a mathematical model for the nonlinear analysis of
TRANSCRIPT
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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0012
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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
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0003
0004
0005
0006
0007
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0010
0011
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0013
0014
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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
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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
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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
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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
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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
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right infringement.
Disagree (Permission not granted) Agree (Permission granted)
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