N

B(K,J)-B(K.J)/B(J.J) 70 CONTINUE 20 CONTINUE

RETURN END

95

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034

C C C C C

C C

c

c c c

SUBROUTINE BSOLVE(B,R)

PERFORMS FORWARD AND BACKWARD SUBSTITUTION TO SOLVE CB]{R}-{R}

THE SOLUTION VECTOR IS STORED IN THE RHS VECTOR

C0MM0N/B2/NX,NY,NUM,NUMl DIMENSION B(576,576),R(576) N-4*((NX+2)*(NY+2)-1)

FORWARD SUBSTITUTION

DO 80 1-2,N SUM-O.0

DO 90 J-1,I-1 SUM-SUM+B(I.J)*R(J)

90 CONTINUE R(I)»R(I)-SUM

80 CONTINUE

BACKWARD SUBSTITUTION

R(N)-R(N)/B(N.N) DO 100 I-N-l.l.-l SUM-O.0

DO 110 J-I+1,N SUM-SUM+B(I.J)*R(J)

110 CONTINUE R(I)-(R(I)-SUM)/B(I.I)

100 CONTINUE

RETURN END

96

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027

C C C C C

c

c

c

SUBROUTINE SELECT(RHS2.U1.V1.U2,V2)

SELECTS THE UI(NUMl).VI(NUMl) U2(NUMl).V2(NUMl) VECTORS

GIVEN THE RHS2 VECTOR

C0MM0N/B2/NX.NY.NUM,NUMl DIMENSION RHS2(576) DIMENSION U1(121).V1(121),U2(121),V2(121)

KK-O KMAX-(NX+2)*(NY+1)-1

DO 10 K-1,KMAX IF(REAL(K/(NX+2)),NE.REAL(K)/(REAL(NX)+2.)) THEN

KK-KK+1 K4-4*K Ul(KK)-RHS2(K4-3) Vl(KK)-RHS2(K4-2) U2(KK)-RHS2(I4-1) V2(KK)-RHS2(K4)

END IF 10 CONTINUE

RETURN END

97

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

C C C

SUBROUTINE RHSIDEA(U1,V1.U2,V2,W,RHS1)

CALCUUTE THE RIGHT HAND SIDE OF A

COMMON/LOAD/Q.QINC.NINC.MAXIT.ERR.AL C0MM0N/B2/NX,NY.NUM.NUMl C0MM0N/BINPUT/GI,EUS,PR,H1.H2.T.HX.HY.G C0MM0N/DER/DU1X(100).DUIY(IOO),DV1X(100),DV1Y(100), 1 DU2X(100),DU2Y(100),DV2X(100),DV2Y(100) COMMON/DERW/WXdOO) ,WY(100) ,WXX(100) ,WYY(100) .WXY(IOO) COMMON/CE/CCl.CC2.CC3 DIMENSION RHSKl) DIMENSION U1(1).V1(1).U2(1).V2(1),W(1)

CALL FIRSTDERU(UI.DUIX.DUIY) CALL FIRSTDERV(V1,DV1X.DV1Y) CALL FIRSTDERU(U2,DU2X.DU2Y) CALL FIRSTDERV(V2,DV2X,DV2Y) CALL FIRSTDERW(W,WX,WY) CALL SECDERW(W,WXX,WYY,WXY)

DO 10 J-1,HY MM-NX*(J-1) DO 10 I-1,NX

JJ-MM+I . WX2-0.5*WX(JJ)*WX(JJ)

WY2-0,5*WY(JJ)*WY(JJ) WXWY-WX(JJ)*WY(JJ)

E1X-DU1X(JJ)+WX2 E1Y-DV1Y(JJ)+WY2

E1XY-DU1Y(JJ)+DV1X(JJ)+WXWY E2X-DU2X(JJ)+WX2 E2Y-DV2Y(JJ)+WY2 E2XY-DU2Y(JJ)+DV2X(JJ)+WXWY

A1-E1X+PR*E1Y B1»E1Y+PR*E1X C1-E1XY*(1.-PR) A2«E2X+PR*E2Y B2»E2Y+PR*E2X C2-E2XY*(1.-PR)

D1-CC1*(A1*WXX(JJ)+B1*WYY(JJ)+C1*WXY(JJ)) D2-CC2*(A2*WXX(JJ)+B2*WYY(JJ)+C2*WXY(JJ)) D3-CC3*(DU1X(JJ)-DU2X(JJ)+DV1Y(JJ)-DV2Y(JJ)) RHSl(JJ)»QINC+D1+D2-D3 IF (I.EQ.l.AND.J.EQ.l) THEN

RHS1(JJ)-0.25*RHS1(JJ)

ELSE

IF (I.EQ.l.OR.J.EQ.l) THEN RHS1(JJ)-0.5*RHS1(JJ)

END IF

98

0055 END IF 0056 10 CONTINUE 0057 C

0058 RETURN 0059 END

99

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035

C C C

C C

c

c

SUBROUTINE FIRSTDERU(U.DUX,DUY) C0MM0N/B2/NX,NY,NUM.NUMl C0MM0N/BINPUT/GI.EUS,PR.H1.H2.T.HX,HY,G C0MM0N/CC/C2X,r2Y,XY DIMENSION U(1).DUX(1),DUY(1)

CALCUUTE DUX

DO 10 J»1,NY DO 10 I-l.NX

JJ-NX*(J-1)+I KK-(NX+1)*(J-1)+I IF (I,EQ.l) THEN

DUX(JJ)=2.*C2X*U(KK+1)

ELSE DUX(JJ)-C2X*(U(KK+1)-U(KK-1))

END IF 10 CONTINUE

CALCUUTE DUY

DO 15 J-1,NY DO 15 I-1,NX

JJ-NX*(J-1)+I KK-(HX+1)*(J-1)+I IF (J.EQ.l) THEN

DUY(JJ)-0.0

ELSE DUY(JJ)-C2Y*(U(KK+NX+1)-U(KK-HX-1))

END IF 15 CONTINUE

RETURN END

100

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035

C—

C C C

C C C

c

SUBROUTINE FIRSTDERV(V.DVX.DVY) C0MM0N/B2/NX,NY,NUM,NUMl C0MM0N/BINPUT/GI,EUS,PR,H1,H2,T,HX,HY,G C0MM0N/CC/C2X,C2Y,XY DIMENSION V(1),DVX(1),DVY(1)

CALCUUTE DVX

DO 20 J-1,NY DO 20 I-1,NX

JJ-HX*(J-1)+I KK-(IX+1)*(J-1)+I IF (I,EQ.l) THEN

DVX(JJ)-0.0 ELSE

DVX(JJ)-C2X*(V(KK+1)-V(KK-1)) END IF

20 CONTINUE

CALCUUTE DVY

DO 30 J-1,HY DO 30 I-1,NX

JJ-NX*(J-1)+I IK-(HX+1)*(J-1)+I IF (J.EQ.l) THEN

DVY(JJ)»2.*C2Y*V(KK+NX+1)

ELSE DVY(JJ)-C2Y*(V(KK+NX+1)-V(KK-NX-1))

END IF 30 CONTINUE

RETURN END

101

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041

C

c c

c c c

SUBROUTINE FIRSTDERW(W,WX,WY) C0MM0N/B2/NX.NY,NUM,NUMl C0MM0H/BINPUT/GI,EUS,PR,H1,H2,T,HX,HY,G C0MM0N/CC/C2X,C2Y,XY DIMENSION W(1),WX(1),WY(1)

CALCUUTE WX

DO 10 J-1,IY DO 10 I-1,NX

JJ-NX*(J-1)+I IF (I.EQ.l) THEN

WX(JJ)-0.0 ELSE

IF (I.EQ.NX) THEN WX(JJ)—C2X*W(JJ-1)

ELSE WX(JJ)-C2X*(W(JJ+1)-W(JJ-1))

END IF END IF

10 corriHUE

CALCUUTE WT

DO 20 j-i,rr

DO 20 I-l.NX JJ-«X*(J-1)+I IF (J.EQ.l) THEN

WY(JJ)-0.0 ELSE

IF (J.EQ.NY) THEN WY(JJ)—C2Y*W(JJ-NX)

ELSE WY(JJ)-C2Y*(W(JJ+NX)-W(JJ-HX))

END IF END IF

20 CONTINUE

RETURN END

102

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

C—

C C C

C C C

C C C

SUBROUTINE SECDERW(W,WXX,WYY.WXY) C0MM0N/B2/NX,NY,NUM,NUMl C0MM0N/BINPUT/GI.EUS.PR,H1.H2,T,HX,HY,G COMMON/BC/CX,CY,CX2.CY2,CXY,CXP,CYP,XOVY.PXOVY.YOVX,PYOVX DIMENSION W(1),WXX(1).WYY(1),WXY(1)

CALCUUTE WXX

DO 10 J-1,NY DO 10 I-l.NX

JJ»NX*(J-1)+I IF (I.EQ.l) THEN

WXX(JJ)»CX2*2.*(W(JJ+1)-W(JJ)) ELSE

IF (I.EQ.NX) THEN WXX(JJ)-CX2*(-2.*W(JJ)+W(JJ-1))

ELSE WXX(JJ)-CX2*(W(JJ+1)-2.*W(JJ)+W(JJ-1))

END IF END IF

10 CONTINUE

CALCUUTE WYY

DC 20 j-i.rr DO 20 I-l.NX

JJ-NX*(J-1)+I IF (J.EQ.l) THEN

WYY(JJ)-CY2*2.*(W(JJ+HX)-W(JJ))

ELSE IF (J.EQ.NY) THEN

WYY(JJ)-CY2*(-2.*W(JJ)+W(JJ-HX))

ELSE WYY(JJ)-CY2*(W(JJ+NX)-2.*W(JJ)+W(JJ-NX))

END IF END IF

20 CONTINUE

CALCUUTE WXY

IT -0.25/(HX*HT) DO 30 J-l.HY DO 30 I-l.HX

JJ -NX*(J-1)+I

IF (I.EQ.l .OR. J.EQ.l) THEN WXY(JJ)-0.0

ELSE IF (I.EQ.NX .AND. J.EQ.NY) THEN

WXY(JJ)-XY*W(JJ-NX-1) ELSE

IF (I.EQ.NX) THEN WXY(JJ)-XY*(W(JJ-HX-1)-W(JJ+NX-1))

0055

103

ELSE °°^^ IF (J.EQ.NY) THEN

WXY(JJ)=XY*(W(JJ-NX-1)-W(JJ-NX+1)) ELSE

0057 0058

nil WXY(JJ)-XY*( W(JJ+NX+1)+W(JJ-NX-1) . _ ° ^ -W(JJ+NX-1)-W(JJ-NX+1)) 0061 EUD jp 0062 END IF 0063 END IF 0064 END IF 0065 30 CONTINUE 0066 C 0067 RETURN 0068 END

104

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

C C C

SUBROUTINE RHSIDEB(W.RHS2)

CALCUUTES THE RIGHT HAND SIDE OF B

C0MM0N/B2/NX,NY.NUM,NUMl

COMMON/DERW/WXdOO) ,WY(100) .WXX(IOO) .WYY(IOO) ,WXY(100) C0MM0N/DERW2/W2X(121).W2Y(121).W2XX(121).W2YY(121),W2XY(121) COMMON/BK/Kl,K2,K3,K4 DIMENSION W(100),RHS2(576)

CALL FIRSTDERW(W,WX,WY) CALL SECDERW(W,WXX,WYY,WXY) CALL DERWTWO(W)

RHS2(l)-0, RHS2(2)-0, RHS2(3)-0. RHS2(4)-0,

CALL KSUB(HX+1,NY+1) RHS2(Kl)-0. RHS2(K2)-0. RHS2(K3)»0. RHS2(K4)-0.

CALL KSUB(NX+2,NY+1) RHS2(Kl)-0. RHS2(K2)-0. RHS2(K3)-0. RHS2(K4)-0.

CALL KSUB(HX+1,NY+2) RHS2(K1)«0. RHS2(K2)-0. RHS2(K3)-0. RHS2(K4)»0.

DO 10 1-2,NX DO 10 J-2,HY CALL R2C0RE(I,J,RHS2)

10 CONTINUE

DO 20 1-2,NX CALL R2B0T(I,RHS2)

20 CONTINUE

DO 30 J«2,HY

CALL R2LEFT(J,RHS2) 30 CONTINUE

DO 40 I-1,NX CALL R2T2(I,RHS2) CALL R2T1(I,RHS2)

105

0055 40 CONTINUE 0056 C

°°57 DO 50 j=i^^ 0058 CALL R2R2(J,RHS2) 0059 CALL R2R1(J.RHS2) 0060 50 CONTINUE 0061 C

0062 RETURN 0063 END

106

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

SUBROUTINE DERWTWO(W)

C0MM0N/B2/NX,NY.NUM,NUMl COMMON/DERW/WXdOO) .WYdOO) .WXXdOO) ,WYY(100) ,WXY(100) C0MM0N/DERW2/W2X(121) .W2Y(121) ,W2XXd21) .W2YY(121) .W2XY(121) C0MM0N/BINPUT/GI,EUS.PR.H1.H2,T,HX,HY,G C0MM0N/CC/C2X,C2Y,XY DIMENSION WdOO)

DO 10 J-1,NY+1 K-(J-1)*(NX+1) LL-(J-1)*NX DO 10 I-1,NX+1 KK-K+I

IF (I.LE.NX .AND, J,LE.NY) THEN JJ-LL+I W2X(KK)-WX(JJ) W2Y(KK)-WY(JJ) W2XX(KK)-WXX(JJ) W2YY(KK)-WYY(JJ) W2XY(KK)-WXY(JJ)

ELSE IF (I.EQ.NX+1,AND.J.NE.NY+1) THEN

W2X(KK)—2.* C2X*W(J*NX) W2Y(KK) -0. W2XX(KK)-0, W2YY(KK)-0.

IF (J.EQ.NY) THEN W2XY(KK)»2.*XY*W((J-1)*NX)

ELSE IF (J,EQ.l) THEN W2XY(KK)-0.

ELSE W2XY(KK)-2.*XY*(W((J-1)*NX)-W((J+1)*NX))

END IF END IF

ELSE IF (J.EQ.NY+1.AND.I.NE.NX+1) THEN

W2Y(KK) —2.*C2Y»W((NY-1)*NX+I)

W2X(IK) -0. W2XX(KI)-0. W2YY(KK)-0. IF (I.EQ.NX) THEN

W2XY(KK)-2.*XY*W((HY-1)*NX+I-1)

ELSE IF (I.EQ.l) THEN W2XY(KK)-0.

ELSE W2XY(KK)-2.*XY*(W((NY-1)*NX+I-1)-W((NY-1)*NX+I+1))

ENDIF ENDIF

END IF END IF

107

0055 EJTO jp

0056 10 CONTINUE 0057 C 0058 RETURN 0059 END

108

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

0011

0012

0013

0014

0015

0016

0017

0018

0019

0020

0021

C-

C SUBROUTINE R2C0RE(I,J,RHS2)

C0MM0N/B2/NX,NY,NUM,NUMl

C0MM0N/DERW2/W2X(121),W2Y(121).W2XX(121),W2YY(121),W2XY(121)

C0MM0N/BINPUT/GI.EUS.PR,H1,H2.T,HX,HY,G

COMMON/BK/Kl.K2,K3,K4

COMMDN/BD/CA.CB,CD,CE

DIMENSION RHS2(576)

CALL KSUB(I,J)

JJ-(J-1)*(NX+1)+I Al—(W2X(JJ)*W2XX(JJ)+CA*W2Y(JJ)*W2XY(JJ)+CB*W2X(JJ)*W2YY(JJ))

A2—(W2Y(JJ)*W2YY(JJ)+CA*W2X(JJ)*W2XY(JJ)+CB*W2Y(JJ)*W2XX(JJ))

RHS2(K1)=A1-CD*W2X(JJ)

RHS2(K2)=A2-CD*W2Y(JJ)

RHS2(K3)-A1+CE*W2X(JJ)

RHS2(K4)=A2+CE*W2Y(JJ)

RETURN

END

109

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017

C-

C SUBROUTINE R2B0T(I,RHS2)

C0MM0N/DERW2/W2X(121) ,W2Yd21) ,W2XXd21) ,W2YY(121) ,W2XY(121) COMMON/BK/Kl,K2,K3,K4 COMMON/BD/CA,CB,CD,CE DIMENSION RHS2(576)

CALL KSUB(I,1) Al—(W2X(I)*W2XX(I)+CA*W2Y(I)*W2XY(I)+CB*W2X(I)*W2YY(I))

RHS2(K1)=A1-CD*W2X(I) RHS2(K2)=0. RHS2(K3)=A1+CE*W2X(I) RHS2(K4)-0.

RETURN END

110

0001

0002

0003

0004

0005

0006

0007

0008

0009

0010

0011

0012

0013

0014

0015

0016

0017

0018

0019

C-

C

SUBROUTINE R2LEFT(J,RHS2)

C0MM0N/B2/NX,NY,NUM,NUM1

C0MM0N/DERW2/W2X(121) ,W2Yd21) ,W2XX(121) .W2YY(121) .W2XY(121)

COMMON/BK/Kl,K2.K3.K4

COMMON/BD/CA.CB.CD,CE

DIMENSION RHS2(576)

CALL KSUB(1,J)

JJ-(J-1)*(NX+1)+1 A2—(W2Y(JJ)*W2YY(JJ)+CA*W2X(JJ)*W2XY(JJ)+CB*W2Y(JJ)*W2XX(JJ))

RHS2(Kl)-0.

RHS2(K2)=A2-CD*W2Y(JJ)

RHS2(K3)-0.

RHS2(K4)-A2+CE*W2Y(JJ)

RETURN

END

I l l

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019

C-

C

SUBROUTINE R2T2(I,RHS2)

C0MM0N/B2/NX,NY,NUM,NUM1 C0MM0N/DERW2/W2X(121) ,W2Y(121) .W2XXd21) ,W2YY(121) ,W2XYd21) COMMON/BK/Kl,K2.K3.K4 COMMON/BD/CA,CB.CD,CE DIMENSION RHS2(576)

CALL KSUB(I,NY+1) JJ»NY*(NX+1)+I

RHS2(K1)—CA*W2Y(JJ)*W2XY(JJ) RHS2(K2)—CD*W2Y(JJ) RHS2(K3)= RHS2(K1) RHS2(K4)- CE*W2Y(JJ)

RETURN END

112

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018

C-

C SUBROUTINE R2T1(I,RHS2)

C0MM0N/B2/NX,NY.NUM.NUMl C0MM0N/DERW2/W2X(121) .W2Yd2l) .W2XXd21) ,W5YY(121) ,W2XY(121) COMMON/BK/Kl,K2,K3.K4 DIMENSION RHS2(576)

CALL KSUB(I,NY+2) JJ=NY*(NX+1)+I

RHS2(Kl)-0. RHS2(K2)—0.5*(W2Y(JJ))**2 RHS2(K3)-0. RHS2(K4)= RHS2(K2)

RETURN END

113

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019

C-

C SUBROUTINE R2R2(J,RHS2)

C0MM0N/B2/NX,NY,NUM,NUMl C0MM0N/DERW2/W2X(121) ,W2Y(121) .W2XX(121) ,W2YY(121) ,W2XYd21) COMMON/BK/Kl,K2,K3,K4 COMMON/BD/CA,CB,CD,CE DIMENSION RHS2(576)

CALL KSUB(NX+1,J) JJ-J*(NX+1)

RHS2(K1)—CD*W2X(JJ) RHS2(K2)—CA*W2X(JJ)*W2XY(JJ) RHS2(K3)- CE*W2X(JJ) RHS2(K4)= RHS2(K2)

RETURN END

114

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018

C-

C

SUBROUTINE R2R1(J,RHS2)

C0MM0N/B2/NX,NY,NUM,NUMl C0MM0N/DERW2/W2X(121) ,W2Yd21) ,W2XXd21) ,W2YY(12l) ,W2XYd21)

CJMMON/BK/Kl,K2,K3.K4 DIMENSION RHS2(576)

CALL KSUB(NX+2,J) JJ-J*(NX+1)

RHS2(K1)—0.5*(W2X(JJ))**2 RHS2(K2)-0. RHS2(K3)= RHS2(K1) RHS2(K4)-0.

RETURN END

115

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011

C C C

SUBROUTINE COPY(A,B,N)

TO SET VECTOR {A}= VECTOR{B}

DIMENSION A(N).B(N) DO 10 I-l.N A(I)-B(I)

10 CONTINUE RETURN END

116

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024

C C C C

SUBROUTINE CHECK(WNEW.WOLD.NUM.NCONV)

TO CHECK CONVERGENCE OF VECTORS {WNEW}.W{OLD} OF DIMENSION NUM. IF CONVERGENCE SATISFIED NCONV IS SET TO 1

COMMON/LOAD/Q.QINC.NINC,MAXIT,ERR,AL DIMENSION WNEWdOO),WOLD(100)

SUMl-0. WMAX-0. DO 10 1=1,NUM

SUMl-SUMl+ABS(WNEW(I)-WOLD(I)) IF (ABS(WNEW(I)).GT.WMAX) WMAX=ABS(WNEW(I))

10 CONTINUE

IF (SUMl.LT. ERR*WMAX*NUM) THEN

NCONV-1 ELSE

NCONV-O ENDIF

RETURN END

117

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015

C C C

SUBROUTINE RBINTERP(WOLD,WNEW,W,AL)

TO CALCUUTE THE INTERPOUTED VALUE OF {W} GIVEN WOLD AND WNEW

C0MM0N/B2/NX,NY,NUM,NUMl DIMENSION WNEW(1),W0LD(1),W(1)

BE-1.-AL DO 10 1=1.NUM W(I)=BE*WOLD(I)+AL*WNEW(I)

10 CONTINUE

RETURN END

118

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024

C C c

c c c c

SUBROUTINE RAINTERP(Ul.Vl.U2.V2.UIOLD.VIOLD,U20LD,V20LD, 1 U1NEW.V1NEW.U2NEW,V2NEW)

TO CALCUUTE THE INTERPOUTED VALUE OF U1,U2.V1.V2

C0MM0N/B2/NX.NY.NUM.NUMl DIMENSION Uld) .U2(l) .Vl(l) .V2(l) .UlOLD(l) ,V10LD(1) . 1 U20LDd) .V20LD(1) .UlNEWd) ,V1NEW(1) ,U2NEW(1) ,V2NEW(1)

THE VALUE OF BETA MAY BE CHANGED TO IMPROVE THE ITERATIVE PROCEDURE

BETA-1.0 BETA2=1. -BETA DO 10 I»1,NUM1 U1(I)=BETA2*U10LD(I)+BETA*U1NEW(I) V1(I)=BETA2*V10LD(I)+BETA*V1NEW(I) U2(I)=BETA2*U20LD(I)+BETA*U2NEW(I) V2(I)=BETA2*V20LD(I)+BETA*V2NEW(I)

10 CONTINUE

RETURN END

119

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038

C C C

C

C

SUBROUTINE PRINTRES(WNEW,U1,V1.U2.V2,INC,ITER)

PRINT RESULTS IN CASE DISPLACEMENTS CONVERGE

C0MM0N/B2/NX,NY.NUM,NUMl COMMON/LOAD/Q,QINC.NINC,MAXIT.ERR.AL COMMON/INTERP/ALPHAdS) ,W0VERT(15) .HAV

DIMENSION WNEW(l),Ul(l),U2(l),Vl(l),V2d)

WRITE(8,*) QINCWNEWd) WRITE(7,*) WNEW(1)/HAV,AL.ITER

WRITE(6.10) INCITER 10 FORMATC INCREMENT « ' . I 3 /

1 ' CONVERGENCE ACHIEVED AFTER',13.' ITERATION(S)'// 2 ' UTERAL DEFLECTION'/ 3 ' « 4 i « « » * * » 4 [ » W 4 t « 4 i » » * « ' / / )

DO 30 I-l.NUM WRITE(6,20) I.WNEW(I)

20 FORMAT(2X,13.2X. E12,5) 30 CONTINUE

WRITE(6.40) 40 FORMAT(/' 1 2 3

Ul-DISPUCEMENT Vl-DISPLACEMENT'. U2-DISPLACEMENT V2-DISPLACEMENT'/

mmmm0**0*»mm»»* *»•**••*••*•*•*»//)

DO 60 I-1,NUM1

WRITE(6.50) I.U1(I).V1(I).U2(I).V2(I) 50 F0RMAT(X,I3,X.2(E12,5,5X,E12,5,10X)) 60 CONTINUE

RETURN

END

120

0001 0002 0003 0004 0005 0006 0007 0008 0009

C'

c c c

SUBROUTINE PRINTDIV(INC)

PRINT MESSAGE IN CASE DISPUCEMENTS DIVERGE

WRITE(6,10) INC 10 FORMATC **DIVERGENCE OCCURED AT',I3.'TH ICREMENT**')

RETURN END

121

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026

C—

C C

c c

c

c

c

1

1 1

10

20

SUBROUTINE GUESS(UIG.VIG.U2G,V2G.WG.

U1PR.V1PR,U2PR.V2PR.WPR.

UIPP.VIPP.U2PP.V2PP.WPP)

TO CALCUUTE THE GUESS DISPLACEMENT VECTOR FOR THE NEXT

INCREMENT

C0MM0N/B2/ NX.NY.NUM.NUMl

DIMENSION U1PR(121) .V1PR(121) .U2PR(121) .V2PRd21) .WPR(IOO)

DIMENSION U1PP(121),V1PP(121),U2PP(121),V2PP(12l),WPP(100)

DIMENSION U1G(121),V1G(121),U2G(121),V2G(121).WG(IOO)

DO 10 1=1,NUMl

U1G(I)=2,*U1PR(I)-U1PP(I)

V1G(I)»2.*V1PR(I)-V1PP(I)

U2G(I)=2,*U2PR(I)-U2PP(I)

V2G(I)-2.*V2PR(I)-V2PP(I)

CONTINUE

DO 20 I-1,NUM

WG(I)-2,*WPR(I)-WPP(I)

CONTINUE

RETURN

END

122

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018

C-

C C

C

C

SUBROUTINE XALPHA(DEFL.AL) TO CALCUUTE THE RELAXATION PARAMETER

COMMON/INTERP/ALPHAdS) .W0VERT(15) ,HAV

DO 10 1=2,14 Al-DEFL/HAV IF(A,L£.W0VERT(I).AND.A1.GT.W0VERT(I-1)) THEN

A-A1-WOVERT(I-1) B-W0VERT(I)-W0VERT(I-1) AL-ALPHA(I-1)*(B-A)/B+ALPHA(I)*A/B

END IF 10 CONTINUE

WRITE(•,*)DEFL

RETURN END

123

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 00.29 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054

C C C C

SUBROUTINE STRAINM(UI.VI.U2.V2.W,E1X,E1Y,E1XY, 1 E2X,E2Y,E2XY)

TO CALCUUTE PLATES MEMBRANE STRAINS FOR A (NX+1)*(NY+1) MESH GIVEN THE DISPLACEMENT VALUES

C0MM0N/B2/ NX,NY,NUM,NUMl

COMMON/DER/DUlXdOO) ,DU1Y(100) ,DV1X(100) ,DV1Y(100) , 1 DU2Xd00) ,DU2Y(100) ,DV2X(100) ,DV2Y(100) COMMON/DERW/WXdOO) ,WY(100) ,WXX(100) ,WYY(100) ,WXY(100) COMMON/BINPUT/ GI,ELAS,PR,H1,H2,T,HX,HY,G

DIMENSION UI (121) , VI (121) ,U2d21) .V2(12l) .W(IOO) DIMENSION E1X(121),E1Y(121),E1XY(121) DIMENSION E2X(121),E2Y(121).E2XY(121)

CALL FIRSTDERU(U1,DU1X,DU1Y) CALL FIRSTDERV(Vl.DVIX.DVIY) CALL FIRSTDERU(U2.DU2X.DU2Y) CALL FIRSTDERV(V2.DV2X.DV2Y) CALL FIRSTDERW(W.WX,WY)

DO 10 J-l.NY+1 DO 10 I-l.NX+1

K-(J-1)*(NX+1)+I IF (J.NE.NY+1,AND,I.NE.NX+1) THEN

L-(J-1)*NX+I E1X(K)-DU1X(L)+0.5*WX(L)*WX(L) E2X(K)-DU2X(L)+0.5*WX(L)*WX(L) E1Y(K)-DV1Y(L)+0.5*WY(L)*WY(L) E2Y(K)-DV2Y(L)+0,5*WY(L)*WY(L) E1XY(K)-DU1Y(L)+DV1X(L)+WX(L)*WY(L) E2XY(K)-DU2Y(L)+DV2X(L)+WX(L)*WY(L)

ELSE

IF(I,EQ,NX+1,AND.J.NE,NY+1) THEN IF (J.EQ.l) THEN

DV0NE-Vl(2*NX+2)/HY

DVTW0-V2(2*NX+2)/HY

ELSE DVONE-(VI(K+NX+1)-VI(K-NX-l))/(2.*HY) DVTW0-(V2(K+NX+1)-V2(K-NX-1))/(2.*HY)

ENDIF EIX(K)—PR*DVONE EIY(K)-DVONE ElXY(K)-0,0 E2X(K)—PR*DVTWO

E2Y(K)-DVTW0 E2XY(K)-0.0

ENDIF IF(J,EQ,NY+1.AND,I.NE.NX+1) THEN

IFd.EQ.l) THEN DU0NE-U1(K+1)/HX DUTW0-U2(K+1)/HX

124

0055 ELSE

0056 DU0NE=(U1(K+1)-U1(K-1))/(2.*HX) °05^ DUTW0=(U2(K+1)-U2(K-1))/(2.*HX)

0058 ENDIF 0059 E1X(K)=DU0NE 0060 EIY(K)—PR*DUONE 0061 ElXY(K)-0.0 0062 E2X(K)=DUTW0 0063 E2Y(K)=-PR*DUTW0 0064 E2XY(K)-0.0 0065 ENDIF 0066 ENDIF 0067 10 CONTINUE 0068 C 0069 RETURN 0070 END

125

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038

C C C

C

C

c

c

c

SUBROUTINE BENDSTRESS(W,B1SIGX,B1SIGY,B1SIGXY, 1 B2SIGX,B2SIGY,B2SIGXY)

TO CALCUUTE THE BENDING STRESSES FOR THE TWO PLATES

C0MM0N/B2/NX,NY,NUM,NUMl COMMON/DERW/WXdOO) ,WY(100) ,WXX(100) ,WYY(100) ,WXYd00) C0MM0N/DERW2/W2X(121) ,W2Yd21) ,W2XX(121) ,W2YY(121) ,W2XY(121) C0MM0N/BINPUT/GI,EUS,PR,H1,H2,T,HX,HY,G C0MM0N/CC/C2X,C2Y,XY

DIMENSION W(IOO) DIMENSION B1SIGX(121) ,BlSIGYd21) ,BlSIGXYd21) DIMENSION B2SIGX(121) ,B2SIGYd21) ,B2SIGXY(121)

CALL SECDERW(W,WXX,WYY,WXY) CALL DERWTWO(W)

D1-ELAS*H1/(2.*(1.-PR*PR)) D2-D1*H2/H1

DO 10 J-1,NY+1 DO 10 I-1,NX+1 K-(J-1)*(NX+1)+I AA-W2XX(K)+PR*W2YY(K) BB-W2YY(K)+PR*W2XX(K) CC-d.-PR)*W2XY(K) BISIGX(K)—D1*AA BISIGY(K)—D1*BB BISIGXY(K)—D1*CC B2SIGX(K)—D2*AA B2SIGY(K)=-D2*BB B2SIGXY(K)—D2*CC

10 CONTINUE

RETURN

END

126

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052

C C C

SUBROUTINE PRINCIP(XM,YM,XYM,XB.YB,XYB,A,SIGMAX.SIGMIN, 1 STRESSMAX,XMAX,YMAX)

TO CALCUUTE PRINCIPAL STRESSES

C0MM0N/B2/NX,NY,NUM,NUMl C0MM0N/BINPUT/GI,EUS,PR,H1,H2.T,HX.HY.G DIMENSION XMd21),YM(121),XYM(121) DIMENSION XB(l21).YB(121).XYBd2l) DIMENSION SIGMAX(121).SIGMIN(121)

DO 10 J-1,NY+1 DO 10 I-1,NX+1

K=(J-1)*(NX+1)+I X=XM(K)+A*XB(K) Y«YM(K)+A*YB(K) XY-XYM(K)+A*XYB(K) XMEAN-0.5*(X+Y) RAD-SQRT(0.25*(X-Y)**2+XY**2) ' SIGMAX(K)-XMEAN+RAD SIGMIN(K)=XMEAN-RAD

10 CONTINUE

STRESSMAX-0,0 IF (A,EQ.l.) THEN

DO 20 J-1,NY+1 DO 20 I-1,NX+1

K-(J-1)*(NX+1)+I IF (SIGMAX(K).GT. STRESSMAX) THEN

STRESSMAX-SIGMAX(K) II-I JJ-J

END IF 20 CONTINUE

ELSE DO 30 J-1,NY+1 DO 30 I»1,NX+1

K=(J-1)*(NX+1)+I IF (SIGMIN(K),LT. STRESSMAX) THEN

STRESSMAX-SIGMIH(K)

II-I JJ-J

END IF 30 CONTINUE

END IF

XMAX-REAL(II-1)*HI YMAX-REAL(JJ-1)*HY

RETURN END

127

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021

C c c

SUBROUTINE STRESSM(EX,EY,EXY.SIGXM.SIGYM.SIGXYM) C0MM0N/B2/NX,NY.NUM.NUMl C0MM0N/BINPUT/GI,EUS.PR.H1,H2,T,HX.HY.G

" CALCUUTE MEMBRANE STRESSES vilVEN STRAINS

DIMENSION EXd21) .EYd21) ,EXY(121) , 1 SIGXMd21) ,SIGYM(121) ,SIGXYMd21)

CONST-ELAS/(1.-PR*PR) DO 10 J-1,NY+1 DO 10 I-1,NX+1

K*(NX+1)*(J-1)+I SIGXM(K)-CONST*(EX(K)+PR*EY(K)) SIGYM(K)=CONST*(EY(K)+PR*EX(K)) SIGXYM(K)-G*EXY(K)

10 CONTINUE

RETURN END

128

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050

C C C

C

C

c c c

c c c

c c c

c c c

c

SUBROUTINE STRESS(UI,VI,U2,V2,W)

CALCUUTE STRESSES

COMMON/MSTRNl/EIX(121),E1Y(121),E1XY(121) C0MM0N/MSTRN2/E2X(121),E2Y(121),E2XY(121) C0MM0N/MSTRESl/SIGXMl(12l),SIGYMl(121),SIGXYMld21) C0MM0N/MSTRES2/SIGXM2(121).SIGYM2(121),SIGXYM2(12l) C0MM0N/B1STRS/B1SIGX(121),B1SIGY(121),B1SIGXY(121) C0MM0N/B2STRS/B2SIGX(121),B2SIGY(121),B2SIGXY(121) C0MM0N/MAX/SIGMAX(121),SIGMIN(121)

DIMENSION Ul(121),Vl(l21),U2d21).V2(121),W(100)

CALL STRAINM(U1.V1.U2.V2,W,E1X,E1Y,E1XY,E2X.E2Y,E2XY) CALL STRESSM(E1X,E1Y,E1XY.SIGXM1,SIGYM1,SIGXYM1) CALL STRESSM(E2X,E2Y,E2XY,SIGXM2,SIGYM2,SIGXYM2) CALL BENDSTRESS(W.B1SIGX.B1SIGY.B1SIGXY,B2SIGX,B2SIGY.B2SIGXY)

TOP PUTE-BOTTOM FACE

CALL PRINCIP(SIGXM1,SIGYM1.SIGXYM1,B1SIGX.B1SIGY,B1SIGXY.1.. 1 SIGMAX.SIGMIN,STRESSMAX.XMAX.YMAX) CALL PRINTSTRESS(SIGMAX.SIGMIN.STRESSMAX.XMAX.YMAX.l.. 1 10.14.18)

TOP PUTE-TOP FACE

CALL PRINCIP(SIGXM1,SIGYM1,SIGXYM1.B1SIGX,B1SIGY,B1SIGXY,-1., 1 SIGMAX,SIGMIN,STRESSMAX,XMAX.YMAX) CALL PRINTSTRESS(SIGMAX,SIGMIN,STRESSMAX.XMAX.YMAX,-1,. 1 9,13.17)

BOTTOM PUTE-BOTTOM FACE

CALL PRINCIP(SIGXM2,SIGYM2,SIGXYM2,B2SIGX,B2SIGY.B2SICXY.l., 1 SIGMAX,SIGMIN,STRESSMAX,XMAX,YMAX) CALL PRINTSTRESS (SIGMAX, SIGMIN, STRESSMAX. XMAX.YMAX.l. . 1 12,16.20)

BOTTOM PUTE-TOP FACE

CALL PRINCIP(SIGXM2,SIGYM2,SIGXYM2,B2SIGX,B2SIGY,B2SIGXY,-1., 1 SIGMAX,SIGMIN,STRESSMAX,XMAX.YMAX)

CALL PRINTSTRESS (SIGMAX. SIGMIN. STRESSMAX, XKAX,YMAX,-1..

1 11.15.19)

RETURN END

129

0001

0002

0003

0004

0005

00*. 6

0007

0008

0009

0010

0011

0012

0013

0014

0015

0016

0017

0018

0019

0020

0021

0022

0023

0024

0025

0026

0027

0028

0029

0030

0031

0032

0033

0034

0035

0036

C

C

SUBROUTINE OPENFILE

OPEN FILES FOR INPUT AND OUTPUT

OPEN(UNIT-

OPEN (UNIT»

OPEN(UNIT"

OPEN(UNIT"

OPEN(UNIT"

OPEN(UNIT*

OPEN(UNIT=

OPEN(UNIT'

OPEN (UNIT-

OPEN (UNIT=

OPEN (UNIT'

OPEN (UNIT'

OPEN (UNIT'

OPEN(UNIT=

OPEN(UNIT=

OPEN(UNIT^

4.FILE-'

6. FILE-'

:7.FILE='

=8.FILE='

'9.FILE='

=10.FILE"

=11,FILE=

'12.FILE"

'13.FILE"

'14.FILE"

'15,FILE"

=16,FILE=

"17,FILE"

=18,FILE=

"19,FILE'

=20,FILE'

ALPHA.',STATUS='OLD')

RESULT. OUT', STATUS- 'UNKNOWN')

WALPHA.DAT',STATUS-'UNKNOWN')

WQINC.DAT',STATUS-'UNKNOWN')

SIGl.DAT',STATUS-'UNKNOWN')

^'SIG2.DAT'.STATUS-'UNKNOWN')

••' SIG3 . DAT'. STATUS- 'UNKNOWN') ='SIG4.DAT'.STATUS-'UNKNOWN')

='MAXl,DAT'.STATUS-'UNKNOWN')

='MAX2,DAT'.STATUS-'UNKNOWN')

' 'MAX3,DAT'.STATUS-'UNKNOWN')

='MAX4,DAT•,STATUS-'UNKNOWN')

' 'MAXl,OUT'.STATUS-'UNKNOWN')

='MAX2.OUT'.STATUS-'UNKNOWN')

''MAX3.OUT'.STATUS-'UNKNOWN')

='MAX4.OUT'.STATUS-'UNKNOWN')

WRITE(8.*) 0..0.

WRITE(9.*) 0..0.

WRITEdO.*)0..0.

WRITE(11,*)0..0.

WRITE(12.*)0..0.

WRITE(13.*)0..0.

WRITE(14.*)0..0.

WRITEd5.*)0.,0.

WRITEd6.*)0.,0.

WRITE(17,*)0.,0,

WRITE(18,*)0.,0.

WRITE(19,*)0.,0.

WRITE(20,*)0,.0.

RETURN

END

130

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039

SUBROUTINE COMMENT WRITE(7,*)'WMAX/T VERSUS ALPHA' WRITE(7.*)'WMAX/T ALPHA t OF ITERATIONS' WRITE(8.*)'WMAX VERSUS PRESSURE' WRITE(8.*)'PRESSURE MAXIMUM DEFLECTION'

WRITE(9,*)'TOP PUTE TOP FACE' WRITE(10.*)'TOP PLATE BOTTOM FACE' WRITEdl.*)'BOTTOM PLATE TOP FACE' WRITE(12.*)'BOTTOM PLATE BOTTOM FACE'

WRITE(9.*) 'PRESSURE VALUE- PRINCIPAL STRESS(CENTER)' WRITE(10.*)'PRESSURE VALUE- PRINCIPAL STRESS(CENTER)' WRITEdl.*)'PRESSURE VALUE- PRINCIPAL STRESS (CENTER) ' WRITE(12.*)'PRESSURE VALUE- PRINCIPAL STRESS(CENTER)'

WRITE(13.*)'T0P PLATE TOP FACE' WRITE(14,*)'T0P PLATE BOTTOM FACE' WRITE(15.*)'BOTTOM PLATE TOP FACE' WRITE(16.*)'BOTTOM PLATE BOTTOM FACE'

WRITEd3,*)'PRESSURE VALUE- MAXIMUM PRINCIPAL STRESS' WRITE(14.*)'PRESSURE VALUE- MAXIMUM PRINCIPAL STRESS' WRITE(15.*)'PRESSURE VALUE- MAXIMUM PRINCIPAL STRESS' WRITE(16.*)'PRESSURE VALUE- MAXIMUM PRINCIPAL STRESS'

WRITE(17,*)'TOP PLATE TOP FACE' WRITEd8,*)'T0P PLATE BOTTOM FACE' WRITE(19,*)'BOTTOM PLATE TOP FACE' WRITE(20,*)'BOTTOM PLATE BOTTOM FACE'

WRITE(17,*)'PRESSURE -MAXIMUM PRINCIPAL STRESS-LOCATION(X.Y)' WRITE(18',*)'PRESSURE -MAXIMUM PRINCIPAL STRESS-LOCATION(X,Y) ' WRITEd9!*)'PRESSURE -MAXIMUM PRINCIPAL STRESS-LOCATION(X.Y) ' WRITE(20*.*)'PRESSURE -MAXIMUM PRINCIPAL STRESS-LOCATION(X.Y) '

RETURN END

131

0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016

SUBROUTINE PRINTSTRESS (SIGMAX, SIGMIN. STRESSMAX, XMAX.YMAX.AA, 1 N1,N2,N3) COMMON/LOAD/Q,QINC,NINC,MAXIT,ERR,AL DIMENSION SIGMAX(121),SIGMIN(121)

IF (AA.EQ.-l.) THEN WRITE(N1,*) QINC.SIGMAXd)

ELSE WRITE(N1,*) QINCSIGMINd)

END IF WRITE(N2,*) QINC.STRESSMAX WRITE(N3,*) QINC,STRESSMAX,XMAX.YMAX

RETURN END

PERMISSION TO COPY

In presenting this thesis in partial fulfillment of the

requirements for a master's degree at Texas Tech University, I agree

that the Library and my major department shall make it freely avail­

able for research purposes. Permission to copy this thesis for

scholarly purposes may be granted by the Director of the Library or

my major professor. It is understood that any copying or publication

of this thesis for financial gain shall not be allowed without my

further written permission and that any user may be liable for copy­

right infringement.

Disagree (Permission not granted) Agree (Permission granted)

Jl/i^^^ Student's signature StudenC's signature

o^loi/'^o Date Date