,nonlinear analysis of bonded joints...chapter l. introduction ... 2. 3. literature survey basic...
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,NONLINEAR ANALYSIS OF BONDED JOINTS
WITH THERMAL EFFECTS/
by
Edward A. ,,Humphreys1,
Thesis submitted to the Graduate Faculty of the
Virginia Polytechnic !nstitute and State University in
partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
APPROVED:
H. F. Brinson
in
Engineering Mechanics
Carl T. Herakovich, Chairman
~ M. P. Kamat
May, 1977
Blacksburg, Virginia
, .
ACKNOWLEDGEM::NT
The author wishes to thank
- NASA's Langley Research Cent ~r and
for supporting this work und·~r NASA Grant NGR 47-004-090
and
for their assistance, advice, and guidance
for her excellent typing of this
manuscript and perserverance
- his fiancee, , for her understanding and support
and Austin
, for their invaluable consultation and
moral support.
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENT .
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF TABLES
CHAPTER
l. INTRODUCTION ...
2.
3.
LITERATURE SURVEY
BASIC CONSIDERATIONS
3.1 Geometric Restrictions
3.2 Quasi 3-Dimensional Formulation
3.3 Bonded Joints ......... .
3.3.l 2-Dimensional Joint Formulation
3.3.2 Quasi 3-Dimensional Joint
Formulation
3.4 Material Properties .
3.4.1 Prediction of Laminate Stress-Strain
Response . . . . . . . . . . .
3.4.2 Prediction of Laminate Thermal
Properties . . . . . .
3.5 Finite Element Formulation
3.6 Boundary Conditions for Joints
3.7 Qualifying Notes
iii
ii
iii
vii
xv
l
3
7
7
7
10
10
11
12
12
14
22
28
30
4.
5.
iv
NONLINEAR ANALYSIS ..... .
4.1 Modifications of NONCOMl
p~
31
31
4.1.l Increased Finite Element Capacity 31
4.1.2 Solution of Simultaneous Equations . 32
4.1.3 Three Dimensional Properties
4.1.4 Hygroscopic Analysis ....
4.1.5 Elevated Temperature and Moisture
Content Analysis
4.2 Nonlinear Analysis
4.2.1 Incremental Loading P~ocedure
4.2.2 Nonlinear Data Input
RESULTS AND DISCUSSIONS .
5.1 Materials Properties·
5.2 Averaging of Finite Element Fesults .
5~3 Finite Element RepresentatiOi.s
5.4 Stress Free Temperature
5.5 Linear Elastic Results ,·
5.5.l Single Lap Joints with Adhesive
5.5.1.l [OJ Graphite-Polyimide
Adherends .
5.5.1.2 [0/±45/90]s Graphite-
Polyimide Adherends .
5.5.1.3 [90] Graphite-Polyimide
Adherends . . . . . . .
35
35
36
36
37
37
41
41
41
44
44
47
47
47
55
60
v
5.5.1.4 [OJ and [90J Graphite-
Polyimide Adherends . .
5.5.2 Double Lap Joints with Adhesives .
5.5.2.1 [OJ Graphite-Polyimide
60
66
Adherends . . . . . • . . . . . . 66
5.5.2.2 [90J Graphite-Polyimide
Adherends . . . . . . . . . . . . 68
5.5.2.3 [OJ and [90J Graphite-
Polyimide Adherends . . . . . . 68
5.5.3 Single Lap Joints without Adhesive 71
5.5.3.l [OJ Graphite-Polyimide
Adherends . . . . . . . . . . . . 73
5.5.3.2 [90J Graphite-Polyimide
Adherends . . . . . . . . . . . . 73
5.5.3.3 [OJ and [90J Grcphite-
Polyimide Adherends . . 75
5.5.4 Double Lap Joints without Adherends 79
5.5.5 Elastic Loading Comparisons . . . . . 84
5.6 NONLINEAR RESULTS . . . . 86
5.6.l Single Lap Joints 86
5.6.1.l Lap Shear Test 86
5.6.l.2 [0/90/0/90/0J Boron-Epoxy Adher-
ends and AF-126-2 Adhesive . . . 97
6.
vi
5.6.1.3 [OJ Graphite-Polyimide Adher-
ends and Metlbond 1113 Adhesive .. 103
5.6.1.4 [±45Js Graphite Polyimide Adher-
ends and Metlbond 1113 Adhesive. 110
5.6.2 Double Lap Joints . . . . . . . . . . . • 116
5.6.2~1 Titanium Adherends and Metlbond
1113 Adhesive .......... 116
5.6.2.2 [OJ and [90J Graphite-Polyimide
Adherends and Metlbond 1113
Adhesives
SUMMARY AND CONCLUSIONS .
BIBLIOGRAPHY
APPENDICIES
. 116
129
. 132
A
B
c D
VITA
ELEMENT STIFFNESS MATRIX
ADDENDA TO REFERENCE [lJ
MATERIAL PROPERTIES . . . . .
COMPUTER PROGRAM NONCOMl
. . . . . . . . . . . . . . .
. 135
140
148
166 247
Figure No.
1
2
3
4
5
6
7
8
9
10
11
12
13
LIST OF FIGURES
Lap Joint Geometries .
Laminate Geometry .
Joint Boundary Conditions
Block Storage of Stiffness Matrix and
Load Vector . . . . . . .
Determi nati 011 of Current Modulus with
Ramberg-Osgocd Approximations
Averaging of Finite Element Results
Partial Finite Element Grid for Joint
with Adhesive Layer .
Partial Finite Element Grid for Joint
without Adhesive Layer
Elastic Mechanical and Curing Adhesive
Stresses of a Single Lap Joint with [OJ Gr/Pi
Adherends and Metlbond 1113 Ad; esive ..
Free Body Diagrams of Single Loip Joints
Under Two Sets of Boundary Conlitions
Deflected Shape of the Lower Ajherend of a
Single Lap Joint Under ~echanical Loading
Elastic Mechanical and Curing Adhesive Stresses
of a Single Lap Joint with [0/±45/90]s Gr/Pi
Adherends and Metlbond 1113 Adhesive .
Elastic Mechanical and Curing Adhesive Stresses
vii
Page
8
13
23
34
39
43
45
46
48
50
54
56
Figure No.
14
15
16
17
18
19
20
viii
of a Single Lap Joint with [0/±45/90Js Gr/Pi
Adherends and Metlbond 1113 Adhesive, £x=O . 58
Elastic Mechanical and Curing Adhesive Stresses
of a Single Lap Joint with [90J Gr/Pi Adherends
and Metlbond 1113 Adhesive
Elastic Mechanical Adhesive Stresses of a
Single Lap Joint with [OJ and [90J Gr/Pi
Adherends and Metlbond 1113 Adhesive ..
Relative Displacements of a Single Lap Joint
with [OJ and [90J Gr/Pi Adhere.ids and Metlbond
1113 Adhesive Normalized with Respect to the
Adhesive Thickness ....•..
Elastic Curing Adhesive Stresses of a Single Lap
Joint with [OJ and [90J Gr/Pi Adherends and
61
62
63
Metlbond 1113 Adhesive . . . . . . . . . . . 65
Elastic Mechanical and Curing Adhesive Stresses
of a Double Lap Joint with [Oj Gr/Pi Adherends
and Metlbond 1113 Adhesive
Elastic Mechanical and Curing Adhesive Stresses
of a Double Lap Joint with [90J Gr/Pi Adherends
and Metlbond 1113 Adhesive
Elastic Mechanical Adhesive Stresses of a Double
67
69
Lap Joint with [OJ and [90J G·/Pi Adherends and
Metlbond 1113 Adhesive . . • . . . . . . . . . . 70
Figure No.
21
22
23
24
25
26
27
28
ix
Elastic Curing Adhesive Stresses of a Double
Lap Joint with [OJ and [90J Gr/Pi Adherends
and Metlbond 1113 Adhesive ..
Elastic Mechanical Interfacial Stresses of
a Single Lap Joint with [OJ Gr/Pi Adherends
and no Adhesive ............ .
Elastic Mechanical Interfacial Stresses of
a Single Lap Joint with [90J Gr/Pi Adherends
and no Adhesive
Elastic Mechanical Interficial Stresses of a
Single Lap Joint with [OJ and [90J Gr/Pi
Adherends and no Adhesive
Elastic Curing Interfaciai Stresses of a Single
Lap Joint with [OJ and [90] Gr/Pi Adherends
and no Adhesive .
Elastic Mechanical Interfacial Stresses of a
Double Lap Joint with [OJ Gr/Pi Adherends and
no Adhesive ...
Elastic Mechanical Interfacial Stresses of a
Double Lap Joint with [90] Gr/Pi Adherends and
no Adhesive . . . . . . . . .....
Elastic Mechanical Interf< cial Stresses of a
Double Lap Joint with [OJ and [90] Gr/Pi
Adherend and no Adhesive
72
74
76
77
78
80
81
82
Figure No.
29
30
31
32
33
34
35
36
37
x
Elastic Curing Interfacial Stresses of a Double
Lap Joint with [O] and [90] Gr/Pi Adherends and
no Adhesive . . . . . . . .
Nonlinear Mechanical Adhesive Tyz Stresses of a
Lap Shear Joint with Aluminum Adherends and
Metlbond 1113 Adhesive
Nonlinear Mechanical Adhesive Yyz Strains of a
Lap Shear Joint with Aluminum Adherends and
. 83
. 88
Metlbond 1113 Adhesive . . 89
Comparison of Shear Stress-Strain Response of
Metlbond 1113 Adhesive
Nonlinear Mechanical Adhesive crz Stresses of a
Lap Shear Joint with.Aluminum Adherends and
Metlbond 1113 Adhesive ...
Nonlinear Mechanical Adhesive cry Stre:;ses of a
Lap Shear Joint with Aluminum Adherends and
Metlbond 1113 Adhesive
Nonlinear Mechanical Adhesive ax Stresses of a
Lap Shear Joint with Aluminum Adherends and
Metl bond 1113 Adhesive
. . 92
94
. 95
• 96
Nonlinear Mechanical Adhesive Tyz Stresses of a
Single Lap Joint with [0/90/0/90/0] B/E
Adherends and AF-126-2 Adhesive . . . . . . 99
Nonlinear Mechanical Adhesive a Stresses z . of a
Figure No.
38
39
40
41
42
xi
Single Lap Joint with [0/90/0/90/0] B/E
Adherends and AF-126-2 Adhesive .... . 100
Nonlinear Curing Adhesive Stresses of a Single
Lap Joint with [0/90/0/90] B/E Adherends and
AF-126-2 Adhesive . . . . . . . . . . . . . . · 102
Nonlinear Mechanical and CJring Adhesive oy
Stresses of a Single Lap J)int with [0/90/0/90/0]
B/E Adherends and AF-126-2 Adhesive
Nonlinear Mechanical and Curing Adhesive ox
Stresses of a Single L2p Jiont with [0/90/0/90/0]
B/E Adherends and AF-126-2 Adhesive
104
105
Nonlinear Mechanical and Curing Adhesive Tyz
Stresses of a Single ·Lap Joint with [OJ Gr/Pi
Adherends and Metlbond 1113 Adl1esive . . . 107
Nonlinear Mech,rnical and Curinq Adhesive o . . z Stresses of a Single Lap Joint with [OJ Gr/Pi
Adherends and Metlbond 1113 Adhesive . .. 108
43 · Nonlinear Mechanical and Curing Adhesive oy
Stresses of a Single Lap Joint with [OJ Gr/Pi
Adherends and Metlbond 1113 Adhesive . . . 109
44
45
Nonlinear Mechanical and Curing Adhesive Tyz
Stresses of a Single L.1p Joint with [±45Js Gr/Pi
Adherends and Metlbond 1113 Adhesive . . . 112
Nonlinear Mechanical and Curing Adhesive o2
xii
Figure No. Page
46
47
48
49
50
51
52
53
Stresses of a Single Lap Joint with [±45]s Gr/Pi
Adherends and Metlbond 1113 Adhesive . . . . ·113
Nonlinear Mechanical and Curing Adhesive ay
Stresses of a Single Lap Joint with [±45] Gr/Pi . s Adherends and Metlbond 1113 Adhesive ..
Nonlinear Mechanical anrl Curing Adhesive ax
Stresses of a Single Lap Join1 with [±45]s Gr/Pi
Adherends and Metlbond 1113 Achesive ..
Nonlinear Mechanical and Curi1g Adhesive T - yz Stresses of a Double Lap Join1 with Titanium
· 114
.115
Adherends and Metlbond 1113 Achesive .. . .. 118
Nonlinear Mechanical and Curirg Adhesive az
Stresses of a Double lap Joini with Titanium
Adherends and Metlbond 1113 Achesive ..
Nonlinear Mechanical and Curir J Adhesive ay
Stresses of a Double Lap Joini with TitaniLm
. . .119
Adherends and Metlbond 1113 Aci,1esive ....... 120
Nonlinear Mechanical and Curing Adhesive ax
Stresses of a Double Lap Joint with Titanium
Adherends and Metlbondll13 Ad:1esive ....... 121
Nonlinear Curing Adhesive Stresses of a Double
Lap Joint with [OJ and [90] Gr/Pi Adherends
and Metlbond 1113 Adhesive ...... . . . .122
Nonlinear Mechanical and Curirig Adhesive ':1z
Figure No.
54
55
56
xiii
Stresses of a Double Lap Joint with [OJ
and [90J Gr/Pi Adherends and Metlbond
1113 Adhesive ........... .
Nonlinear Mechanical and Curing Adhesive a . z
. . . 125
Stresses of a Double Lap Joint with [OJ and [90J
Gr/Pi Adherends and Metlbond 1113 Adhesive 126
Nonlinear Mechanical and Curing Adhesive cry
Stresses of a Double Lap Joint with [OJ and [90J
Gr/Pi Adherends and Metlbond 1113 Adhesive 127
Nonlinear Mechanical and Curing Adhesive ax
Stresses of a Double Lap Joint with [OJ and [90J
Gr/Pi Adherends and Metlbond 1113 Adhesive 128
B.l Compression Stress-Strain Behavior of [±20Js
Boron/Epoxy Lami nate-Sandvli ch Beam Data . . . . . 141
B.2 Tensile Stress-Strain Beh<lvior of [±30] B/E s Laminate-Sandwich Beam Data ...
B.3 Compressive Stress-Strain Behavior of [±30Js
B/E Laminate-Sandwich Beam Data
B.4 Tensile Stress-Strain Behavior of [±45Js B/E
Laminate . . . . .
B.5 Tensile Stress-Strain Beh~vior of [±60Js B/E
Laminate Sandwich Beam Data .
B.6 Compressive Stress-Strain Behavior of [±60Js
B/E Laminate-Sandwich Beam Data ...
. . 142
.. 143
. . 144
. . 145
. . 146
Figure No.
B.7
xiv
Tensile Stress-Strain Behavior of [0/±45] 5
Bs/Al Laminate . . . . . . . . . . . . . . 147
C. l Stress-Strain Response of Unidirectional Gr/Pi .. 149
C.2 Stress-Strain Response of [±45]s Laminate Gr/Pi .. 150
C.3 Stress-Strain Response of T0/±45/90]s Laminate
Gr/Pi . . . . . . . . . . . . . . . . . . . . 151
C.4 Stress-Strain Response of Unidirectional B/E
Ref. [15] . . . . . . . . . . . . . . . . . . 152
C.5 Stress-Strain Response of [0/90/0/90/0]
Laminate B/E .......... . . . . . . 153
C.6 Stress-Strain Response of Metlbond 1113
Adhesive ............ . . 154
C.7 Stress-Strain Respons~ of AF-126-2 Adhesive . . 155
Table No.
l
2
3
4
5
6
7
8
LIST OF TABLES
Title
Comparison of Various Boundary Conditions
for Single Lap Joints
Results of Equilibrium Checks of Single
Lap Joint with [OJ Gr/Pi Adherends and
Metlbond 1113 Adhesive ....... .
Comparison of Peak Adhesive Stresses of
[0/±45/90Js Joint Under Mechanical Loadings
for 2-Dimensional and Quasi 3-Dimensional
29
52
Analysis . . . . 59
Force-Displacement Result Comparisons for
Elastic Joint Solutions ..... . . . . . 85
Dimensions for Lap Shear Joint and
Single Lap Joint with [0/90/0/90/0J B/E
Adherends . . . . . .
Comparison of Numerical and E;:perimental RE~-
sults for Lap Shear Joint .
Comparison of Elastic and Nonlinear Adhesive
Stresses for a Single Lap Joint with [OJ Gr/Pi
Adherends and Metl bond 1113 Acihes i ve
Comparison of Elastic and Nonlinear Adhesive
Stresses for a Single Lap Joint with [±45Js
. . 87
. . 91
. . l 00
Gr/Pi Adherends and Metlbond 1113 Adhesive~~ ... 111
xv
Table No.
9
10
C-1
C-2
xvi
·Title
Comparison of Elastic and Nonlinear Adhesive
Stresses for a Double Lap Join·: with Ti
Adherends and Metlbond 1113 Adhesive .. • • · 117
Comparison of Elastic and Nonlinear Adhesive
Curing Stresses for a Double Lap Joint with
[OJ and [90] Gr/Pi Adherends and Metlbond 1113
Adhesive .....
Ramberg-Osgood Coefficients
Thermal Properties .....
. 124
.156
. 162
Chapter 1
INTRODUCTION
With the development of advanced fiber reinforced composites as
viable structural materials the adhesive bonded joint has become of
primary importance. The bonded joint does not require that the stru-
ctural members being joined (adherends) be perforated to facilitate
bolts. Without the bolt holes and the stress concentrations associated
with. them, a substantial weight savings can be realized which is a major
reason for selecting composite materials for the structural component.
In order to fully realize the strength of the composite adherends,
the adhesive bonded joint must be efficiently desig;1ed and this requires
an adequate prediction of the stress distributions in the adhesive layer
of the joint. The study of stresses in the adhesive layer has been
approached by researchers in the past using one of two types of analysis
procedures.
Many researchers have attempted to predict the stresses using a
closed form analytical solution. However, when using this approach, the
equations that need to be solved become exceedingly complicated and this
leads to the need for simplifying assumptions. These assumptions have
included linearity, material isotropy, restrictions on the geometry of
the joint, and neglect of thermal effects.
Other researchers have approached the problem through the use of
numerical techniques such as finite element analysis. They have usually
found it necessary to use some or all of the assumptions made for the
1
2
closed form type solution procedure. The motivation for the present
study is to show the capability of a finite element computer analysis
program developed by Renieri and Herakovich [l] to adequately predict
these stress fields. The program has the capability for material ortho-
tropy, material nonlinearities, and temperature dependent properties.
Modifications to the computer program for this study include increased
element capacity, improved execution time, and capability for hygrother-
mal analysis.
Chapter 2
LITERATURE SURVEY
The first investigation into the behavior of bonded joints found in
the literature was presented in 1944 by Goland and Reissner [2]. They
obtained an analytical solution by assuming a state of plane strain,
prescribing the distributions of the shear and peel stresses to be
parabolic and linear, respectively, and applying restrictions on the
ratios of adherend moduli and thickness to adhesive modulus and thick-
ness. The solution is based on the principle of minimum potential
energy and is restricted to linear isotropic materials and identical
adherends. Thermal effects were neglected for this analysis.
Erdogan and Ratwani [3] approached the problem of an orthotropic
plate bonded to an isotropic plate with an isotropic ac'hesive. They
obtained closed form solutions for stepped lap and smocthly tapered
joints. Their solution was based on a su!TITiation of forces in the ad-
hesive layer and adherends and assumed plane stress anc linear material
behavior. This sol·Jtion predicted stress singularities at the edges of
each step in the ad~esive in the stepped lap joint, anc at the ends of
the overlap in the adhesive for the smoothly tapered jcint.
Barker and Hatt [4] used a linear elastic finite element computer
analysis program to compare results with the work of Erdogan and Ratwani
[3]. The adherends were modeled using four noded isoparametric elements
and material orthotropy was considered. The adhesive layer was modeled
using a special element developed for that purpose which was formulated
3
4
to have no thickness and used modified stiffnesses derived from the
moduli, thickness, and length of the adhesive layer. Their results
compared favorably with Erdogan and Ratwani.
Sainsbury-Carter [5] solved the stepped and linearly tapered
bonded joints by assuming linear isotropic materials and solving the
equations of equilibrium. It was assumed that the moduli of the adher-
ends are much larger than those of the adhesive. Also the analysis was
one-dimensional and thermal effects were neglected. It was shown that
the thickness of the adherends greatly affected the magnitude of the
peak shear and peel stresses and an iterative technique was developed to
modify these thicknesses within stress design criteria.
Wah [6] investigated non-symmetric single lap joints with composite
adherends and isotropic adhesives. Laminated plate theory was used to
develop stress and moment resultants and relcte them to mid-plane
strains and curvatures. The laminated adhercnds were required to be
mid-plane symmetric in order to uncouple the bending-stretching terms in
the previously mentioned relationships. A solution was presented for
the joints under shear loadings as well as axial loads. The solutions
were restricted to the elastic range and thermal effects were neglected.
Hart-Smith [7] was the first researcher to consider the non-lin-
earity of the adhesive layer by assuming its stress-strain response to
be elastic-perfectly plastic. This effort presents solutions and
design aids, including thermal effects, for single, double and stepped
lap joints and linear tapered joints. While a thermal mismatch is
considered all material properties are considered temperature indepen-
5
dent. This research notes the increased failure strength by allowing
for plastic deformation in the adhesive.
A comparison of theoretical and experimental shear stress in a
single lap joint was presented by Sharpe and Muha [8]. The joint
modeled had plexiglass adherends and an epoxy adhesive. Good correla-
tion was obtained with the work of Goland and Reissner [2] and with a
linear computer analysis program, BOND4, of the University of Delaware.
Renton and Vinson [9-10] performed parametric studies on single lap
joints as well as fatigue testing and thick adherend lap shear testing.
The specimens were comprised of mid-plane symmetric composite adherends
and elastic behavior only was studied. Linear thermal effects were also
included. The parameters studied were over-lap length, adhesive thick-
ness, and ply orientation in the composite adherends. Comparisons were
made with the work of Goland and Reissner [2] with Renton and Vinson's
work showing better satisfaction of stress free boundary condition at
the edge of the adhesive layer.
Grimes, Greimann et al [11] approached the analysis of single,
double, and stepped lap joints from both the finite element method and
numerical integration of the governing differential equations. Their
analysis included full material nonlinearity in the adherends and
adhesive layer. The development for both solutions was based on the
deformation theory of plasticity with the finite element analysis
utilizing an iterative procedure until the solution converged. In both
forms of analysis, solutions were presented for room temperature only
and curing stresses were neglected.
6
DasGupta and Sharma [12] used an analysis similar to Goland and
Reissner [2] to predict stresses in lap joints with prebent adherends.
The work showed a decrease in peak stresses with the use of bent ad-
herends.
Renton [13] provided an analysis of the thick adherend lap shear
test using the work of Renton and Vinson [9-10]. This research verified
the validity of the test.
Other researchers have investigated the effects of moisture [14],
and the reliability [15] of lap joints.
While this survey is by no means all-inclusive, it is representa-
tive of the research that has been performed and from this survey the
need for fully-nonlinear mat2rial behavior and temperature dependent
properties can be seen as these physical realities have been consis-
tently neglected.
Chapter 3
BASIC CONSIDERATIONS
The bonded joints selected for analysis in this study are single
and double lap joints with and without adhesive layers. Typical geo-
metries for joints with adhesive layers are shown in Figure 1. Composite
and isotropic adherends are considered with nonlinear material properties
and thermal stresses as well as temperature dependent properties.
Hygro·scopic analysis capabilities are presented but no results are
included in this investigation due to a lack of complete consistent
data.
3.1 Geometric Restrictions
For the present study it is assumed that the joint is in a state of
plane strain (i.e. Ex= 0, or Ex= const). This is a valid assumption
if the x dimension of the joint (Fig. l) is 1arge and the cross-section
under consideration is some distance removed from contraints that are
dependent upon the x coordinate.
The analysis is also restricted to balanced, mid-plane symmetric
composite adherends and laminate material properties are used for these
components.
3.2 Quasi 3-Dimensional Analysis
The analysis of reference [l] co~siders a long prismatic bar under
the influence of a uniform applied strain or temperature change to have
strains independent of the x coordinate. With this assumption, the
7
8
0.045
(a) SINGLE LAP JOINT
(ALL DIMENSIONS INCHES)
ADHESIVEJ
.t4---l.9376
( b) DOUBLE LAP JOINT
Figure 1. Lap Joint Geometries
9
strain displacement relations can be written as
_ aw _ f ( ) av aw ( ) e:z - az-- 3 y,z 'Yyz = az-+ ay = f4 y,z (3.1)
_ au + aw _ f ( ) _ au + av _ f ( ) y xz - al ax - 5 y ,z ' y xy - ay ax - 6 y ,z
where· u, v, and ware x, y, and z displacements, respectively, and f1 through f6 are unknown functions of y and z coordinates only. With the
use of suitable mathematical manipulation, (3.1) can be integratec yield-
ing the following form for the displacement fields
u(x,y,2) = x(C1y+C2z+c3) + U(y,z)
2 v(x,y,z) = x(C4z+C6) - c1 ~ + V(y,z)
2 w(x,y,z) = x(-C4y+C5) - c2 ~ + l~(y,z)
(3.2)
where c1 through c6 are unknown constants and U, V, and Ware unknown
functions of y and z only. With this assumption, and ne~lecting body
forces, the equilibrium equations can be written as
aTxy .dTxz . -·- + -- = 0 a.v az
aa y aT yz --+~=O ay az
~ + aaz = 0 ay az
(3.3)
10
3.3 Bonded Joints
The analysis developed in [l] can be used as two different formula-
tions for the solution of bonded joints. The first of these corresponds
to a classical plane strain solution in which £ = y = y = 0. This x xz xy type of solution is entirely 2-dimensional and i~; the type of plane
strain analysis used in a majority of joint analyses previously in the
literature.
The second formulation is a more general plane strain solution
where Ex ; 0 but is equal to some constant ~x· For this procedure the
components of strain Yxz and Yxy are assumed to be zero. The principal
difference between the two formulations is that the second accounts for
the transverse stiffness (Ex) of the adherends and adhesive while the
first does not. This second formulation is used for this analysis and
comparisons between results obtained fron, the two solutions are pre-
sented in Chapter 5. In the following sc!ctions and chapters the first
formulation is referred to as a 2-dimens;onal formulation while the
second is referred to as a quasi 3-dimen;ional analysis as it corresponds
closely to the analysis of section 3.2.
3.3.l 2-Dimensional Joint Formulation
In the classic plane strain solutio(1 the displacements, strains,
and therefore stresses, are independent •)f the x-coordinate (Fig. 1).
Under these assumptions the displacement fields (Equ. 3.2) reduce to
u = 0
v = V(y,z)
w = W(y,z)
(3.4)
11
The nonzero strain components (ey' ez and Yyz) have the same definitions
as in Equ 1 s. ( 3. l ) .
3.3.2 Quasi 3-Dimensional Joint Formulation
If, in a bonded joint, it can be assumed that ex is a nonzero con-
stant and that Yxz and Yxy are zero, then the displacement fields
(Equ's. 3.2) reduce to
u = s x x v = V(y,z)
w = W{y,z)
(3.5)
In these equations the only nonzero constant from Equ's. (3.2), c3, has
been renamed sx and corresponds to the uniform normal strain ex. The
remaining strain components (ey' ez, and Yyz) again have the same form
as in Equ's. (3.1).
The assumption that the strain comp< nent normal to the plane Jf the
analysis is constant is strictly valid f•,r the case of single lap
joints with identical orthotropic adhere; ds. It is also valid for
double lap joints where the outer adhere;:ds are identical and exhibit
orthotropy. The assumptions that Yxz = Yxy = 0 are valid if the
adherends are orthotropic.
The restriction that all adherends be orthotropic is satisfied
by all joints analyzed in this study. It should be noted however
that two of the joints analyzed do not satisfy the first condi-
tion as they are single lap joints with differing adherends.
12
For these joints, the solution n.eglects the effects of bending out of
the plane of the joint and is similar to a membrane solution in this
respect.
3.4 Material Properties
In order to adequately model a layered adherend layer by layer an
excessive number of finite elements would be needed. Because of this,
it is necessary to use laminate material ~roperties and consider the
composite adherends to be homogeneous orthotropic materials for the
joint analysis.
Obtaining laminate material properties from the literature proved
to be impossible thus making it necessa~· to generate these properties
analytically. For this generation of priperties two different approaches
were used. The stress-strain response o = a laminate was predicted
following the work of Renieri and Herako1ich [l], while thermal proper-
ties were predicted using classical lamination theory.
3.4.l Prediction of Laminate Stress-Strain Response
The details of the analysis of ref. [l] will not be presented as to
do so would be overly repetitious; however a brief outline will be pre-
sented for completeness.
The analysis utilizes the displacement fields of Equ's. (3.2).
Because the laminates in question are balanced and midplane symmetric
the analysis can be reduced to the quarter section shown in Figure (2b)
with certain symmetry and anti-symmetry conditions. The displacement
13
T 2H
1
(a)
z
ho ...... -2~~~-..--..-..-..-..+-...._.-+ ..... y
3 k
( b)
Z,3
Figure 2. Laminate Geometry
(c)
14
fields (Equ's. 3.2) reduce to
u = ~xx + U(y,z)
v = V(y,z)
w = W(y,z)
(3.6)
and again the constant c3 has been renamed ~x and corresponds to a uni-
form applied strain. The displacement fields (3.6) along with the stress-
free boundary conditions along the free edges, top and bottom surfaces
and certain restrictions imposed upon the displacements by the symmetry
and anti-symmetry conditions mentioned previously represent the boundary
value problem to be solved by the finite element analysis. With this
analysis and the nonlinear finite element program, the moduli Exx and
EYY can be predicted as functions of strain level.
3.4.2 Prediction of Laminate Thermal Properties
Laminate thermal properties including coefficients of thermal ex-
pansion and moduli as functions of temperature are predicted using
lamination theory and unidirectional material properties as functions
of temperature. Lamination theory as presented here cannot directly
predict temperature dependent laminate properties, however, if the uni-
directional properties used as input correspond to an elevated tempera-
ture, then the laminate properties generated will also correspond to
this temperature. Therefor~, laminate properties can be predicted at
discrete temperatures corresponding to the input data.
The constitutive relations for a single, orthotropic lamina in the
principal material coordinates are
15
C1 l c,, c12 c13 0 0 0 e: l
(12 c12 c22 c23 0 0 0 e:2
C13 c13 c23 C33 0 0 0 e:3 = (3. 7) T23 0 0 0 C44 0 IJ Y23
Tl3 0 0 0 0 C55 0 Y13
Tl2 0 0 0 0 0 c66 yl2
where the principal coordinates are shown in Fig. 2a as the 1-2-3 system.
These equations can be written in an abbreviated form as
(3.8)
For a coordinate rotation about the 3 axis through an angle e (Fig. 2a)
the stresses and strains are transformed according to the following
relations,
{cr}x = [T1]{cr}1 and {e:}x = [T2]{e:J1 (3.9)
where
C1 m2 n2 0 0 0 2mn x cry n2 m2 0 0 0 -2mn
C1 0 0 l 0 0 0 {cr} = z ' [Tl] = x 0 0 0 m -n 0 Tyz
T 0 0 xz 0 n m 0
0 0 0 2 2 TXY -mn mn (m -n )
16
2 n2 E m x 0 0 0 mn 2 2 0 0 0 Ey n m -mn
Ez 0 0 1 0 0 0 {E} ;::: ' [T 2J = x 0 0 0 m -n 0 Yyz
Yxz 0 0 0 n m 0
Yxy -2mn 2mn 0 0 0 (m2-n2)
and m =case, n = sine
Combining Equ's. (3.8) and (3.9) yields
{cr}x = [T,J[CJ[T2]-1{E}x
or
fo} = [c]{d x x (3.10)
where [CJ is defined as
[CJ = [T1J[CJ[T2J-1 (3.11)
and has the form
- - - -ell c12 c, 3 0 0 c,6
-c12 C22 C23 0 Cl c26
[cJ = c,3 c23 C33 0 Cl c36 (3.12) 0 0 0 C44 Ce,.5 0
- -0 0 0 C45 C55 0
cl6 c26 C35 0 0 c66
17
Equ's. (3.10) are the coristitutive relations for an orthotropic lamina
rotated through an angle e.
For lamination theory it is assumed that a lamina is in a state of
plane stress. It should be noted that this lamination theory development
is the only case in which plane stress will apply while plane strain is
assumed in all other developments.
The mathematical statement of the plane stress assumption is
a = T = T : Q z xz yz (3.13)
which can be used to reduce the constitutive relations (Equ's. 3.7) to
=
0
0
0 (3.14)
This simplified form of the stiffness matrix [CJ is known as the reduced
stiffness matrix [Q]. If the transformation matrices (Equ's. 3.9) are reduced similarly, a
rotated plane stress constitutive relation [Q] can be formed in the
same manner as Equ's. (3.10). Thus
fo}x = [Q]{dx (3.15)
where - - -
ax i::x o,, 012 Ql6 - - -{a} = cry {E:} = E:y , and [Q] = Ql2 Q22 Q26 x ' x
'xy Yxy Q16 Q26 Q66
18
Now, taking the standard plate theory assumption that normals to
the mid-plane of the plate remain normal after loading, the strains of
Equ's. (3.15) may be written as
where
{e:o} =
{K} =
{E: } = { e: o } + z{ K} x
{e:} = total strains x
e: 0
x e: 0 y = mid-plane strains
= plate curvatures
and z = distance from the mid-plane
Defining stress resultants
Nx = LH ax
{N} = Ny ay jz
Nxy 'xy
and combining Equ's. (3.15), (3.16), and :3.17) yields
1 H _ ( H {N} = [Q]{e: 0 }dz + j [Q]Z{K}dz
H -H
(3.16)
(3.17)
(3. 18)
or
where
and
19
N [A] = I [Q](k)(h -h ) k k-1 k=l
(3.19)
(3.20)
where the k denotes the number of the ply and the h's are as in Fig.
2b.
For symmetric lamiantes [BJ = 0 and
Inverting this relationship yields
Noting that
- l {a} = - {N} x 2H
and combining Equ's. (3.22) and (3.23), leads to
{ e: 0 } = [a * ]{ a }
(3.21)
(3.22)
(3.23)
(3.24)
20
where
[a*] = 2H[AT 1
These relations can be used to define the laminate properties
-E = crx = _1 -x e: 0 a* x 11
(3.25)
For an orthotropic material the coefficients of thermal expansion
are
°'1
°'2
{a}l = °'3 (3.26) 0
0 I
I
0 I These coefficients transform, under the ·otation defined earlier, in
the same manner as the strains (Equ. 3.9,.
( 3. 27)
21
For the 2-dimensional analysis Equ. (3.27) reduces to
= (3.28)
Now define laminate coefficients of thermal expansion such that
{e 0} = {a}t.T (3.29)
where 6T is a uniform temperature change and
Combining Equ's. (3.18), (3.22), and (3.29), and considering symmetric
laminates only yields
or
{e 0} = {a}6T = [AJ-11 H [Q]{a}dzt.T
-H
- [A]-1 N - k k ) {a} = k:l [Q] {a} (hk-hk-1
(3.30)
( 3. 31 )
When the moduli and thermal coefficient~ used to calculate the [CJ
matrix (Equ's. 3.7) are those corresponding to an elevated temperature,
then the moduli of Equ's. (3.25) and coefficients of expansion of Equ's.
(3.31) will be laminate properties also corr~sponding to that tempera-
22
ture. These reduced moduli will be used as input in the finite element
program.
These methods of generating material properties were resorted to
because of the lack of consistent experimental data found in the
literature.
3.5 Finite Element Formulation
As in section 3.2.l the presentation of the complete formulation
would be a duplication of the work of Renieri and Herakovich [l],
therefore only the highlights will be given here.
The finite element solution process involves the subdivision of a
structure into a finite number of smaller elements (Fig. 3). This
process is known as discretization. For each of these finite elements
a set of interpolation functions are chosen tc represent the displace-
ments at any point in the element as functions of the displacements at
the corners or nodes of the element. Using the strain-displacement
relations (Equ. 3.1) the strains can also be calculated as functions of
nodal displacements. Now with the use of a variational principle, such
as the principle of minimum potential energy, a set of equations re-
lating nodal forces to nodal displacements can be obtained for each
element,
(3.32)
where {F} = nodal forces
{u} = nodal displacements
[K] = element stiffness matrix
23
8 ELEMENT
(a) SINGLE LAP JOINT
8
(b) DOUBLE LAP JOINT
Figure 3. Joint Boundary Conditions
24
and the superscript (2) refers to the individual element. These elemental
relations (Equ's. 3.32) are then combined or assembled into a larger
system of equations relating forces to displacements for the entire
structure. The solution of these equations yields the displacements and
therefore the strains and stresses over the entire body.
The finite element scheme developed by Renieri and Herakovich [l]
utilizes the constant-stress, constant-strain triangular element with
three nodes. The interpolation functions used are
(3.33)
As can be seen from the form of Equ's. (3.33) these are linear relations
and will yield constant strains when substituted into Equ's. (3.1). The
constants a1 through a9 are functions of the spatiitl coordinates and
nodal displacements of the individual elements and ~x is the applied
uniform strain.
Manipulation of Equ's. (3.33) and substitution into Equ. (3.1)
yields the following strain-displacement relations for an element.
25
e:x E; A x e:y av1 + cv2 + ev3 e:z l bw1 + dw2 + gw3 (3.34) - A 'Yyz bv1 + dv2 + gv3 + aw1 + cw2 + ew3
Yxz bu1 + du2 + gu3
Yxy au1 + cu2 + eu3
where A = area of the element
u, ' u2' u3 = x-displacements at nodes l ' 2, and 3 respectively
v, ' v2' V3 = y-displacements at nodes 1 ' 2' and 3 respectively
wl' w2' W3 = z-displacements at nodes 1 ' 2, and 3 respectively
and a, b, c, d, e, g are known constants involving spatial coordinates
only.
For the case of a uniform thermal load the strains are
{e:}l = {e:*}1 - T { e: } l (3.35)
where {e:*} l = to ta 1 strain
{ e:}l = mechanical strain
and T { e: } l = thermal strain
which consists of {a}1 (Equ. 3.26) multiplied by the temperature change
~T. Transforming Equ's. (3.35) to an. arbitrary coordinate system yields
an individual element.
(3.36)
26
Noting Equ 1 s. ( 3. 34) , these strains can be written as
2 2 e:x i; - (m a.1 + n o.2)6T x
(av1 + cv2 + ev3)/A - 2 2 E:y ( n a.1 + m a.2)6T
e:z (bw1 + dw2 + gw3)/A - a.36T ( 3. 37) =
Yyz (bv1 + dv2 + gv3 + aw1 + cw2 + ew3)/A
Yxz (bu1 + du2 + gu3)/A
Yxy (au1 + cu2 + eu3)/A + 2mn6T(a1-a.2)
The preceding formulation for thermal strains is completely analogous
for that of hygroscopic strains. For an orthotropic material the coef-
ficients of hygrosopic expansion are
~:,
f;2
{ rn 1 = 83
(3.38) 0
0
0
Following exactly the development of Equ's. (3.35) through (3.37) and
substituting {S}l for {a.}1 and 6M for 6T where 6M is a uniform percent
weight change due to moisture absorption or desorbtion, yields
27
e:x sx - (m2sl + n2s2)AM
E:y (av1 + cv2 + ev)/A - (n2Bi + m2s1)AM
e:z (bw1 + dw2 + gw3)/A - B3AM (3.39) ;:::;
Yyz (bv1 + dv2 + gv3 + aw1 + cw2 + ew3)/A
Yxz (bu1 + du2 + gu3)/A
Yxy (au1 + cu2 + eu3)/A + 2mnAM(B1-s2)
The hygroscopic expressions are presented here because the capability
for this type of analysis has been included in the computer program.
Results will not be presented because of the lack of data as stated
earlier. It should also be noted that the derivation is for a uniform
temperature or moisture change and that analysis should be limited to
cases where uniformity is a valid assumption.
The principle of minimum potential en2rgy states that a body is in
equilibrium when the total potential energy$ is minimum where
(3.40)
Ue = internal strain energy
and
We = potential energy of the applied loads
The internal strain energy for an element is
(3.41)
which for an element with constant strains and unit thickness reduces
to
28
(3.42)
In both Equ's. (3.41) and (3.42) the strains involved depend upon whether
the loading is mechanical (Equ's. 3.34), thermal (Equ's. 3.37) or
hygroscopic (Equ's. 3.39). The potential energy of the applied loads is
given by the negative of the applied forces multiplied by their respective
displacements.
Minimization of Equ's. (3.40) with respect to displacements yield
the elemental stiffness matrix plus strain, thermal and hygrosopic re-
lated vectors. The forms for these can be four.d in Appendix A.
3.6 Boundary Conditions for Joints
The boundary conditions applied to single and double lap joints for
the present study are shown in 1=i g. 3. A number of dif Fe rent boundary
conditions and loadings were in~estigated and compariso~s were made.
These are summarized in Table 1. Noting Table 1 it is seen that
all conditions except the fourth yield comparable results. The lower
peak stresses for this condition can be attributed to an overly flexible
model. It was reasoned that this does not correspond to physical reality
as a real joint would not be free to deflect up and down where the load-
ing is applied. The first set of conditions was eliminated from consider-
ation as they can only be applied to symmetric single lap joints. The
second set was eliminated becau~e during the solution process a negative
diagonal in the global stiffness matrix was encountered. The third set
of conditions was disregarded b~cause the stress distribution in the
TABLE 1
Comparison of Various Boundary Conditions for Symmetric Single Lap Joints
Type of Constraint Numerically Symmetric Stress Applied I Max Max Max Max End and Loading Stable Distribution Load Tyz cry oz Displacement
- =· ~~·-.=F-~===l========l=====l===========I
1 ~ Yes Yes 300 2.39 .84 1.13 .91 x 10-3 ~ ! " D. F (lbs) (KS!) (KSI) (KSI) .82 x lo-3
F...., ~· m. ( in ) .~
2 ~ Jf I I 1-F No Yes 300 2.39 .84 1.13 .91x10-3 F7; ·, ~ (lbs) (KSI) .82 x 10-3 --------- -
3-i t 2.52 .97 1.32 .179 x 10-2 I I Yes No 300 2.25 .72 .. 93 .165 x 10-2
F ~ · (lh~) (KST) (~~T) (K~!) (in)
4 ~ Yes Yes .165 x 10-2 1. 91 .66 .88 .165 x 10-2
2 ~ A (in) .165 x 10-
5
fJi:t i:::=::::fll Yes Yes . 165 x 1 o-2 2. 32 . 83 1. 11 . 165 x 10-~ ~ ~ (in) (KSI) (KSI) (KSI) .165 x 10-
N ~
30
adhesive layer was not symmetric for a symmetric joint while it is in-
tuitively obvious that it should be. The fifth set of constraints and
loading was chosen because it provided symmetric results, numerical
stability and is applicable to nonsymmetric joints.
3.7 Qualifying Notes
Due to the lack of data present in the literature a number of
assumptions concerning material properties have been made. For lamina
data input, it is assumed that E22 is identical to E33 . It is also
assumed that the three shear moduli G12 , G13 and G23 are identical.
Poisson ratios are considered constant and percent modulus retentions
are assumed identical in tension and compression
When inputting laminate material properties, much of the data is
generated according to the analyses.presented earlier. It is assumed
that the 02
-£2
laminate response is identical to the 0 2-£ 2 response of
a unidirectional lamina. It is also assumed that the 'yz-Yyz and
'xz-Yxz curves are the same as the , 12-y12 response of a lamina and
that the laminate poisson ratios vxz and vyz are the same as the lamina
poissons ratios v 13 and v23 , respectively. Other restrictions are the
same as for lamina data.
It should be mentioned here that the material properties used in
this analysis (Appendix C) are not consistent. The data has been taken
from a number of sources and is not all realted to identical material
systems. Even with these limitations it is felt that the investigation
still fulfills its goal of showing the capability to analyze joints if
consistent properties were available.
Chapter 4
NONLINEAR ANALYSIS
A significant portion of the research effort involved in this study
was devoted to the modification of the finite element analysis program
NONCOM [l]. The improved version, NONCOMl will be briefly described
here.
4.1 Modifications of NONCOMl
For the analysis of bonded joints, a number of modifications to the
existing finite element program were implemented. These included:
(1) Increased finite element capacity
(2) Inclusion of a more efficient equation solver
(3) Capability for input of fully 3-dimensional orthotropic
material properties
Other modifications included:
(4) Capability for hygroscopic analysis
(5) Capability for elevated temperature and moisture
content analysis
4.1.1 Increased Finite Element Capacity
In order to model an adhesive bonded jcint a large number of finite
elements are needed because of the large stress gradients and inherent
large aspect ratio of the adhesive layer. r:or this reason the maximum
number of elements was increased from 100 tc 400 elements. This was
done with an increase of high speed storage of approximately 50 percent.
31
32
This increase was held this low through the use of storage addressing
schemes where many arrays are stored in the same col'Mlon storage loca-
tions and by the block storage schemes of the equation solver chosen.
4.1.2 Solution of Simultaneous Equations
The equation solver in NONCOM is not suitable for large systems of
equations as its solution time becomes excessively large for such
systems. For this reason a new equation solver, SESOL [16] was selected
for NONCOMl. This equation solver offers a fast solution time and high
speed storage reduction schemes. In brief the solution algorithm
considers the system of linear equations
[K]{X} = {R} ( 4.1)
where [K] is the assembled stiffness matrix, {X} is the nodal displace-
ment vector and {R} is the applied nodal load vector. The stiffness
matrix is factored into an upper and lower triangular matrix
(4.2)
where [G] is upper triangular and [L]T is lower triangular and normalized
such that Lii = 1 (i not summed). Since [K] is symmetric
G •• = G •• L. . ( i not summed) lJ 11 lJ
and equation (4.2) can be written as
[K] = [L]T[D][L]
(4.3)
(4.4)
where
[DJ =
Defining
G •• 11
33
(i not summed)
{V} = [D][L]{x}
and combining equations (4.1) and (4.6) yields
(4.5)
(4.6)
(4.7)
The vector {v} in equation (4.7) is first found by Gauss reduction of
the load vector and then the nodal displacements are found through back
substitution into equation (4.6). The stiffness matrix and load vector
are assembled and stored on low speed storage in block form as in Fig.
4. During the solution process, reductions are performed on non-zero
terms only and only two blocks need be in high speed storgage at any
time.
In the computer program NONCOMl the number of equations per block
is determined as a function of the maximum half oandwidth plus the
diagonal of the assembled stiffness matrix. This is done to minimize
the number of blocks necessary and maximize the number of equations per
block within high-speed storage limitations. By making the number of
blocks a minimum, I-0 operations performed by the computer· with the
elemental stiffness matrices are also minimized. To further this re-
duction ~f operations the blocks are assembled two at a time.
Half Band Width !Plus Diagonal
x x x x 0 0\0 T x x x o o o'~
x 0 x 0 0 T\ x 0 x 0 T x',
x x 0 T 0 \ \
xxTxx \ x T x 0 x ',
T T T T T T\T T
Symmetric
x x 0 x 0 (f '° x o x x o o',
x 0 x 0 x x x x x
x x 0
x x x
Stiffness Matrix [K]
r r r 0 0 0
r
r 0 0 0 r 0 r r r
Load Vector {R}
Figure 4
.... xxxxOOOTr x x x 0 0 0 T 0 r x 0 x 0 0 T 0 0 0 x 0 x 0 T x 0 0 0 x x 0 T 0 0 0 0 0 x x T x x 0 0 0 r x T x 0 x 0 0 0 r T T T T T T T T 0 x x 0 x 0 0 0 0 0 x 0 x x 0 0 0 0 0 x 0 x 0 x 0 O 0 r x x x x 0 0 0 0 0 x x 0 0 0 0 0 0 r x x 0 0 0 0 0 0 r x 0 0 0 0 0 0 0 r 0 0 0 0 0 0 0 0 0
Block Storage
Block Storage of Stiffness Matrix and Load Vector
l
2
3
4
w -I==>
35
It should be noted the size of the maximum half bandwidth plus
diagonal is not only a function of node numbering, as is usually the
case with the finite element analysis, but is also a function of the
type of loading applied. When the applied loading is thermal, hygro-
scopic or an average force applied in the x direction an equation re-
lating a uniform strain to an average force in the x direction is
required in addition to the equations relating nodal forces and dis-
placements. This equation is related to all of the elements and may
involve all the nodal displacements of the finite element model. To
minimize the effect of this equation on the maximum half bandwidth plus
diagonal it is assembled in the center of the stiffness matrix. The
average force equation and its effect on the bandwidth can be seen in
Fig. 4 where the T's represent the average force terms. The half band-
width plus diagonal without the average force equation is shown by the
dotted line. These two bandwidths are for identical finite element
models under.different loadings.
4.1.3 Three Dimensional Properties
The program NONCOMl was given the capability for fully three
dimensional orthotropic material properties because of the restriction
that composite adherends must be modeled as homogeneous orthotropic
laminates (Chapter 3). This means that a plane of transverse isotropy
cannot in general be assumed as done in NONCOM [l].
4.1.4 Hygroscopic Analysis
A capability for hygroscopic loadings has been included in NONCOMl.
36
It is modeled in the computer program in exactly the same fashion as the
thermal analysis and is therefore subject to the same limitations. The
most notable of these is the restriction that the moisture distribution
be uniform throughout the finite element model. Other restrictions will
be mentioned in the section dealing with the nonlinear analysis.
4.1.5 Elevated Temperature and Moisture Content Analysis
This modification allows for mech~nical loading at elevated tempera-
ture and moisture content or thermal loading at elevated moisture
content or hygroscopic loading at elevated temperature. Insight into
the interactions between thermal and hygroscopic material response could
not be obtained from the literature. For this reason two assumptions
are made about these interactions. First, it is assumed that hygro-
scopic properties are independent of temperature and that thermal proper-
ties are independent of moisture content. It is also assumed that
changes in mechanical properties due to temperature and moisture content
are cumulative. By cumulative it is meant that when a modulus is to be
modified to correspond to both temperature and moisture content it is
first modified for the temperature and then this new modulus is then
modified to correspond to the moisture content. The process of changing
material properties will be more fully described in the following section.
4.2 Nonlinear Analysis
In order to simulate nonlinear material behavior in a computer
program with the linear elastic finite·elerrent analysis described in
Chapter 3 two separate problems must be dealt with. First a method of
37
accounting for material nonlinearities within the analysis procedure and
second a method of representing the nonlinear material properties.
4.2.l Incremental Loading Procedure
The finite element computer program NONCOMl deals with varying
material properties through the.use of a incremental solution procedure.
With this type of procedure, the load whether mechanical, thermal, or
hygroscopic is applied as a series of increments. This yields a series
of linear solutions with total stresses, strains, and displacements
formed by sunmation of the linear increments of these quantities. When
applying the load incrementally the material properties are updated to
correspond to the current levels of strain, temperature, and mositure
content. With the finite element method an individual element can have
material properties varying independently of other elements in the
model.
4.2.2 Nonlinear Data Input
Material stress-strain response for an othrotropic material in the
principal material coordinates are represented in the form of modified
Ramberg-Osgood [17] approximations which have the form
a K n. E = E + icr l i = l or 2 (4.8)
In equation (4.8) E is the elastic modulus and Ki and ni are Ramberg-
Osgood coefficients. A method for calculating the four coefficients K;
and ni is described in Ref. [l]. A tangent modulus can be defined as
38
E= _dcr = ___ E_-=-_ d£ n.-1 K.En.cr 1 +l
1 1
i = 1 or 2 (4.9)
where t is the tangent modulus corresponding to the principal stress a.
Noting Fig. (5) the value of the stress ap corresponding to the strain
at the end of load increment P is
p O' =
p I:
k=l J .. i /:,£ 1:: (4.10)
where 6£j is the increment of strain during the jth load increment.
Combining equations (4.9) and (4.10) yields for the P + 1th increment
EP+l = ____ ._E __ _ p
K.En.[ I: 6£jtj]n-l+l 1 l . 1 J=
i = 1 or 2 (4.11)
With equation (4.11) anJ principal material strains, the tangent moduli
are calculated at the end of each increment to be used for the next
increment. Moduli determined are t 11 , E22 , E33 , G23 , G13 , and G12 .
It can be seen in Fig. 5 that the strain £~-O for which the tangent
modulus is calculated differs from the strain £p ~here the modulus
should be calculated. This difference is a function of the size of the
load increment and can be made negligible by choosing an appropriately
small increment. For the computer analysis it is assumed that the shear
response is independent of sign while ~xtensional behavior can be
different in tension and compression.
Temperature and moisture dependent properties are represented as
linearly segmented curves. These· properties consist of percent modulus
retention curves which represent the change in stiffness of a material
en en w a:: r-en
-ll&J \II <J
39
j_ f.:_o ~ 1f(I 2f f,2 J f,p 16( 6( 6f! I
-- RAMBERG - OSGOOD (R-0) APPROXIMATION
----- PATH FOLLOWED BY CORRESPONDING STRESS METHOD
e ·POINTS WHERE E'1
ARE DETERMINED
STRAIN
Figure 5 · Determination of Current Modulus with Ramberg-Osgood Approximations
40
due to variations in temperature or moisture levels and coefficients of
thermal and hygroscopic expansion as functions of temperature and
moisture content respectively. During thermal or hygroscopic loading,
the moduli and coefficients of expansion are calculated at the mid-point
of the increment using linear interpolation. For input to the computer
program it is assumed that percent modulus retentions are identical for
tension and compression.
Chapter 5
RESULTS AND DISCUSSIONS
The analysis procedures presented in Chapters 3 and 4 were used to
generate laminate properties and analyze various bonded joints. The
joints investigated included single and double lap joints with, and
without adhesive layers. Both elastic and non-linear results are pre-
sented. The material systems considered were graphite-polyimide, boron-
epoxy, titanium, and aluminum for the adherends, and Metlbond 1113 and
AF-126-2 for the adhesives.
5.1 Materials Properties
The mechanical and thermal properties for the materials used in
this study were taken from the literature whenever possible. However,
as was stated in p~evious chapters,·complete properties were not always
available for a given material system. Therefore, laminate properties,
with the exclusion of uni-directional laminates, were predicted in
accordance with the analysis procedures presented earlier and the
results can be found in Appendix C which contains all of the material
properties of this study.
5.2 Averaging of Finite Element Results
The finite element analysis presented in Chapter 3 is based upon a
displacement formulation. This approach yields results in the form of
displacements at the node points and stresses and strains which are
constant over each element. Because the stresses are constant for an
41
42
individual element, a distribution of stresses over a series of elements
may appear to be discontinuous. In many cases, however, it is known
that the stresses must be continuous. For this reason, stress averaging
was used to produce the desired smooth distributions.
Noting Fig. 6a, stresses are presented along the line A-A which
corresponds to the mid-plane of the adhesive layer. The stresses pre-
sented at point E would correspond to an average of the stresses in
elements 3 and 4. This method of averaging was used for all joints with
adhesive layers.
When considering bonded joints without adhesive layers, the stresses
in question are along the interface between the two adherends. This
interface is shown as line C-C in Fig. 6b. For the stresses 'yz and
a2
which must be continuous across the interface: the results presented
correspond to an average of elemental stresses above and below the inter-
face. Thus, these stresses at point G would consist of an average of
elements 13, 14, 21, and 22. The normal stress components ax and cry are
not necessarily continuous across this interface so these stress
components are averaged along both line B-B, and line D-D. At point F,
an average of element 11 and 12 is presented and at point H, the stresses
are averaged between element 19 and 20.
The stress components 'xy and 'xz have not been mentioned as they
do not occur either at the mid-plane of the adhesive layer in adhesive
bonded joints, nor at the interface between the adherends in non-
adhesive bonded joints since the adherends in this study, when composite
laminates, are considered to be homogeneous, orthotropic materials. If
43
z ___ y
x A--~-~ ..
(a) JOINT WITH ADHESIVE
B 8
c c
D D
( b) JOINT WITHOUT ADHESIVE
Figure 6. Averaging of Finite Element Results
44
these components were present it could easily be shown that Txz must be
continuous across an adherend-adherend interface while Txy would need not
be.
5.3 Finite Element Representations
For the analysis of bonded joints, t~o finite element models were
used. Fig's. 7 and 8 sl1ow partial plots of the finite element models of
joints with, and withou·: adhesive layers, respectively. Both of these
models were generated u·;ing a mesh generator described by Bergner, Davis,
and Herakovich [18]. In both figures the scaling of the model for the
figure is not uniform. In Fig. 7, the aspect ratio of an element in the
adhesive layer ranges from 2.5 at the fre£ edge to 15 at the center of
the adhesive layer. The aspect ratio's o~ the elements in the adherends
range from 1.1 to 9.0. For the joints without adhesives (Fig. 8) all
aspect ratios are 1.0.
5.4 Stress Free Temperature
Bonded joints are, in general, cured with a combination of elevated
temperature and pressure. The maximum temperature involved in this
process is known as the cure temperature. The temperature at which
curing stresses begin to form is the stress free temperature and, in
general, the cure and stress free temperatures are not the same. The
stress free temperature of th~ adhesives used in this study was chosen
to be 270°F. This value was selected because both adh2sives are epoxy
based and 270°F was the value used in [l] for epoxy matrix material
systems.
z I
\
\ NOT TO
SCALE
~ y
Figure 7. Partial Finite Element Grid for Joint with Adhesive Layer
y
~ (J'1
I
• z
I I/ 'I II I
NOT TO SCALE ~
°' I 1/ I/ 1/ I/ I/ I/ I/ I/ I/ II I I I Ill /'"" """ """
II I/ II I/ II I/ I/ I/ I
zt • •vi fl fl /I /I /I /I /I /I /I ~ /I I I l/~/!1/!1 /!1/~/11 ~///// u ~u ~,'., :
1
:1
1
u nu :, : y ..
Figure 8. Partial Finite Element Grid for Joint without Adhesive Layer
47
For bonded joints without adhesives, the stress free temperature
was chosen to be 350°F. This is the value reported in [19] as the
stress free temperature of graphite-polyimide laminates. This is ap-
propriate as polyimides were the only composite lamiantes used in joints
without adhesives.
5.5 Linear Elastic Results
This section contains the linear thermoelastic results for various
joints. The dimensions for adhesive bonded joints are shown in Fig. 1.
For joints without adhesives the dimensions are identical to those of
joints with adhesives except the adhesive layer is removed. Most of the
curves in this section were drawn by the VPI & SU computer plotter.
In the figures that follow the superscripts M ard T are used to
differentiate stresses. Mechanically induced stresses are indicated by
the superscript M while thermal, or curing, stresses are denoted by the
superscript T. These are also used in combination irdicating a super-
position of mechanical and curing stresses. In some instances a curing
stress is referred to, while the corresponding figure presents only the
mechanical, and combined mechanical and curing stresses. The magnitude
of the curing component can, of course, be determined by taking the
difference of the combined, and mecranical stresses.
5.5.1 Single Lap Joints with Adhesives
5.5.1.1 [OJ Graphite-Polyimide Adherends
The adhesive stresses of a single lap joint with [OJ graphite-
polyimide adherends and Metlbond 1113 adhesive are shown in Fig. 9. The
.--.. (/) ~ ......... (/) (/) w 0::: ~
48
2.0
AZ SFT=270°F
A __ [OJ --A
~ CO] y 1.6 8=0.0005 in. L
l
1.2 a;:M+T y
axM+T
0.8
Ty~.AND T.M+T yz 0.4
-0.4 '------'------'------'-----' 0.0 0.125 0.25
J1L
0.375 0.5
12.0
9.0
6.0
3.0
Figure 9. Elastic Mechanical and Curinq Adhesive Stresses of a Single Lap Joint with [OJ Gr/Pi Adherends and Metlbond 1113 Adhesive
.--.. 0
0.... ~
49
loading consists of a thermal increment of -200°F and an applied dis-
placement. The resulting stresses are shown to t/L = 0.5 as the stresses
are symmetric about this line.
Upon examination of Fig. 9 it is seen that the stress free boundary
conditions
cr I = 0 and • I = O y t/L = 0 yz t/L = 0 (5.1)
are not satisfied by the finite element solution. This is due to the
nature of the constant stress finite elements used and the limitations
on the maximum number of element available. In order to check the
finite element analysis 1 ability to meet such stress free boundary
conditions, an analysis was performed on a small portion of the adhesive
layer from t/L = 0.0 to t/L = 0.05. The displacements predicted along
the upper and lower interfaces of the adhesive in the joint solution
were used as loading for the partial adhesive analysis. The stress
distributions produced by this analysis exhibited the proper trends
with crz reaching a peak value near t/L = 0 and cry and Tyz tending
towards zero. These distributions are not presented, however, as they
appeared very erratic. It is believed that this was caused by round-
off error in the applied displacements. This error may have become
significant after subtracting rigid body motion from the joint analysis
displacements.
In order to check the validity of the finite element solution
presented in Fig. 9, a number of static equilibrium calculations were
L
50
(a)
y r . ~~ ,,._-- f Tyz dy = V2
o' v •• rj
(b)
Figure 10. Free Body Diagrams of Single Lap Joints Under Two Sets of Boundary Conditions
51
made. The equilibrium equations for one half of a single lap joint
corresponding to the free body diagram of Fig. lOa are
EF = 0 = V - Rb' y 1
EF = 0 = N - R z 1 a'
and (5.2)
The finite element program NONCOMl does not back substitute the nodal
displacements to solve for the nodal forces. Therefore it is not
possible to determine the reactions Ra or Mb. However, the reaction Rb
can be determined as the average of the cry stresses of the elements
adjacent to the edge where Rb acts multiplied by the adherend thickness
and assuming a unit depth. A compa~ison )f v1 and Rb determined from
the finite element solution indicates a four percent error as shown in
Table 2.
Since the unknown reactions severely limit the equilibrium calcula-
tions for the previous joint, similar calculations were also performed
on a more simply constrained joint. This joint corresponds to the joint
shown in Table 1, condition 4, and a free body diagram of one half of
the joint is shown in Fig. lOb. For this free body diagram the equili-
brium equations are
(5.3) I:F = 0 = N2 z '
52
TABLE 2
Results of Equilibrium Checks of Single Lap Joint with [OJ Gr/Pi Adherends and Metlbond 1113 Adhesive
Under Two Sets of Boundary Conditions
Joint F.B.D. I:Fy l: F z (Fig. 10)
vl = 92.6 lb
a Rb = 89.0 lb
ERROR = 4.0%
v2 = 71.8 lb N~E = -0.25 lb
b Rd = 78.3 lb N~XACT = 0.0 lb
ERROR = 8.3%
53
and
As in the previous joint, the reaction moment Md cannot be determined
from the finite element solution. However, by removing the constraint in
the z direction, the integral of the oz stresses, N2, must now equal zero.
The results of these equilibrium calculations are also shown in Table 2.
Returning to Fig. 9, it can be seen that under mechanical loading
only~ the shearing stresses dominate the stress fields. This is a
function of the overlap lenth, L. If the overlap were longer, for a
given loading, the shearing stresses would be reduced while the peak
a~ stresses would increase. This will be shown later in the section
containing nonlinear results and can be verified by considering the
force and moment equilibrium equations (5.2). It is interesting to note
that the normal stresses cr~, a~, and cr~ are very close in magnitude for
this joint.
Since the stresses presented are produced by the displacements of
the joint, its deflected shape would provide significant insight into
the physics of the problem. Fig. 11 shows the deflection of the upper
edge of the lower adherend under mechanical loading only. The dashed
line signifies the beginning of the overlap (2 = 0). In this figure
it is difficult to distinguish any curvature of the adhere~d in the
region of the overlap because of the relatively small distance involved.
A plot of the overlap only showed a nearly straight line distribution
also indicating very little bending in this region. It is interesting
0.4
+-'
' z 0 0.2 l-o w _J LL w 0 I
~ 0.0
~ tL\ COJ I COl ' i I ... y aL ~RI
2.1875---1
-0.2 I . . • 0.0 0.3125 0.625 0.9375 1.25 1.5625 1.875
Y (in)
Figure 11. Deflected Shape of the Lower Adherend of a Single Lap Join Under Mechanical Loading
2.1875
(n
+'>
55
that the maximum bending occurs just before the overlap region. However,
due to the layout of the finite element model, which was designed for
adhesive studies, the stresses in this high bending region cannot be
accurately obtained.
Again returning to Fig. 9 it is seen that the only nonzero components
of curing stresses are o~ and o~. Actually, the finite element solution
did predict other components of curing stresses but their magnitudes
were insignificant. It is difficult to make a valid comparison of the
relative magnitudes of the mechanical and curing stress components as
the mechanical loads were produced by a small load increment while the
curing stresses are due to the full temperature change from the stress
free temperature to room temperature. The a; curing stresses represent
approximately 15 percent of the ultinate strength of the adhesive and
while this magnitude is not exceedingly large, its contribution should
be included in a failure theroy.
5.5.l.2 [0/±45/90]s Graphite-Polyimide Adherencs
Fig. 12 represents the mechanical and curing stresses for a single
lap joint with [0/±45/90]s graphite-polyimi de adherer,ds and Metl bond
1113 adhesive. The loadings are identical with those of the previous
joint. Comparing Fig's. 9 and 12 it can be seen that under mechanical
loading only, the magnitude of the peak adhesive strEsses decrease with
decreasing adherend stiffness EY. Thus, for the samt~ loading, the joint
with [OJ adherends has higher stresses than the joint with [0/±45/90]s
adherends. However, when considering curing stresses, this is no longer
the case. It can be seen that the magnitudes of the stresses o~ and
...--Cf) ::::s:::: -Cf) Cf) w 0:: t:;
56
2.0
o::M+T
1.6 ~._y_a:_xM_.f..:::T::::==::=::=::=::=::=:::::=== 1.2 z SFT=270°F
I l
[ 0/±4f>/90J 5 A -- ---A
[Q/±45/90Js y 8:: 0.0005 in. L
0.8 i
0.4 AND r.M+T yz
-CJ;MAND 0:t. +T z
.::r.:M y
0.0
-0.4 "--------·----------------0.0 0.125 0.25
~/L
0.375 0.5
12.0
9.0
6.0
3.0
0.0
Figure 12. Elastic Mechanical and Curina Adhesive Stresses of a Single Lap Joint with [0/±45/90] Gr/Pi Adherends and Metlbond 1113 Adhesive s
...--a 0.... ~ -
57
a; are larger for the [0/±45/90]s joint than for the [OJ joint. This
is due to the effects of the transverse stiffness and coefficient of the
thermal expansion. For the [0/±45/90]s adherends the longitudinal (y)
and transverse (x) directions have identical modulus and coefficient of
expansion. This quasi-isotropy leads to curing stresses, a~ and
a~, that are identical for the joint with [0/±45/90]s adherends. This
is not the case for the joint with [OJ adherends as the transverse
direction has a much lower modulus and higher coefficient of expansion
than the longitudinal direction.
Because the adherends of this joint have identical transverse and
longitudinal stiffness it is appropriate to determine what effects the
transverse stiffness have upon the mechanically induced stresses. For
this purpose the joint of Fiq. 12 was also loaded under a classical
plane strain assumption. The resulting stresses are shown in Fig. 13.
The loading and materials were identical for both joints except that for
the joint of Fig. 13 the average normal stress acting perpendicular to
the plane was not specified and the strain normal to the plane, Ex' was
zero. Also, ax stresses are not presented for the classic plane strain
solution. Comparing the two figures (12 and 13) it can be seen that the
mechanically induced stresses are reduced slightly for the quasi three-
dimensional analysis. This indicates that a 2-D solution would under-
estimate the strength of this joint. Comparisons of the stress com-
ponents for the two joints can be found in Table 3. The curing stresses
are identical for the two joints because both analyses were performed
under the 3-D analysis.
--
58
2.0 ------------------------------------....,
~a-rr I. 6 . """-----=
~------~~---~~-----
SFT= 270° F 1.2 z
L------BA~:=[~C 1:_~5/90 ]s [0/:1:45/90]s ~y
8=0.0005 In. L~~ -----en 0.8 en
j
w 0::
~
O"zM __ __,
a;M y
-0.4._ ______ __. ________________ _,,..._ ______ _...
0.0 0.125 0.25
9-;L 0.375 0.5
12.0
9.0
6.0
3.0
Figure 13. Elastic Mechanical and Curing Adhesive Stresses of a Single Lap Joint with [0/±45/90]s Gr/Pi Adherends and Metlboncl 1113 Adhesive, E: =O x
-0 a.. ::E -
59
TABLE 3
Comparison of Peak Adhesive Stresses of [0/±45/90]s Joint Under Mechanical Loading for
2-Dimensional and Quasi 3-Dimensional Analysis
Type of Peak T~z Peak a~ Peak a~ Analysis (ksi) (ksi) (ksi)
2-D 1.44 .826 .654
3-D 1.33 .762 .60
Percent 8% 8% 8~'. Difference
60
5.5.1.3 [90] Graphite-polyimide Adherends
The mechanical and curing stresses of a single lap joint with [90]
graphite-polyimide adherends and Metlbond 1113 adhesive are shown in
Fig. 14. Comparing Figs. 9, 12, and 14 the trends pointed to earlier
concerning adhesive stresses and adherend stiffness are again confirmed.
It is seen that the [90] adherends produce the lowest mechanical stress-
es of the three joints. Comparing Fig's. 9 and 14 only it is revealed
that the a! curing stresses for the [OJ joint are identical to the cr~
stresses for the [90J joint. This is as would be expected and the same
correspondence is also seen between the cr~ of the [90J joint and the cr~
of the [OJ joint.
5.5.1.4 [OJ and [90J Graphite-Polyimide Adherends
The adhesive stresses for a single lap joint with [OJ and [90]
graphite-polyimide adherends and Metlbond 1113 adhesive due to mechani-
cal loading only are presented in Fig. 15. The most striking aspect of
these stresses is the lack of symmetry present in their distributions.
This is due, of course, to the unsymmetric nature of this joint. It is
also interesting to note that the peak values of stress occur near the
line i/L ~ 1.0 This correspondence of the peak stresses with the more
flexible adherend seems inconsistent as the trends of the previous
joints pointed to higher stresses with stiffer adherends. This can be
explained upon examination of Fig. 16 which presents displacements of
the upper adherend-adhesive interface relative to the displacements of
the lower adherend-adhesive interface. The displacements are normalized
with respect to the thickness of the adhesive layer. It can be seen
.......... Cf) ~ ......... Cf) Cf) w 0::: ~
61
2.0
. Z SFT= 270°F [90J
1.6 (90J
----~--1----..a.. y 8 = 0. 0005 in. L
1.2 a:;M+T x
0.8 ~ C°'M>T
0.4 T.M AND T.M+T yz yz
0::M y
0.0
Lo;MAND a:,M+T z
-o. 4 .._ ___ ....... ______ .._ ____ ...._ ___ __,
0.0 0.125 0.25
j/L
0.375 0.5
12.0
9.0
6.0
3.0
0.0
Figure 14. Elastic Mechanical and Curing Adhesive Stresses of a Single Lap Joint with [90] Gr/Pi Adherends and Metlbond 1113 Adhesive
.......... ~ ~ .........
62
0.55r-------------------------------------,
z [90)
1---_;.,.;A:.......=-===~ -A [OJ
'------.--~t----.Y 8=0.0005 in.
0.45 3.0
0.35 ......... Cf)
-- 2.0 ~ M ........ Tyz en en 0.25 w a::
I-Cf)
M ay -0.15
0:M x
0.05
------ 0.0 0:M ----z
-0.05
0.0
Figure 15.
0.25 0.5 0.75 1.0 ~/L
Elastic Mechanical Jl/hesive Stresses of a Single Lap Joint witJ [OJ ancl [90] Gr/Pi Adherends and Metl bo11d 1113 Adhesive
0 Q_
2 ..........
0.1
0.08
0.06
1-Z · w ~ w u j0.04 Cl. ~ Cl
0.02
0.0
63
z l [90]
i..........;:(~O=J~-r-~--1~~•Y 8=0.0005in. L
i
Vrlt ----
Writ
-0.02.._ ______ _._ ______ __.i...... ______ . _________ ~ ______ _.
0.0 0.2 0.4 0.6 0.8
~/L
Figure 16. Relative Displacements of a Single Lap Joint with [OJ and [90] Gr/Pi Adherends and Metlbond 1113 Adhesive Normalized with Re5pect to the Adhesive Thickness
1.0
64
that the relative displacements vr (y-component) reach a much higher
value at t/L = 1 than at t/L = 0, while the wr displacements (z-com-
ponent) are nearly symmetric about the center of the joint (t/L = 0.5).
The normalized component v can be considered as a shear strain thus in-r . dicating why the shearing stresses are much larger near ~/L = l in Fig.
15. It is interesting that the wr component is nearly symmetric. This
indicates that bending is not a major factor. This corresponds to the
results seen in Fig. 11 where the curvature of adherend was nearly zero
in the overlap region. From Fig. 16 it is seen that the increased
stresses are due, almost entirely, to the increased flexibility (EY) of
the [90] adherend.
Returning to Fig. 15 a discontinuity in the cr~ stress distribution
can be observed at t/L = 0.2. This can be explained as a change in
adherend stiffness corresponding to a change in the finite element
representation of the adherend. Referring back to FiJ. 7, this change
in representation is seen as the point at which the adherend in the
model is changed from two, to one layer of elements. This situation was
unavoidable due to the large number of elements required to model the
adhesive layer and a limitation on the maximum number of elements
available. This change in stiffness is recognizable in many of the
stress distributions presented for adhesive bonded joints.
Curing stresses for the single lap joint with [O] and [90] ad-
herends are presented in Fig. 17. These stresses are, not surprisingly,
different than those for the joints presented earlier. For this case,
all of the components of stress induced by mechanical loading are
-(f) ~ ......... (J) (f) w a:: I-(f)
65
4.5 z - 30.0
[90] A A [Q] y
3.0 L SFT= 270° F 20.0 Ty'T.z l
0:T J(
1.5 10.0
a:T y
0.0 0.0
0:T z
-1.5 -10.0
-3.0 -20.0
-4.5 ..._ ___ ....... ________ .• '------4·-30.0 0.0 0.25 0.5 0.75 1.0
~/L
Figure 17. Elastic Curing Adhesive Stresses of a Single Lap Joint with [OJ and [90] Gr/Pi Adherends and Metlbond 1113 Adhesive
...-. 0
0... ~ .........
66
present due to cure. From observing the '~z and a~ distributions it can
be seen that the integrals of these stresses appear to be zero, as would
be required by equilibrium considerat1ons. It is interesting to note
the magnitude of the '~z stress peaks near t/L = 0 and t/L = 1. The
values of shearing cure stresses are nearly 75 percent of the ultimate
shear strength of the adhesive. Because the mechanical and curing
stresses have opposite signs near t/L = l it can be seen that curing
counteracts the large mechanical stresses in this region. It should be
mentioned, however, that the percent modulus retentions, used for de-
termining moduli as functions of temperature, are only strictly valid
through a limited portion of the stress-strain curve of a material.
Because of this, the peak shear cure stresses may not be quite as ac-
curate as the other components of cure stress which correspond to points
lower on their respective stress-strain curves.
5.5.2 Double Lap Joints with Adhesives
5.5.2.l [OJ Graphite-Polyimide Adherends
The adhesive stresses due to curing and mechanical loading for a
double lap joint with [OJ graphite-polyimide adherends and Metlbond 1113
adhesive are presented in Fig. 18. Here, as with the single lap joints,
the mechanically induced shearing stresses don:inate the mechanical
stresses. It is interesting to note the lack of symmetry present in
these stress distributions due to the restrictions upon the w displace-
ments along the midsurface of the inner adherend. These restrictions
are induced by the symmetry of joint about the midsurface of the inner
-CJ) ~ -CJ) CJ) w a:: I-CJ)
67
2.0 [Q] SFT=270°F
z - - - [OJ - - 12.0 A
1.6 (OJ A
8= 0.0005 in. L v.
9.0 I. 2 a;:M+T y
a:M+T 6.0 0.8
x
T.MAND -r,fl+ T yz yz
0.4 3.0
0:M x
-o. 4 "-----~-----l...---_ _.. ____ _, 0.0 0.25 0.5
R;L
0.75 1.0
-c Q_ ~ -
Figure 18. Elastic Mechanical and Curing Adhesive Stresses of a Double Lap Joint with [OJ Gr/Pi Adherends and Metlbond 1113 Adhesive
68
adherend. Comparing Fig's. 9 and 18 it can be seen that the mechanically
induced stresses are smaller for the double lap joint than for the
single lap joint. The distributions are more uniform for the double lap
joint indicating reduced bending in the adherends. This is also due to
the restrictions on the w displacements.
The curing stresses for this double lap joint follow the trends
exhibited by the single lap joints where the adherends were identical,
with oT and oT being the only significant curing stresses. As expected, . x y the magnitudes of the curi_ng stress compo11ents are i dent i ca 1 to those of
the single lap joint with [OJ adherends.
5.5.2.2 [90J Graphite-Polyimide Adh' rends
Fig. 19 represents the curing and mechanical loading induced stress-
es of a double lap joint with [90J adherends. Co1,1paring Fig's. 18 and
19 it is seen that the peak stress values decreas~ with increased flexi-
bility of the adherends, as was the case with sir gl e 1 ap joints. Nati ng
the curing stresses of both of these joints it c11n be seen that the
o~ cure stresses are higher for the [90J adherencl joint while the o~
cure stresses are higher for the [OJ adherend joint. This is exactly
the same as with the single lap joints and the reason for it is also the
same.
5.5.2.3 [OJ and [90] Graphite-Polyimide Adherends
Mechanically induced adhesive stresses of a double lap joint with
[OJ and [90] graphite-polyimide adherends and Metlbond 1113 adhesive are
represented in Fig. 20. As was the case with the two previous double
lap joints, comparisons made with a single lap of the same adherends
.......... Cf) ~ ......... Cf) Cf) w 0::: I-Cf)
69
1.5 .----------------.
1.2
0.9
z
,---- o:M+T x
(90) SFT=270°F
--[90J--A_ - - ~-A
(90) y 8=0.0005 in. L
~
0::M+T y
10.0
8.0
6.0
0.6 4.0~
0.3 2.0
Ty"'i AND Ty~+ T
0::M y a:M
0.0 0.0 a;_M AND ClzM+T
-o. 3 ..._ ___ ...__ __________ ....&, ____ __._2.0 0.0 0.25 0.5 0.75 1.0
9/L
Figure 19. Elastic Mechanical and Curing Adhesive Stresses of a Double Lap Joint with [90] Gr/Pi Adherends and Metlbond 1113 Adhesive
n.. ~ -
70
0.5
(OJ 3.0 z - -(90J - -0.4
A --A [Q] y
8 = 0.0005 in. ~L
.R. 0.3 2.0
,....., Cf) ~ ......... Cf) f3 0.2 a::
Ty~ ti 1.0
0. I
-0.1 0.0 0.25 0.5 0.75 1.0
Figure 20. Elastic Mechanical Adhesive Stresses of a Double Lap Joint with [OJ and [90] Gr/Pi Adherends and Metlbond 1113 Adhesive
,....., a
a_ ~ .........
71
(Fig. 15) shows lower peak stress values and a more uniform distribution
of stresses for the double lap joint. A striking similarity between
these two joints is the location of the peak stress values corresponding
to a line near t/L = 1.0. In the double lap joint, as in the single
lap, this is due to the increased relative displacement vr at the line
t/L = 1.0. It is interesting to note that the restrictions on the
displacements in the z direction in the double lap do not greatly de-
crease the relative difference in peak stresses near t/L = 0 and t/L =
1.0 in comparison to the single lap joint, indicating that the effects
of bending are also small for the double lap joint.
The curing stresses for this double lap joint are presented in Fig.
21. This figure shows that the cr~ curing stresses are very small while
the '~z stresses are approximately 90 percent of the ultimate shear
strength. Comparing these stresses ·to those for a single lap joint of
the same adherends (Fig. 17) indicates that for curing stresses, the
restrictions.upon displacements at the midsurface of the inner adherend
in the double lap joint are not necessarily hc~lpful. The double lap
joint produces higher shearing stresses, but ·1ower peel stresses than
those of the single lap joint. The only difference between the two
joints is the increased bending stiffness of the inner adherend in the
double lap. This indicates that bending stiffness is an important
factor in these curing stresses. This must also be the cause for
differences in the cr~ and cr~ curi.ng stresses of the two joints.
5.5.3 Single Lap Joints without Adhesives
......... Cf) ~ .._.. Cf) Cf) w a:: I-en
72
6.0
[OJ
z - -(90] - - 30.0 4.0 A ---A
[OJ y L i SFT=270°F
2.0 CJXT 15.0
O:T y
0.0 ~
0.0 ~ ~ .._..
Oi.T
-2.0 -15.0 -
T tyz
-4.0 - -30.0
-6.0 -------·--" ____ __. ___ _ 0.0 0.25 0.5
~/L
0.75 1.0
Figure 21. Elastic Curing Adhesive Stresses of a Double Lap Joint with [OJ and [90] Gr/Pi Adherends and Metlbond 1113 Adhesive
73
Before presenting the first set of results in this section it should
be pointed out once more exactly how the following stresses were
averaged (see section 5.2). For the case of stresses which must be
continuous across the interface between the adherends, the stresses pre-
sented are an average of the elemental stresses above and below this
interface. For stresses which need not be continuous, the results
presented consist of elemental stresses averaged either above or below
the interface, thus two crx and cry curves are presented for each joint.
This will also be the case for the stress results in the next section.
5.5.3.l [OJ Graphite-Polyimide Adherends
Fig. 22 presents the interfacial stresses of a single lap joint
with [OJ graphite-polyimide adherends and no adhesive. These stress
distributions are interesting in that the shear stress Tyz is not the
dominant stress as it was in the joints with adhesives. For this joint
it is seen that the cr~ upper and lower stres,;es are of greater magni-
tude and that cr~ attains higher peak values :han the shearing stresses.
For this joint T~z appears to be nearly unif )rm along the interface at
a relatively low value. Due to the strength of the [OJ laminates in
the fiber direction it would appear that failure would initiate as a
result of the peel stresses a~.
Curing stresses are not presented for t1is joint as they do not
exist. The finite element solution was also checked on this point with
the results being zero as required.
5.5.3.2 [90] Graphite-Polyimide Adherends
Mechanically induced interfacial stresses of a single lap joint
....... (/) ~ .._.. (/) (/) w 0:: I-(/)
74
2.5 z
(OJ
\ [OJ
2.0 8=0.0005in. L ~
oyMLOWER
1.5
I. 0 ayM UPPER
0.5
~~M Ty~
1 0.0 ~
-------- Cf"xM UPPER AND LOWER ,
-o. 5 .__ ___ __... ______________ ___.
0.0 0.25 0.5 ~/L
0.75 1.0
16.0
12.0
8.0
4.0
Figure 22. Elastic Mechanical Interfacial Stresses of a Single Lap Joint with [OJ Gr/Pi Adherends and no Adhesive
....... 0 a.. ~ ........
75
with [90J graphite-polyimide adherends are presented in Fig. 23. These
distributions show the same trends as the previous joint with the o~
upper and lower stresses being the largest, o~ reaching a relatively
high peak, and T~z showing a nearly smooth distribution. Comparing
Fig's. 22 and 23 it is seen that the larger stress values occur with the
stiffer adherends as was the case for all of the adhesive bonded joints
in previous sections.
5.5.3.3 [OJ and [90J Graphite-Polyimide Adherends
Fig. 24 is a plot of interfacial stre~ses for a joint with [OJ and
[90J graphite-polyimide adherends under mechanical loading. As with the
previous two joints, the internal stresses o~ upper and lower are the
largest in magnitude. It is interesting tc' note that for this joint the
peak stresses do not necessarily occur with the more flexible adherend,
as was true for adhesive bonded joints. For this joint o: and o~ lower have peak values near ~/L = 0 while ~~z and cr~ upper have peaks
near t/L = 1.0. Further examination of Fig. 24 indicates that the
internal stresses are highest in the direction of the fiber in these
unidirectional laminates. Thus o~ is larg=r for the [OJ adherend than
in the [90J adherend while o~ is larger for the [90J adherend.
Curing stresses for this joint are presented in Fig: 25. The
magnitudes of the o~ stresses in the [OJ adherend are nearly 60 percent
of the ultimate strength of laminate which represents a significant
curing stress. The a; curing stresses in the [90J adherend are also
large at 40 percent of ultimate. The other internal components of
76
0.5 ------------------.
a:,.MLOWER
-0.1
0.0 0.25 0.5
~/L
a;_M UPPER
0.75 LO
Figure 23. Elastic Mechanical Interfacial Stresses of a Single Lap Joint with [90] Gr/Pi Adherends and no Adhesive
......... Cf) ::::.:::: .......... Cf) Cf) w a:: I-(f)
77
1.0 z (90)
[OJ y 6.0 0.8 8= 0.0005 in. L
~
- 5.0 o::M y LOWER
0.6 4.0
0.4 3.0 ......... 0
()_
~ 2.0 ........
0.2 cr,M UPPER
I_ y
~ 1.0
0:M M ~MLOWER z Tyz
_-.·~ 0.0
a;_M UPPER -1.0 -Q2i...-~~~-'-~~~~-"-~~~_...~~~--
O.O 0.25 0.5
.R/L
0.75 I. 0
Figure 24. Elastic Mechanical Interfacial Stresses of a Single Lap Joint with [OJ and [90] Gr/Pi Adherends and no Adhesive
78
12.0 z [90]
[OJ y SFT=350°F 60.0
8.0 T - C-~ LOWER __,..,
J cr.l UPPER 30.0 4.0 :,...--' -
......... en ~ - T en Tyz en 0.0 0.0 w 0::: I- T en Tyz
-4.0 UPPER -30.0
-a.o -GO.O
-12. 0 L..-----'-----~-----'-----0.0 0.25 0.5
~/L
0.75 1.0
Figure 25. Elastic Curing Interfacial Stresses of a Single Lap Joint with [OJ and [90] Gr/Pi Adherends and no Adhesive
-0 0.. ::;? -
79
stress oT upper and oT lower are of comparable numerical value, but are x y in the fiber direction of their respective adherends and therefore do
not represent such significant percentages of the ultimate strengths.
This figure shows that equilibrium appears to be satisfied by the o~
and T~z curing stresses.
5.5.4 Double Lap Joints without Adhesives
·Fig's. 26 and 27 represent mechanically induced interfacial stress-
es fo~ double lap joints with [OJ and [90J graphite-polyimide adherends
respectively. These joints show higher st)~esses with the stiffer adher-
ends as with all other joints presented. It can be seen that with these
joints the a~ stresses are the largest as has been shown for all joints
without adhesives.
Comparing Fig's 22 and 26 it can be seen that the double lap joint
has higher stresses for the same displacemc~nt loading. This can only be
caused by the increased bending stiffness of the inner adherend in the
double lap joint. This trend is also presEnt in comparison of Fig's. 23
and 27 representing joints with [90J adherends. For joints with ad-
hesive layers the effect of the increased bending stiffness of the inner
adherend was to smooth and reduce slightly the stress distributions for
double lap joints in comparison to single ·1ap joints.
Fig's. 28 and 29 represent the mechanical and curing stresses
respectively for a double lap joint with [OJ and [90J graphite-polyimide
adherends. As with the single lap with [OJ and [90] adherends and no
adhesive, the peak stresses for mechanical loading do not occur with the
,...... Cf) ~ ......... Cf)
~ a::: I-Cf)
80
5.0 [OJ
z - -[OJ - -
4.0 tOJ y
8= 0.0005 in. L i
3.0 a:/11 LOWER
0::M y INNER
2.0
I. 0
0:M z
a;_M INNER -M '1 lyz
0.0
-1.0 ____ __....._ ___ ~ ____ ...._ ___ _
0.0 0.25 0.5· .2/L
0.75 I. 0
28.0
21.0
14.0c; 0... ~ .........
7.0
0.0
Figure 26. Elastic Mechanical Interfacial Stresses of a Double Lap Joint with [G] Gr/Pi Adherends and no Adhesive
-(/) ~ -(/) (/) w a:: I-(/)
0.5
0.4
0.3
0.2
0.1
-0.1
0.0
81
[90]
z - -[90J- -
[90] y 8 = 0.0005 in. L
>j
oyM LOWER
ayM INNER.
0.25
0:M __ z
0.5 ~/L
~-/
0.75
3.0
2.0
-c n.. ~ 1.0 -
1.0
Figure 27. Elastic Mechanical lnterfacial Stresses of a Double Lap Joint with [90] Gr/Pi Adherends and no Adhesive
82
5.0 [O]
- -(90J- -
4.0 [Q] i--------.--+--- y 28.0
8= 0.0005 in.
3.0 '4--- G°yMLOWER
L R
21.0
~ 2.0 14.0..--. w a::: ~
I. 0
0.0
7.0
M a;M INNER Tyz CTMLOWER x
0.0
-1. 0 0.0 0.25 0.5 0.75 1.0
~/L
Figure 28. Elastic Mechanical Interfacial ~,tresses of a Doub 1 e Lap Joint with [OJ and [~lO] Gr/Pi Adherend and no Adhesive
0 (L
~ -
-Cf) ~ ........ Cf) Cf) w a::: ~
83
9.0--------------------, 60 .0
6.0 40.0 oyT INNER
[Ol
- - [90J- -3.0 20.0 [OJ y
SFT=350°F
0.0 -"-- 0.0 ~ ~ ........
-3.0 -20.0
-6.0 -4C:O
0:,.T INNER
-9. 0 '------..&.-..----'-----~-----' -60. 0 0.0 0.25 0.5 0.75 1.0
Figure 29.
~/L
Elastic Curing Interfacial Str~sses of a Double Lap Joint with [OJ and [90] Gr/Pi Adherends and no Adhesive
84
more flexible adherend. This is ag<dn contrary to the results presented
for lap joints with adhesive layers. The curing stress components cr~
for the [O] adherend and o~ for the [90] adherend represent approxi-
mately 60 percent of ultimate strength for the laminates.
5.5.5 Elastic Loading Comparisons
After reviewing the stress distributions of the four previous sub-
sections, interesting comparisons can be made by considering the force
loading corresponding to the displacement applied for each of the joints.
The results can be seen in Table 4. In this table the force load for
double lap joints corresponds to the total load carried by the joint.
The forces for all of the joints are calculated by averaging stresses in
the elements at the ends of the adherends, m11ltiplying by the thickness
of the adherend, and assuming a unit depth.
Making comparisons of single and double lap joints with the same
adherends it can be seen that the double lap joints carry more than
twice as much force as the single lap joints. In the cases where the
joints have adhesives, this can be seen as a beneficial effect of the
increased bending stiffness of the inner adh1~rend, as the peak stresses
for double lap joints of this catf~gory are lower than for the single lap
joints. The increased load carrying capacity is due to the more uniform
shear stress distribution of the double lap joints, which creates a
larger resultant force opposing the load.
Comparisons of single and double lap joints without adhesives also
show a more than doubled load carrying capacity for the double lap
Type of Joint
Single
Single
Single
Single
Double
Double
Double
Single
Single
Single
Double
Double
Double
85
TABLE 4
Force-Displacement Result Comparisons for Elastic Joint Solutions
Adherends Adhesive Applied Bonded Displacement
[OJ, [OJ Yes 0.0005 in.
[90J' [90J Yes 0.0005 in.
[0/±45/90Js, Yes 0.0005 in. [0/±45/90J . s
[OJ, [90J Yes 0.0005 in.
[OJ, [OJ, [OJ Yes 0.0005 in.
[90J, [90J, [90J Yes 0.0005 in.
[OJ, [90J, [OJ Yes 0.0005 in.
[OJ, [OJ No 0.0005 in.
[90J' [90J No 0.0005 in.
[OJ, [90] No 0.0005 in.
[oJ, [oJ, ro1 No 0.0005 in.
[90J, [90J, [90] No 0.0005 in.
[OJ, [90J, [OJ No 0.005 in.
Corresponding Force
88.8 lb.
11.6 lb.
38. 9 lb.
20. 9 lb.
202. 0 lb.
24.4 lb.
44. l lb.
86. 3 lb.
10.9 lb.
20. 2 lb.
244.2 lb.
34. 7 lb.
159.0 lb.
86
joints. However, in this case, the peak stresses are higher for the
double lap joints making it unclear as to which joints are the more
efficient.
5.6 Nonlinear Results
Bonded joint stress distributions presented in this section were
predicted using the analysis of Chapter 3 and the nonlinear formulation
of Chapter 4. Mechanical and thermal loading was applied as a series of
increments. Where curing stresses are presented, they correspond to
the total curing load. Mechanically induced stresses are presented at
three load levels for each individual joint. The load levels for an
individual joint do not necessarily correspond to those of other joints.
The dimensions of the joints of this section are identical to those of
the elastic results (Fig. 1) except. where otherwise stated.
5.6.1 Single Lap Joints
5.6.1.l Lap Shear Test
The adhesive shear stress and strain distributions for a lap shear
test joint with aluminum adherends and Metlbond 1113 adhesive are shown 1 'A f ( I
in Fig's. 30 and 31 respectively. This joint corresponds to that used i)- ! j;.
in Ref. [20] for determining the adhesive shear properties as used in
this study. The dimensions of this joint are presented in Table 5. 11
·-:-1 '
These two figures (30 and 31) point to the 1~ffects of the nonlinear
shear behavior of the adhesive. Examination of Fig. 30 reveals that as
the displacement loading increases, the shEar stress distribution
becomes more uniform. Fig. 31, however, shows that the shear strain
87
TABLE 5
Dimensions for Lap Shear Joint and Single Lap Joint with [0/90/0/90/0] B/E Adherends
.Joint Overall Overlap Adherend Adhesive Length Length Thickness Thickness
(In) (In). (In) (In)
Thick Adherend Lap Shear Test 5. l 0.308 0. 1 £'.5 0.003 Joint
/' '
[0/90/0/90/0] Adherends 6.25 0.75 o. o:~6 0.005
88
6.0 z
A -
5.0 L 35.0 ~
4.0 \ 8=0.0064 in. 28.0
-Cf) ~ - 8=C.004 in. 21.0 3.0
;:EN .--. ~::io. 0
2.0
I. 0
0.0 0.125
8= C.002 in.
0.25 ~/L
0.375
14.0
7.0
0.5
Figure 30. Nonlinear Mechanical Adhesive Tyz Stresses of a Lap Shear Joint with Aluminum Adherends and Metlbond 1113 Adhesive
Q_ 2!
25.0
20.0
-~ 15.0 0 -::::IE N
~
10.0
5.0
0.0 0.0
Figure 31.
z
0.125
89
---8=0.0064 in.
Al A -- -A
Al y L l
8=0.004 in.
,.---8=0.002 in.
0.25
R/L
0.375 0.5
Nonlinear Mechanical Adhesive y Strains of a yz Lap Shear Joint with Alum~num Adherends and Metlbond 1113 Adhesive
90
distribution does not become more unfiorm as loading increases. This
can be explained by considering the shear stress-strain response of the
adhesive (Appendix C, Fig. C.6). The slope of this curve becomes much
smaller as the strain is increased. Therefore, in the bonded joint, the
increment of stress corresponding to an increment of strain, at high
strain levels, is smaller than at low strain levels. Thus, the shear
stress distribution in the center of the joint is increasing more
rapidly as loading increases than at the edges (t/L = 0 and t/L = 1),
because the strains are higher at the edges. It is interesting to note
that the largest shear stress presented (Fig. 30) is nearly a constant
value. This value corresponds to the ultimate shear strength of the
adhesive.
As was stated earlier, this joint corresponds to a joint used in
the lap shear tests [20]. Therefore, comparisons between numerical and
experimental work were made and the results can be found in Table 6 and
Fig. 32.
The experimental stresses and strains presented in Table 6 are
values corresponding the maximum stress of the adhesive shear curve.
These stress and strain values are chosen for the comparison because of
the nature of both the Ramberg-Osgood [17] approximations, and the
finite element analysis. The Ramberg-Osgood parameters cannot model the
stress-strain curve beyond the point at which the slope becomes zero and
the finite element formulation does not produce a positive definite
stiffness matrix when a negative modulus is used.
In Table 6, the numerical strain chosen for the comparison was near
91
TABLE 6
Comparison of Numerical and Experimental Results for Single Lap Shear Joint
Max Adhesive Max Adhesive Failure Load Shear Strain Shear Stress (kips)
(%) ( ksi)
Numerical 27 4.4 1.35
Experimental 29* 4.4* 1.36 [20]
Percent 7% 0% 0.7% Difference
* Corresponds to maximum stress value
-. Cf) ~ .......... Cf) Cf) w 0::: ~ 0::: <! w I Cf)
92
5.0 MAXIMUM STRESS 28.0
AVERAGE
4.0 F. E.
~ 21. 0 •
3.0 ~ F. E. STRAIN NEAR UL= O, ..--...
AVEF AGED STRESS 0 0.... ~
-14.0 .......... 2.0
I. 0 7.0
0.0 ._ ______ _. _________________ ~-------
0.0 10.0 20.0 30.0 t:.·0.0
SHEAR STRAIN (%)
Figure 32. Comparison of Shear Stress-Strain Response of Metlbond 1113 Adhesive
93
the line i/L = O. This is also the region where the strain was de-
termined in Ref. [20]. This table shows good correlation between ex-
perimental results and the numerical prediction. In Fig. 32 two finite
element stress-strain responses are presented. In both of these finite
element curves, the stress corresponds to an average of elemental
stresses throughout the adhesive layer.
The upper numerical curve (AVE.F.E.) presents strains that are also
aver~ged over the entire adhesive layer. The lo~er numerical curve
(F.E. STRAIN NEAR i/L = 0, AVERAGED STRESS) shows strains corresponding
to the finite elements adjacent to the line i/L = 0 in the adhesive.
The upper curve shows a better stress correlation while the lower curve
shows a better ultimate strain correspondence with experimental data.
Fig's. 33, 34, and 35 present the cr~, a~, and a~ adhesive stresses
of the same lap shear test joint under identical loadings. These
figures do not show the effects of adhesive nonlinearity in such a
pronounced fashion as Fig. 30. The reason for this is two fold. First,
the extensional stress-strain response of the adhesive is not as non-
1 inear as the shear response. Secondly, the extensional stress values
produced are not as large in magnitude relative to the ultimate strength.
The maximum crz and cry stresses correspond to approximately 15 percent of
the ultimate while the maximum ax component is approximately 10 percent
of ultimate. Upon examination, Fig's. 34 and 35 reveal the discon-
tinuity at ~/l = 0.2 that was discussed in the elastic results (section
5.5.1.4). Thermal stresses are not presented for this joint as they
would have no bearing on the comparisons made. This is because the
-(f) ~ -
:E 0
94
2.0
z A 12.0
Al y 1.6 8 L
9-
9.0 I. 2
8= 0.0064 in.
6.0 0.8
8= 0.004 In. -0 0.... ~ -0.4 3.0
8=0.002 in.
-o.4 L.-~~~~~~~~--~
0.0
Figure 33.
0.125 0.25
.R/L
0.375 0.5
Nonlinear Mechanical ~dhesive a Stresses of a z Lap Shear Joint with Aluminum Adherends and Metlbond 1113 Adhesive
-en ~ -b>-
95
1.0 8= 0. 0064 In.
6.0
0.75 8=0.004 in.
4.0
0.5
2.0 0.25
QO --~~~--"~~~~-'-~~~~--~·~~__. 0.0
0.0 0.125
'
0.25 VL
0.375 0.5
Figure 34. Nonlinear Mechanical Adhesive ay Stresses of a Lap Shear Joint with Alumi 1Um Adh2rends and Metlbond 1113 Adhesive
.......... 0
0.... ~ -
-(/) :::.:::
:::E b"'
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
0.0
Figure 35.
8
z
Al
0.125
96
A
9.
8=0.0064
8=0.004
(): 0.002
0.25
R!L
in.
in.
in.
Al A
y 6.0
4.5
3.0 .......... a
0.... ~ ..........
1.5
0.375 0.5
Nonlinear Mechanical Adhesive o Stresses of a x Lap Shear Joint with Aluminum Adherends and Metlbond 1113 Adhesive
97
shear properties produced from the experimental work on this joint [20]
included any effects of curing.
An interesting comparison can be made concerning results obtained
from the two different solution formulations presented in Chapter 3. In
order to produce similar stress and strain distributions, it was neces-
sary to apply a larger displacement load to the joint analyzed under the
quasi 3-dimensional formulation. The additional load corresponded to
1/15. of the total displacement applied for the 2-dimensional solution.
The resulting ultimate force loads were identical for the two analyses
indicating that under the quasi 3-dimensional formulation the joint was
more flexible and therefore capable of withstanding a larger displace-
ment load.
5.6.l.2 [0/90/0/90/0] Boron-Epoxy Adherends and
AF-126-2 Adhesive
This adhesive bonded joint was selected for analysis in order to
compare the results of this study to those of Ref. [11]. For this
comparison it was necessary to use a force loading instead of the dis-
placement loading used for all other joints in this study. The force
loading was required in order to exactly match the loading in [11]. In
[11], two solution procedures are used. In the first, an iterative
finite element analysis is used to account for material nonlinearities.
The second procedure utilizes direct numerical integration of the
governing differential equations for the joint. Results obtained by the
second procedure are labeled as theoretical in the following figures.
The dimensions of this joint are given in Table 5.
98
Fig's. 36 and 37 present the adhesive stress components <~z and
a~ respectively. In Fig. 36 it can be seen that the distributions of
the shear stresses predicted by the present study compare favorably with
the two distributions predicted in [11]. The principal difference is
seen to be the magnitude of the peak stresses near the lines t/L = 0 and
t/L = 1. It is interesting that none of the shear stress distri-
butions presented in Fig. 36 satisfy the stress free boundary condition
<yz ~ 0 at t/L = 1. It appears that the areas under each of the three
curves are approximately the same as required by equilibrium considera-
tions.
In Fig. 37 it can be seen that the cr~ distributions predicted by
the present study and theoretical results of [11] have significant
differences. The present analysis predicts a symmetric distribution of
peel stresses while the theoretical results [11] .show peak stresses at
t/L = 0 and t/L = 1 differing by more than 100 percent. The nonsym-
metric nature of these results appears physically inconsistent as the
adherends are identical. The finite element results presented in [11]
do show the symmetry of stresses as predicted by this study. Another
interesting aspect of the theoretical results is the reversal in sign of
the peel stresses near t/L = 1.0. This is not seen in the finite
element results of this study or [11].
Comparisons of the numerical values of the peel stresses cannot
realistically be made as the extensional properties of the AF-126-2
adhesive were not known. The extensional Ramberg-Osgood coefficients
used for this adhesive correspond to the extensional properties of
99
12.0 --------------------.
10.0
2.0
z [ 0/90/0/90/0J]
A 7~0 '---~--.---+-~-.y
P=2000 lbs
THEORETICAL CllJ
AND F. E. [ 11 l
~ / ~ ' ,/'l
/J 15.0
~~ .__ ________________________ oo 0.0
0.0
Figure 36.
0.25 0.5
~/L
0.75 I. 0
Nonlinear Mechanical Adhesive T Stresses of a yz Single Lap Joint with [0/90/0/90/0] B/E Adherends and AF-126-2 Adhesive
-(/) ~ -bN
100
25.0 z
160.0 A
y
20.0 P=2000 lbs ~
120.0
15.0
80.0
40.0
-5. 0 -------'-----~----....__ 0.0
Figure 37. Nonlinear Mechanical Adhesive a Stresses of a z Single Lap Joint with [0/90/0/90/0] B/E Adherends and AF-126-2 Adhesive
-c a... ~
101
Metlbond 1113.
Comparing Fig's. 36 and 37, it can be seen that for all solutions
except the finite element solution of [11], the peel stresses are
larger than the shear stresses. This verifies the statement made about
the effects of the overlap length, L, upon the magnitudes of the shear-
ing and peel stresses in an earlier section (5.5.1.1). For this joint,
with a large adherend overlap, the peel stresses are dominant with
resp~ct to maximum value. If the magnitudes were compared with respect
to ultimate strengths however, it is believed that the shearing stresses
would again dominate. It is not known what these ultimate strenths are
however, and therefore this comparison cannot be made.
The curing stresses for this joint are presented in Fig. 38. As
was shown for the elastic results, the only significant curing stresses
are a! and a;. This is again due to the identical adherends and the
material properties of the adherends and adhesive. It is interesting
that even though the values of the curing stress components a~ and a;z are insignificant, they reach peak values n£arly an order of magnitude
larger than in any other joint with identical adherends. Since these
curing components (a~ and T;z) are negligible, the stresses presented in
the previous two figures (36 and 37) can be considered either mechanical
or combined mechanical and curing stresses.
The magnitudes of the curing components a~ and a; with respect to
the ultimate extensional strength of the adhesive is not known because
as was stated earlier, this ultimate strength is not known. The analy-
ses of [11] ignore the effects of curing, and while this does not affect
-(/') ~
(/') (/') w 0::: I-(/')
102
2.0 o;T y
r "' 12.0
1.6 o;T x
z SFT= 270° F 9.0 I. 2
A ~y
~ 6.0 0.8
0.4 3.0
0.0 Ty1z
0.0
o:}
-0.4 0.0 0.25 0.5 0.75 I. 0
~/L
Figure 38. Nonlinear Curing Adhesive Stresses of a Single Lap Joint with [0/90/0/90] B/E Adherends and AF-126-2 Adhesive
-0 a.. ~ -
103
the az and Tyz stress components, it can be seen that curing is signi-
ficant in the two components of stress not presented in [11] (ax and
cry).
Fig's. 39 and 40 present the combined mechanical and curing stress-M+T M+T . es, cry and ax of this joint. These distributions reveal that the
curing stresses (Fig. 38) are of the same sign as the mechanically in-
duced stresses and therefore would be detrimental to the performance of
the joint under tensile loading.
5.6.l.3 [OJ Graphite-polyimide Adherends and Metlbond
1113 Adhesive
Nonlinear curing stresses for this joint are not plotted as the
significant components (a~ and a~) are uniform throughout the adhesive.
The numerical values for these stresses can be found in Table 7. This
table presents a comparison between.elastic and nonlinear results for
this joint. For these curing stresses, the elastic solution under-
estimates the a~ and a~ components by 10 percent and 5 percent respec-
tively.
The mechanical loading of this joint was analyzed utilizing the 2-
dimensional formulation (Chapter 3). This was done because for this
joint, the two formulations produce negligible differences in the stress M M M components cry, az, and Tyz' The similarity in results is due to the
relatively small difference in magnitudes of the adherend transverse
stiffness (Ex) and the adhesive extensional stiffness. Combined mech-
anically and curing induced adhesive stresses are presented in Fig's.
41, 42, and 43. As in the lap shear joint (Fig. 30) only the adhesive
-Cl) ~ -b:io.
104
SFT= 270° F
75.0
10.0
60.0
ao P= 2000.0 lbs
45.0 6.0 -c
a. ~ -
30.0 4.0 P = 1333.33 lbs
2.0 15.0
P= 666.661bs
0.0 _________ __.. ____ ..L.----~ 0.0
0.0
Figure 39.
0.25 0.5 j/L
0.75 1.0
Nonlinear Mechanical and Curinq Adhesive o . y Stresses of a Single Lap Joint with [0/90/0/90/0] B/E Adherends and AF-126-2 Adhesive
105
2.0 15.0
0.0 ___ ___, ___ ___._ ________ ...... 0.0
0.0
Figure 40.
0.25 0.5 l/L
0.75 1.0
Nonlinear Mechanical and Curing Adhesive a - x Stresses of a Single Lap Joint with [0/90/0/90/0] B/E Adherends and AF-126-2 Adhesive
106
TABLE 7
Comparison of Elastic and Nonlinear Adhesive Stresses for a Single Lap Joint with [OJ Gr/Pi Adherends and Metlbond 1113 Adhesive
Type of Curino Stresses Peak Mechanical Stresses SFT = 270°F (ksi) (ksi)
Analysis ax 0 y cry az T yz
Elastic 0.68 1.3 2.8 3.8 8.0
Non-linear 0.76 1.4 2.8 3.8 4.4
Difference 10% 5% 0% 0% 82%
,...... CJ) ::x::: -N ...,,..
3.0
2.0
1.0
0.0 0.0 0.125
107
0.25 .R/L
~ 8 = 0.003 in.
..__...___ _____
0.375
21.0 ,...... 0 a_ :E
14.0
7.0
0.5
Figure 41. Nonlinear Mechanical and Curing Adhesive oyz Stresses of a Single Lap Joint with [OJ Gr/Pi Adherends and Metlbond 1113 Adhesive
108
-1. 0 ___ __. ________ __.. ____ _
0.0 0.125 0.375 0.5
Figure 42. Nonlinear Mechanical and Curing Adhesive a z Stresses of a Single Lap Joint with [OJ Gr/Pi Adherends and Metlbond 1113 Adhesive
-CJ) :x:
b::..
109
z SFT= 270° F
5.0 35.0
4.0 28.0 8=0.0057 in.
8=0.0045 in.
8=0.003 3.0
in. 21.0
2.0 14.0
I. 0 7.0
_________________ ._ ___ __. 0.0 0.0 0.0
Figure 43.
0.125 0.25
~/L
0.375 0.5
Nonlinear Mechanical and Curing Adhesive o y Stresses of a Sinqle Lap Joint with [OJ Gr/Pi Adherends and Metlbond 1113 Adhesive
0 a... ~ -
110
shearing stresses for this joint (Fig. 41) exhibit nonlinear effects.
Comparisons between elastic and nonlinear mechanically induced stresses
are presented in Table 7. These comparisons show that the cr~ and
a~ stress components behave linearly throughout the range of mechanical
loading applied. The shearing stress is decidedly nonlinear though,
with the elastic solution predicting stresses 82 percent higher than the
nonlinear results indicate.
5.6.l.4 [±45]s Graphite-Polyimide Adherends and
Metlbond 1113 Adhesive
The nonlinear curing stresses for this joint are not plotted for
the same reasons as the previous joint. The numerical values of the two
significant curing stresses are presented in Table 8 which presents a
comparison of elastic and nonlinear results for this joint. This
comparison shows a six percent increase in the curing stresses for the
nonlinear analysis.
Combined mechanical and curing adhesive stresses for this joint are
shown in Fig's 44, 45, 46 and 47. Again, it is seen that nonlinear
behavior is present in only the shear stresses. It is unfortunate that
the maximum loading for this joint did not produce a peak shear stress
corresponding to the ultimate strength. This would have produced much
more pronounced nonlinear effects. Comparisons between elastic and
nonlinear mechanical adhesive stresses are also presented in Table 8.
It is seen that at the maximum load level attained, the nonlinear
results predict a peak shear stress 28 percent below the elastic re-
sults. The elastic mechanical results are determined as the first
111
TABLE 8
Comparison of Elastic and Nonlinear Adhesive Stresses for a Single Lap Joint with [±45]s Gr/Pi Adherends and Metlbond 1113 Adhesive
Type of Curing Stresses Peak Mechanical Stresses SFT = 270°F (ksi) (ksi)
Analysis ax cry ax cry az Tyz
Elastic 1.5 1.5 1.3 2.6 2.9 5. 1
Nonlinear 1.6 1.6 1. 3 2.6 2.9 4.0
l)ifference 6% 6% no;, r'Jo/o 0% 28%
4.25
3.75
0.75
0.0 0.0
z
(:1:45Js
0.125
112
SFT=270°F
L
0.25 ~/L
l
(:1:45Js A
y
0.375
30.0
25.0
5.0
0.5
Figure 44. Nonlinear Mechanical and Curinq Adhesive Tyz Stresses of a Single Lap Joint with [±45] Gr/Pi s Adherends and Metlbond 1113 Adhesive
113
3.75
z SFT=270°F 25.0
A (:1:45Js y
3.0 8 20.0
2.25 8 =0.013in. 15.0
8=0.009 In. - 8=0.005 in. en I. 5 10.0 -~ 0 - a.. 0 ~ -
0.75 5.0
0.0
-0.75 L..----L----..1...-----'------= -5.0 0.0 0.125 0.25 0.375 0.5
J; L
Figure 45. Nonlinear Mechanical and Curing Adhesive a z Stresses of a Single Lap Joint with [±45] Gr/Pi s Adherends and Metlbond 1113 Adhesive
114
4.0 ----8=0.013 in.
35.0
----- 8=0.009in.
c (f) ~ 3.0 -----8 = 0. 005 in. 28.0 ........ - 0 a..
~ -2.0 14.0
I. 0 7.0
_____ __.... ____ ......_ ____ ...__ ___ __. 0.0 0.0
0.0
Figure 46.
0.125 0.25 ~/L
0.375 0.5
Nonlinear Mechanical and Curing Adhesive o - y Stresses of a Single Lap Joint with [±45]s Gr/Pi Adherends and Metlbond 1113 Adhesive
-en ::.::: -b""
3.0
2.5
2.0
1.5
I. 0
0.5
0.0 0.0
z
0.125
115
·SFT=270° F
8=0.013
8=0.005
0.2.5 VL
0.375
20.0
16.0
12.0
-0 a... ~
8.0
4.0
0.0 0.5
Figure 47. Nonlinear Mechanical and Curing Adhesive ax Stresses of a Single Lap Joint with [±45]s Gr/Pi Adherends and Metlbond 1113 Adhesive
116
increment of the nonlinear results scaled to the maximum displacement
load. It is somewhat surprising that the nonlinear analysis does not
predict lower stress components a~, a~, and a~ for this and the previous
joint. It can be explained however by considering the final magnitudes
of these stress components. All are relatively low on the extensional
stress-strain response of the adhesive. At these stress levels the
curve is very linear resulting in very linear stress predictions.
5.6.2· Double Lap Joints
5.6.2.l Titanium Adherends Metlbond 1113 Adhesive
Comparisons of elastic and nonlinear curing stresses for this joint
are shown in Table 9. Once more, the two nonzero components of curing
stress (a~ and a~) were uniform and they are not plotted. Here again,
an increase in the nonlinear results over the elastic case is seen.
Combined stresses for this joint are presented in Fig's. 48, 49, 50
and 51. In these figures only the shear curves show pronounced non-
linearities, as before, but Table 9 indicates that the other stresses
(ax, ay, and az) are slightly reduced for the nonlinear analysis. The
reason this joint should exhibit nonlinear behavior where the previous
joints did not is not immediately discernable. The most obvious dif-
ference between this joint and the previous ones is the restriction
placed upon the w displacements at the midplane of the inner adherend by
the symmetry of the double lap.
5.6.2.2 [OJ and [90] Graphite-Polyimide Adherends and
Metlbond 1113 Adhesive
Curing stresses for this joint are presented in Fig. 52. These
117
TABLE 9
Comparison of Elastic and Nonlinear Adhesive Stresses for a Double Lap Joint with Ti Adherends and Metlbond 1113 Adhesive
Type of Curing Stresses SFT = 270°F (ksi)
Peak Mechanical Stresses (ksi)
Analysis ox O'y ax ay az Tyz
Elastic l.20 l.20 l.63 l.97 2.92 6.18
Nonlinear l. 25 1. 25 l. 58 l. 91 2.8 4.4
Difference 4% 4% 3% 3% 4% 40%
6.0
z
5.0 8
4.0
3.0 c en ~ -N
~
2.0
1.0
0.0 0.0
Figure 48.
118
[TiJ SFT=270° F
- - [Ti] --A A
[TiJ y 35.0
8=0.0065 In. 28.0
21.0
-
0.25
8=0.002
0.5 ~/L
In. 14.0
7.0
0.75 I. 0
Nonlinear Mechanical and Curing Adhesive r yz
0 0.. ::::!:
Stresses of a Double Lap Joint with Titanium Adherends and Metl bond 1113 Adhesive
4.0
3.0
2.0
-(/) 1.0 ~ -b ..
-2.0
0.0
z
8
Figure 49.
119
[TiJ SFT=270° F
-[Ti] - _L A
CTI]
8 =0.0065 in.
8 =0.004
8 =0.002
0.25
in.
In.
0.5 l/L
A y
0.75
21.0
14.0
7.0
0.0
-1.0
1.0
Nonlinear Mechanical and Curing Adhesive a z
-c Q_ ::!: -
Stresses of a Double Lap Joint with Titanium Adherends and Metlbond 1113 Adhesive
-Cf') ~ -b,..
120
6.0
5.0 SFT=270°F 35.0 CTIJ
z --CTIJ
A 4.0 CTiJ A 28.0 y
8
3.0 8 =0.0065 in. 21.0 0 a..
8 =0.004 in. ~
8=0.002 in.
2.0 14.0
1.0 7.0
____ _.... __________ ~---~ 0.0 0.0 0.0
Figure 50.
0.25 0.5 ~/L
0.75 1.0
Nonlinear Mechanical and Curing Adhesive a y Stresses of a Double Lap Joint with Titanium Adherends and Metlbond 1113 Adhesive
3.0
2.5 z
2.0
-en 1.5 '.::lr::: -t)
1.0
0.5
0.0 0.0
Figure 51.
[Ti]
[Til
0.25
121
SFT=270°F
A --CTIJ --
L .. J.
0.5 ~/L
A y
8=0.002 in.
8= 0.004 in.
8=0.0064 In.
0.75
16.0
12.0
-0 a.. ::2: -
- 8.0
4.0
1.0
Nonlinear Mechanical and Curinq Adhesive a ' x Stresses of a Double Lap Joint with Titanium Adherends and Metlbond 1113 Adhesive
-en :::.:::: -en en w a:: ~
122
6.0 [OJ SFT=270°F
z --[90]--
4.0 tOJ A y
L
~
2.0 c-T x
~ 0.0
0:T z
-2.0
T Ly;:
-4.0
-6.0 '------'------'----...._ ___ __, 0.0 0.25 0.5
~/L
0.75 1.0
30.0
15.0
0.0
-15.0
-30.0
Figure 52. Nonlinear uring Adhesive Stresses of a Double Lap lJoint with [OJ ancl [90] Gr/Pi Adherends and Metlbond 1113 Adhesive
-0 a.. 2 -
123
distributions appear very similar to the elastic curing stresses for the
same joint (Fig. 21). Comparisons between these two analyses are shown
in Table 10. It can be seen that the nonlinear curing stresses differ
from the elastic results between 2 percent and 6 percent. The 23
percent difference shown for crz near t/L = 0 is probably exagerated by
the low magnitudes of these stresses. It can be seen in Table 10 that
the nonlinear effects cause an increase in the cr~ and cr~ curing stresses
as shown in previous results. However, the a! and T~z stresses are
decreased by the nonlinear behavior. These two stresses are related
through equilibrium (section 5.5.1.1) and it is therefore appropriate
that both change in the same fashion. This decrease is due to
the high values of shear stress produced by curing and the low adhesive
modulus at these values.
Fig's. 53, 54, 55, and 56 present combined mechanical and curing
adhesive stresses for this joint. Comparing Fig's. 52 and 53 it can be
seen that the curing stresses are very beneficial to the performance of
this joint. In Fig. 53, as the displacement load level increases, the
shear stresses near t/L = 1 are seen to increase more rapidly than at
t/L = O. Thus, the shear stresses near t/L = 1 would be much larger
than the shear stress near t/L = 0 for mechanic.al loading only. Ho1 1-
ever, the curing shear stresses near 1/L = l have the opposite sign of
the mechanical shear stresses in this region and are of relatively large
magnitude. Therefore, a large portion of the mechanically induced shear
stresses are negated by the curing shear stress and thus the joint is
capable of carrying an increased load.
124
TABLE 10
Comparison of Elastic and Nonlinear Adhesive Curing Stresses for a Double Lap Joint with [OJ and [90] Gr/Pi Adherends
and Metlbond 1113 Adhesive
Curing Stresses SFT = 270° (ksi) Type of ax cry oz T yz Analysis t/L=O .e,/L=l t/L=O t/L=l .e,/L=O t/L=l t/L=O .e,/L=l
Elastic 1.47 1.03 1. 26 0.67 0.092 -0.53 3.6 -4.0
Nonlinear 1.53 1.08 1. 32 0. 71 o. 12 -0.51 3.53 -3.8
Difference 4% 5% 5% 6% 23% 4% 2% 5%
(/') ~ -
N ~>.
125
10.0 -------------------
8.0
6.0
4.0
2.0
-2.0 0.0
z
0.25
[OJ .
A [QJ
8=0.005in.
0.5 ~/L
SFT=270°F 60.0
- [90) .___ -A
y
45.0
30.0
15.0
0.75 I. 0
Figure 53. Nonlinear Mechanical C1;1d Curing Adhesive Tyz Stresses of a Double Laop Joint \'Jith [OJ 1113 Adhesive
-0 CL ~ -
0.0
-1.0 0.0 0.25
126
'---- 8=0.015in.
0.5
~/L
0.0
0.75 1.0
Figure 54. Nonlinear Mechanical and Curing Adhesive o2
Stresses of a Double Laop Joint with [OJ and [90] Gr/Pi Adherends and Metlbond 1113 Adhesive
5.0
z
4.0
,-...
en ~ 3.0 -0
2.0
I. 0
0.0
Figure 55.
[OJ
(0)
0.25
127
SFT=270°F
- - (90] --+-A A
8=0.015
8=0.01
8= 0.005
0.5 ~/L
y
0.75
35.0
28.0
21.0
14.0
7.0
0.0 I. 0
Nonlinear Mechanical and Curing Adhesive o y
,-... c Cl. ~ -
Stresses of a Double Laop Joint with [OJ and [90] Gr/Pi Adherends and t1etl bond 1113 Adhesive
-en ~ -t)
6.0
5.0
4.0
3.0
2.0
I. 0
0.0 0.0
z
Figure 56.
0.25
128
SFT = 270° F
[0]
- - (90)- -A A [OJ
8=0.015 in. -
8=0.01 in. -8 = 0.005 in. -
0.5
R;L
y
0.75
35.0
28.0
21.0
14.0
7.0
1.0
Nonlinear Mechanical and Curing Adhesive a x
-0 a_ ~ -
Stresses of a Double Laop Joint with [OJ and [90] Gr/Pi Adherends and Metlbond 1113 Adhesive
Chapter 6
SUMMARY AND CONCLUSIONS
The present analysis has been concerned with the nonlinear analysis
of bonded joints. Upon reviewing the results presented, the following
conclusions can be made with respect to the joint materials and geometries
studied.
1. The effects of adhesive nonlinearities greatly influence
the shear stress predictions in the adhesive layer of
bonded joints.
2. The effects of adhesive nonlinearities have little in-
fluence upon the normal stress components in the
adhesive layer of bonded joints.
3. Adherend nonlinear behavior has little effect upon
the adhesive stresses in bonded joints.
4. Residual curing stresses are significant in adhesive
bonded joints. These curing str1~sses are detrimental
in joints with similar adherends, but may be beneficial
in joints with differing adherends.
5. Residual curing stresses are significant in bonded
joints with differin~ adherends ~nd no adhesive.
6. Residual curing stresses are not significantly in-
fluenced by material nonlinearities or temperature
dependent properties.
7. Adherend stiffness has profound 1 ~ffects upon mechani-
cally induced stresses in bonded joints. Stresses
129
130
produced in the adhesive layer of adhesive bonded
joints and along the adherend-adherend interface in
bonded joints without adhesives are higher for
stiffer adherends. In adhesive bonded joints with
differing adherends, maximum adhesi"e stresses
correspond to the more flexible adhi~rend.
8. Adhesive and interfacial stresses a·e non-uniform
with maximum values produced near tie edges of the
overlap region.
9. Adhesive bonded double lap joints are more ef-
ficient single lap joints due to a more uniform
stress distribution in double lap joint.
10. A quasi 3-dimensional analysis predicts a more
flexible joint response than a 2-dimens onal formu-
lation. A larger displacement load is required in
a quasi 3-dimensio1al analysis to predict adhesive
stresses comparable to those of a 2-dimensional
analysis.
11. A quasi 3-dimensional analysis demonstrates the
effects of adherend transverse stiffness and thermal
coefficient of expansion upon the residual curing
stresses.
12. Adhesive stresses are not significantly influenced
by different syrrmetric loadings. Force loadings
produce results similar to displacement loadings.
131
13. The method of solution presented satisfies static
equilibrium very closely.
This analysis has shown that future areas of study might include
the following:
1. Analysis capability for out of plane bending and
warpage.
2. Better representation of stress-strain response
as a function of temperature, and moisture.
3. Nonlinear analysis of the effects of moisture in
bonded joints.
4. Inclusion of a capability to allow Poisson Ratios
to vary as a function of strain, temperature, and
moisture.
5. Allowing failure strengths· to vary as functions of
temperature and moisture.
6. Consistent modeling of the interactions of tempera-
ture and moisture.
B IBL rDGRAPHY
1. Renieri, G.D., Herakovich, C. T., "Nonlinear Analysis of Laminated Fibrous Composites," VPI&SU Report VPI-E-76-10, June, 1976.
2. Goland, M., Reissner, E., "The Stresses in Cemented Joints," J. Applied Mechanics, Vol. l, No. l, pp. 17-27, March, 1944.
3. Erodogan, F., Ratwani, M., "Stress Distribution in Bonded Joints," J. Composite Materials, Vol. 5, pp. 378-393, July, 1971.
4. Barker, R. M., Hatt, F., "Analysis of Bonded Joints in Vehicular Structures," VPI-E-73-16, 1973.
5. Sainsburg-Carter, J. B., "Automated Design of Bonded Joints, 11 ASME ·paper no. 72-WA/DE-13, July, 1972.
6. Wah, T., "Stress Distribution in a Bonded Anisotropic Lap Joint, 11
J. Engineering Materials and Technology, Vol. 95, pp. 174-181, July, 1973.
7. Hart-Smith, L. J., "Analysis and Design o-,· Advanced Composite Bonded Joints, 11 NASA CR 2218, January, 1973.
8. Sharpe, W. N., Jr., Muha, T. J., 11 fhe Comrarison of Experimental and Theoretical Shear Stress in the Adhesive layer of a Lap Joint Model, 11
Proceedings of the Army Symposium on Solie Mechanics, 1974: The Role of Mechanics in Design of Structural Joints," AMMRC MS 74-8, September, 1974.
9. Renton, J. W., Vinson, J. R., 11 0n the Behcvior of Bonded Joints in Composite Material Structures, 11 Engineerirg Fracture Mechanics, Vol. 7, pp. 41-60, 1975.
10. Renton, W. J., Vinson, J. R., "The Efficient Design of Adhesive Bonded Joints," AIAA, ASME, SAE, 16th Structures, Structural Dynamics, and Materials Conference, AIAA paper no. 75-798, May, 1975.
11. Grimes, G. C., Greimann, L. F., Wah, T., Commerford, G. E., Blackstone, W. R., Wolfe, G. K., "The Devt'lopment of Nori linear Analysis Methods for Bonded Joints in Advanced filament, ry Com-posite Structures," AFFDL-TR-72-97, Septen1ber, 1972.
12. DasGutpa, S., Sharma, S. P., "Stresses in an Adhesive Lap Joint," ASME paper no. 75-WA/DE-18, July, 1975.
13. Renton, W. J., 11 The Symmetric Lap Shear Test - What Good is it?", Esperimental Mechanics, pp. 409-415, November, 1975.
132
133
14. Wetherhold, R., Vinson, J. R., "An Analytical Model for Bonded Joint Analysis in Composite Structures Including Hygrothermal Effects," Fourth National Conference on Composite Materials: Testing and Design, May, 1976.
15. Birchfield, E. B., Cole, R. T., Impel"lizzeri, L. f., "Reliability of Step-Lap Bonded Joints," AFFDL-TR-72-26, April, 1975.
16. Wilson, E. L., Bathe, K., Doherty, W. P., "Direct Solution of Large Systems of Linear Equations," Computers and Structures," Vol. 4, pp. 363-372, 1974.
17. Ramberg, W., Osgood, W. B., "Description of Stress-Strain Curves by Three Parameters," NASA TN 902, 1943.
18. ·Bergner, H. w., Davis, J. G., Herakovich, C. T., "Analysis of Shear Test Methods for Composite Laminates," VPl-E-77-14, April, 1977.
19. Daniel, I. M., Liber, T., "Lamination Residual Stresses in Fiber Composites," NASA CR-134826, May, 1975.
20. Sancaktar, E., "Shear Behavior of a Viscoelastic Structural Adhesive," present work on Ph.D. diss:!rtation.
21. Renieri, M. P., Herakovich, C. T., Brinson, H. F., "Rate and Time Dependent Behavior of Structural Adhe;ives, 11 VPI-E-76-7, April, 1976.
22. Sessler, J. G., Weiss, V., "Aerospace Structural Metals Handbook," AFML-TR-68-115, January, 1968.
23. Gibbs, H. H., "Status Report: NR-150 Polyimide Binders and Ad-hesives," Proceedings SAMPE Conference, Spring 1976.
24. Petit, P. H. Waddoups, M. E., "A Method for Predicting the Nonlinear Behavior of Laminated Composites," J. Composite Materials, Vol. 3, 1969, 2-19.
25. Kaminski, B. E., Lemon, G. H., McKague, E. L., "Development of Engineering Data for Advanced Composite Materials," AFML-TR-70-108, 1972.
26. Viswanithan, C. N., Davis, J. G., Herakovich, C. T., "Tensile and Compressive Behavior of Borsic/Aluminum Composite Laminates," VPI-E-75-12, June, 1975.
27. Marceau, A., Scardino, W., "Durability of Adhesive Bonded Joints," AFML-TR-75-3, February, 1975.
28. Sturgeon, J. B., "Tensile Tests on Lap Joints in Carbon Ffore Reinforced Plastics," RAE-TR-70159, August, 1970.
134
29. Pipes, R. B., Vinson, J. R., Chou, T., "On the Hygrothermal Response of Laminated Composite Systems," J. Composite Materials, Vol. 10, pp. 129-148, April, 1976.
30. Verette, R. M., "Temperature/Humidity Effects on the Strength of· Graphite/Epoxy Laminates," AIAA Paper No. 75-1011, August, 1975.
31. Whitney, J. B., Ashton, J. E., "Effect of Environment on the Elastic Response of Layered Composite Plates," AIAA Journal, Vol. 9, No. 9, September, 1971.
32. Grimes, G. C., Greimann, L. F., "Analysis of Discontinuities, Edge Effects, and Joints," Composite Materials, Vol. 8, Part II, pp. 135-230' 1975.
APPENDIX A
ELEMENTAL STIFFNESS MATRIX
Equations (A.l) represent the equilibrium equations for applied
strain loading. Equ's (A.2) represent the equilibrium equations in
average force loadings. In these equations, [K] is the symmetric
elemental stiffness matrix, ~x{S} and {T} are force vectors correspond-
ing to the applied strain and temperature change respectively, {F} is
the vector of applied forces, and {x} is the vector of unknown nodal
displacements.
[K](i){X}(i) + ~ {S}(i) = x
(9x9) (9xl) (9xl)
{F}(i)
(9xl)
[K](i){x}(i) - {T}(i) = {F}(i)
(lOxlO) (lOxl) (lOxl) (lOxl)
Defining the following terms
a = (Z2-z3)/2
b = (Y3-Y2)/2
c = (z3-z1)/2
d = (Y1-Y3)/2
e = (z1-z2)/2
g = (Y2-Y1)/2
135
(A. l)
(A.2)
136
At = the area of element {t) * F = average normal force
where Y1 through v3 and z1 through z3 are the coordinates of the nodal
points of element t in the Y-Z plane, the element of the matricies of
Equ. (A.1) can be defined as follows.
- 2 - 2 t K11 = (C55b + c66a )/A
K12 = (c55 bd + c66ac)/At
- - t K15 = (c26ca + c45bd)/A
- - t K16 = (c26ea + c45 bg)/A
- - t K17 = (c36ba + c45ba)/A
- - t K18 = (c36da + c45bc)/A
- - t K19 = (c36ga + c45be)/A
K33 = (- 2 - 2 t c55g + c66e )/A
K34 = - - t (C2£iae + c45gb)/A
K35 = - - t (c26ce + c45 bd)/A
K36 = (C26e 2 + C 92)/At 45 - - t K37 = (c36be + c45ga)/A
- 2 - 2 t K22 = (C55d + c66c )/A
K23 = (E55dg + E66ce)/At - - t K24 = (c26ac + c45db)/~ - ? - 2 t
K25 = (C26c + C45d )/~
- - t (c26ec + c45dg)/A
- - t (c36bc + c45da)/A
K44 = (C 2 - 2 t 22a + C44b )/A
K45 = - - t (c22ac + c44bd)/A
K46 = - - t (c.22ae + c44bg)/A
- - t K47 = (c44ba + c23ab)/A
K48 = (e44bc + c23ad)/At
137
- - t K38 = (c36de + c45gc)/A - - i K49 = (c44be + c23ag)/A
K39 = - - i (c36ge + c45ge)/A
K55 = - 2 - 2 t (c22c + c44d )/A K66 = - 2 - 2 i (c22e + c44g )/A
K56 = - - i (c22ce + c44dg)/A K67 = - - i (c44ga + c23eb)/A
K57 = - - i (c44da + c23cb)/A K68 = - - t (C449C + c23eg)/A
K58 = - - t (c44dc + c23cd)/A K69 = - - i (c44ge + c23eg)/A
K59 = - - i (c44de + c23cg)/A
K77 = - 2 - 2 i (C33b + c44a )/A K88 = - 2 - 2 i (c33d + c44c )/A
K78 = - - i (c33bd + c44ac)/A K89 = - - i (c33dg + c44ce)/A
K79 = - - t (C33bg + c44ae)/A Kgg = - 2 - 2 t (c33g + c44e )/A
s, = c16 a s2 = c16 c s3 = c16 e
- - -S4 = c12 a s5 = c12 c s6 = c12 e
- -s7 = c13 b s8 = c13 d s9
= c13 g
x, = u 1 x = 2 u2 X3 = U3
X4 = Vl X5 = V2 x6 = v3
x7 = Wl x8 = w2 Xg = w3
138
F = fl F = f 2 - 3 1 x 2 x F3 - f x
- 1 F = i F = f3 F4 - fy 5 y 3 y
F = fl 7 z F = f 2 8 z F = f3 8 z
where f's are nodal forces.
For Equ's. (A.2) the previously defined terms apply plus the fol-
lowing additional terms
K710 = Cl3b - R,
KlOlO = CllA
F = F* 10
139
where
T £ = x 2 (m a1
2 + n a2 )~T
T 2 2 £ = (n a1 + m a2 )~T y
T £z = a3~T
For moisture analysis the vector {T} is identical except a 1, a 2 and a 3 are replaced by a1, a2 and a3.
APPENDlX B ADDENDA TO RFERENCE [l]
During the process of modifying the program developed by Renieri
and Herakovich [l] it was discovered that a coding error had been made
in the equations for the transformation matrices. (Chap. 3, Equ's.
3.9.) This was corrected and the nonlinear stress-strain response pre-
dictions of Ref. [l] were regenerated using the same data and finite
element model. In the majority of cases the differences proved to be
negligible. For the cases where the differences were significant,
comparisons between results obtained from the corrected program and
experiment are presented. It should be noted that the coding errors
applied only to the nonlinear results presentl~d in Ref. [l] and not the
elastic results. It should also be noted tha~ no attempt to define
failure was made by this investigator and that the last point plotted
does not necessarily correspond to failure.
140
141
-120
I I
I
-100 I I
I j~
I +20 L -20 ry -ao r-b .. , -u;
~
b" b =0.25
.. ho=0,005 Cf) Cf)
-60 w 0:: I-
'en ---- EXPERIMENTAL 1241
-40 PRESENT ANALYSIS SFT=270°F
-20
0 0 -0.2 -0.4 -0.6 -0.8 -1.0
STRAIN, ~ x (0/o)
Figure B.l. Compression Stress-Strain Behavior of [±20] . s Boron/Epoxy Laminate-Sandwich Beam Data
120
100
80
z
b: 0.25 in. ho: 0.005 in.
142
b" .. 60 (/) (/) w 0:: ..... en 40
- -- -- EXPERIMENTAL [24]
20
--- PRESENT ANALYSIS SFT=270°F
0 ""------'-----......1'------.L------'-------.a-----.-0 0.25 0.5 0.75 1.0 1.25 15
Figure 8.2. Tensile Stress-Strain Behavior of [±30]s B/E Laminate-Sandwich Beam Data
143
-so 'z
+30 j_ -50 r--b3~ry
--.. (/) (/) w a:: I- -30 (/)
-20
-10
0 0
b =0.25in. ho= 0.005 in.
'I r;
I/
I 'I
'/
'I ---- EXPERIMENTAL [24] 'I
-0.2
PRESENT ANALYSIS SFT=270°F
-0.4 -0.6 -0.8 STRAIN,~x (0/o)
Figure B.3 Compressive Stress-Strain Behavior of [±30]s B/E Laminate-Sandwich Beam Data
-1.0
30
25
-:::::-20 ~· ...........
10
5
. z +45 I i-------1-L
144
SANDWICH BEAM --DATA
--~-
-45 ho /
~b~ry/// COUPON DATA b =0.25 in. ho = 0.005 in. ~
~
~ )/
EXPERIMENTAL [25]
PRESENT ANALYSIS SFT= 270°F
0 ________________ _. _________________________ .......
0 0.4 0.8 1.2 1.6 2.0
STRAIN, 'x (0/o)
Figure B.4. Tensile Stress-Strain B~havior of [±45] B/E Laminate s
-(/) ~ '-"'·
blC .. (/) (/) w a:: ~
145
25
I~ z
20
+60 l_ ---/ -~~~ry
,,,.,.
15
b =0.25in . ho=0.005 in. /
10 ---- EXPERIMENT AL [24)
-- PRESENT ANALYSIS 5 SFT=270°F
0 ---------a.---------'------__. ..... ________ ..... ______ __ 0 0.2 0.4 0.6 0.8
Figure B.5. Tensile Stress-Strain Behavior of [±60]s B/E Laminate-Sandwich Beam Data
1.0
-so
-50
-40
boc .. (f) -30 (f) w cc: t-(f)
-20
-10
0
I I
0 -1.0
Figure B.6.
I
I I
I
I
/
/ /
/
146
/ /
/
/ /
/ /
/ /
/ /
/
z
t60 l I -so ~y
t--b-1t b =0.25 in. ho= 0. 005 in.
- - - -- EXPERIMENTAL
--- PRESENT ANALYSIS SFT= 270° F
-2.0 -3.0 -4.0 -5.0
Compressive Stress-Strain Behavior of [±60] B/E Laminate-Sandwich Beam Data s
,,...... (/).
~ -blC .. (/) (/) w a:: I-(/)
147
120
100 (
~ _L ho y " 80 rb-ir /
/ /
/ b =0.375 in. " / ho = 0.0075 In. /
60 / COUPON /
DATA / /
/ /
40 ------ EXPERIMENTAL [26]
PRE:5ENT ANALYSIS SFT= 430°F
20
0 ._ ______ ..... .._ _______ ..... ________ ..._ _________ '*-______ _...
0 0.2 0.4 0.6 0.8
STRAIN, ex (0/o)
Figure B.7. Tensile Stress~Strain Behavior of [0/±45]5 Bs/Al Laminate
LO
APPENDIX C
MATERIAL PROPERTIES
This appendix contains all of the material properties used for this
study. The data presented represents typical data from the literature.
References are provided where appropriate. Fig's. C.1 through C.7 repre-
sent the stress-strain response of the materials used. Table C.l con-
tains the Ramberg-Osgood coefficients for these materials and Table C.2
contains the temperature dependent properties.
In Table C.l the symbol a*.refers to the stress at which the Ram-
berg-Osgood coefficients n2 and k2 become applicable.
148
149
180.0 1200.0
150.0 ·1000.0
~--- CJ; - E'x TENS ION
. 120.0 800.0
(/) (/) w 0:: ~
,,__ ___ G;- Ex COMPRESSION
90.0
60.0
CJY- Ey TENSION 30.0 _v-o;- Ey COMPRESSION
'/ CSHEAR
600.0
-0 a.. ~
400.0
200.0
0.0 -~~
~-----'- -------' 0. 0 0.0 2.5 5.0 7.5 10.0
STRAIN (0/o)
Figure C. 1. Stress-Strain Response of Unidirectional Gr/Pi
150
36.0 ~----------------240.0
30.0 200.0
24.0 160.0
-U) ~
U) 18.0 CT.- Ex U) 120.0 -w c a:: a_
t; :! -12.0 80.0
6.0 40.0
0.0 _____ _.__ ____ -L----L-------' 0.0
0.0 1.25 2.5 3.75 5.0 STRAIN (0/o)
Figure C.2. Stress-Strain Response of [ 1:45]s Laminate Gr/Pi
-en ~ -en en w a:: t;
151
75.0
CJ;- Ex TENSION----...,
60.0
/
/ 45.0
o;- Ex COM.:>RESSION
30.0
15.0
0. 0 M..-----L------'-----~ ---~ 0.0 0.3 0.6
STRAIN (0/o) 0.9 1.2
600.0
500.0
400.0
300.0
-c a_ ~ -
200.0
100.0
Figure C.3. Stress-Strain Response of [0/±45/90]s Laminate Gr/Pi
~
en ::::.::: -en en w a::: I-en
152
360.0-------------------.. 2400.0
300.0
. 240.0
180.0
120.0
60.0
- 2000.0
- 1600.0 C1;-Ex
1200.0
~ 800.0
SHEAR~-400.0
~---~----------'------_J 0.0 0.0 2.0 4.0
STRAIN (0/o)
6.0 8.0
Figure C.4. Stress-Strain Response of Unidirectional B/E Ref. [15]
-0 a.. ~ -
-en ~
en en w a:: r-en
153
150.0 r------------------
125.0 875.0
100.0 o;- Ex 700.0
75.0 I 525.0
/ l ' j
50.0 -i 350.0
Cly-Ey I
25.0 - 175.0
0.0 J 0.0 0.2 0.4 0.6 0.8
STRtllN (0/o)
Figure C.5. Stress-Strai~ Response of [0/90/0/90/0] Laminate B/E
-c a.. ~ -
~ en ::::c:: -en en IJJ er: I-CJ)
12·0
10·0
8·0
6·0
4·0
2·0
O·O O·O
154
EXTENSIONAL
'== 6.55x1cl%1sEc
REF. (21]
10·0
-----·-SHEAR
y= 6.88 x 102%/SEC
REF. [20J
20·0 STRAIN (0/o)
30·0
75·0
60·0
45·0
30·0
15·0
O·O 40·0
Figure C.6. Stress-Strain Response of Metlbond 1113 Adhe~ive
-c a.. ~
I
155
15. 0 ,.---------------------100.0
12.5 / EXTENSIONAL---.. ----- 80.0
. 10.0
(/) ~ -(/) (/) w c:: I-(/)
7.5
5.0
2.5
0.0 0.0 10.0
SHEAR
REF. [15]
20.0 30.0
STRAIN (%)
60.0
40.0
20.0
0.0 40.0
Figure C.7. Stress-Strain Response of AF-126-2 Adhesive
-c a.. ~ -
. Elastic r- Curve ~·:odul us .µ \)
It! (MSI) :E:
Exx 19.0 .21 Tension (vxv) Exx 19. 0 .21 Comp. (vxy) Eyy 2.2 .21 Tension (vyz)
...... Eyy .21 c.. 1.8 (vyz) ........ Comp . c::: (!} Ezz . 21 r- Tension 2.2 (vxz) It! s:: Ezz 1.8 .21 0 (vxz) •r- Comp .µ u Q) Gyz 0.85 s... •r-"O
Gxz 0.85 •r-s::
:::::>
Gxy 0.85
TABLE C.1 Ramberg-Osgood Coefficients
Elastic K1 (PSI-nl) Limit * n1 a
( KSI) (KSI)
38.0 9.593 1. 1087 x 10-51 16.8
0.872 l. 253 l. 9509 x 10-8 16.8
0.872 1.253 l. 9509 x 10-8 16.8
0. l 3 .411 2.7474 x 10-16 12.6
0. 1 3.411 2.7474 x lo-16 12.6
0. l 3.411 2.7474 x 10-16 12.6
n2 K2 (PSI-n2)
2. 0661 7.181 x 10-12
2.0661 7. 181 x 10-12
5.316 4.254 x 10-24
5.316 4.254 x 10-24
5.316 4.254 x 10-24
Ultimate Stress (KSI) 176.0
100.0
9.7
21.8
9.7
21.8
15.0
15.0
15.0
__, c.n °'
. r- Elastic +-> Curve Modulus v rtJ ~ (MSI)
Exx 2.95 .75 Tension (vxy) Exx 2.95 .75 Comp. (vxy) Eyy Tension 2.95 . 21
(vyz) ;;::'. . Eyy ~ Comp. 2.95 .21
(vyz) t!l Ezz .21
(/) Tension 2.2 (vxz) ,..., LO Ezz .21 """ 1.8 Como. (vxz) 8
Gyz 0.85
Gxz 0.85
Gxy 5 .1
TABLE C.l Cont.
Elastic K1 (PSI-n1) Limit n1
(KSI) 1.2 4.064 1. 856 x 10-20
1.2 4.064 1. 856 x 10-20
1.2 4.064 1. 856 x 10-20
1.2 4.064 1. 856 x 10-20
1 .253 1. 9509 x 10-8
0. l 3.411 2.7474 x 10-16
0 .1 3. 411 2.7474 x 10-16
* K2 (PSCn2) CJ n2 (KS!)
16.8 2.0661 7. 181 x 10-12
12.6 5.3158 4. 254 x 10-24
12.6 5.3158 4.254 x io-24
Ultimate Stress (KSI) 30. 1
30. 1
30. 1
30. 1
9.7
21.8
15.0
15.0
15.0
__, (.11 ........
. Elastic Elastic ,.... Curve Modulus Limit +l \)
'° (MSI) ( KSI) :::E
Exx 7.75 .30 2.0 Tension (vxy) Exx 7.75 .30 2 .16 Como. {vxy) Eyy . 7.75 .21 2.0 ...... (v.vz) c... Tension ......... Eyy .21 0:::
C.!:l Comp. 7.75 ( v yz) 2. 16 VI
r-i Ezz 2.2 ( . 21 ) C)
°' Tension vxz ......... LO Ezz 1.8 .21 0.87 o::t" (vxz) +I Como. ......... C)
Gyz 0.85 0. 1 L......I
Gxz 0.85 0. 1
Gxy 2.97
TABLE C. l Cont.
K1 (PSI-nl) * n1 (J
(KSI)
2.422 5. 332 x 10-16
5.538 2. 243 x 10-29 44.7
2.422 5. 332 x 10-16
5.538 2.243 x 10-29 44.7
1.253 1 . 9509 x 10-8 16. 8
3.411 2.747 x 10-16 12.6
3.411 2.747 x 10-16 12.6
n2 K2 (PSI-n2)
3.904 8. 872 x 10-29
3.904 8.872 x 10-29
2.0661 7.181 x 10-12
5.316 4. 254 x 10-24
5.316 4. 254 x 10-24
Ultimate Stress (KSI)
70.6
55.6
70.6
55.6
9.7
21.8
15.0
15.0
15.0
__. (.11 CX>
. Elastic r- Curve Modulus +' \) ttS (MS!) :E
Exx 29.6 .21 Tension ( Vxy) Exx 29.6 .53 Comp. ( Vxy)
* Eyy .35 L.LJ 2.75 ( Vyz) ........ Tension co r- Eyy 2.75 .35 ttS Comp. ( Vyz) c 0 Ezz .35 .,.... 2.75 +' Tension (vxz) u Cl> Ezz .35 s.. 2. 75 .,.... Comp. (vxz) "O ...... c Gyz 0.933 ::J
Gxz 0.933
Gxy 0.933
* Ref [11].
TABLE. C.l Cont.
Elastic Limit K1 (Psi-n1) n1 (KS!)
1.0 4.463 1. 8534 x 1 o-27
1.0 4.463 1.8534 x lo-27
1.0 2.541 8.1604 x 10-14
1.0 2 .541 8. 1604 x 10-14
1.0 2 .541 8. 1604 x 10- 14
1.0 2.541 8.1604 x 10-14
0 .1 2.991 7.8797 x 10-15
0. 1 2.991 7 .8797 x 10-15
0 .1 2.991 7 .8797 x 10-15
a* K2 (Psr-n2) n2 (KS!)
Ult imate 1
Stress (KS!)
200.5
350.0
9.6
45.0
9.6
45.0
9.5
9.5
9.5
_. (J"I \0
. Elastic ,...... Curve Modulus +-> \) tO (MSI) ~
Exx 18. 9 .031 Tension ( Vxy) Exx 18. 9 .108 Comr:J. (\iv") '
' • '•J '
Eyy 13. 5 .35 Ll.J Tension ( Vyz) ......... Eyy .35 co 13.5 ,....., Comp. ( Vyz) 0
Ezz .35 ' ......... 0 2.75 °' Tension C.1xz1 ......... 0 Ezz 2.75 .35 ......... (vxz) 0 Comp. O'I ......... 0 Gyz 0.933 L.....J
Gxz 0.933
Gxy 0.933
TABLE C.l Cont.
Elastic Limit K1 (PSCnl) nl (KSI)
5 .1 e 3.351 6.0822 x 10-21
5. 16 3.351 6.0822 x 10-21
5.44 3.05 6.7331 x 10-19
5.44 3.05 6.7731 x lo-19
l. 0 2.45 8. 1604 x 10-14
1.0 2.45 8. 1604 x 10- l 4
0. l 2.991 7.8797 x 10-15
0. l 2.991 7. 8797 x 10-15
0. l 2.991 7.8797 x 10-15
K2 (PSI-n2) * rJ n2 (KSI)
Ultimate Stress ( KSI)
129.0
129.0
90.7
90.7
9.6
45.0
9.5
9.5
9.5
O'I 0
. Elastic ,...... Curve Modulus .µ <tl (MSI) ::E
-0 Exten. * 0. 325 c:: 0 (Y')
..0,...... ,...... ,...... .µ,......
Shear** QJ 0.0363 ::E
N Exten. 0.455 I l.O N ,......
I Shear*** 0.175 u.. ex: E .~ ~IExten. ~ _1_~_ <tl ,...... +J c:x:: Shear + 6.0 .,.... l.O I-
E Ex ten. + 10.6 ::iv c:: I-
•r- I
E o::::i-::JN Shear + 4.0 r--0
CX:: N
*Ref. [21].
** Ref. [20].
***Ref. [11].
TABLE C. 1 Cont.
Elastic K2 (PSI-n2)
Ultimate Limit K1 {PSI-nl) * Stress \) ni (J n2 (KSI) (KSI) (KSI)
.366 l.58 7.907 2. 039 x 10-33 7.902
l.32 10.45 1. 032 x 10-39 4.14 16.67 3. 162 x 10-62 4.39
.30 1.58 7.907 2. 039 x 10-33 12.0
o. 1 2.684 2. 88 x 10-12 13.0 ~ __,
.34 130.0 --=r- 68.9
.33 44.0
25.0
+Ref. [22].
r-- Temp. % +'
'° OF Exx :E:
0.0 100.0 ,...... '° c: 60.0 0
.,.... 1--t +' 0.. u-...... 400.0 Q) 0:: s... '-" .,.... -c 500.0 .,.... c:
:::> 550.0 115 .0
0.0
60.0 1--t 0.. ........ 70.0 100.0 0:: '-"
I/) 400.0 94 ,.() .., LO o:;t' +I 500.0 78.0 L-.J
550.0
* Ref. [19].
** Ref. [23].
% % Eyy Ezz
100.0 100.0
50.0 50.0
100.0
100.0
94.0
78.0
50.0
TABLE C.2 Thermal Properties
% % % Cl 1 Gyz Gxz Gxy (in/in/°F)
0.0*
100.0** 100.0** 100.0**
93.0** 93.0** 93.0**
75.0** 75.0** 75.0**
0.0*
100.0 100.0
100.0 1. 7xl0-6
93.0 93.0 107.0 1. 04x10-6
75.0 75.0 111. 0 .864x10-6
a2 (in/in/°F) 14. lx10-6*
14. lxio-6*
1. 7xl0-6
1. 04xl0-6
. 864x10-6
a3 ( i n/in/°F) 14.lx10-6*
14. lx10-6*
14. lx10-6
14. lxl0-6
_,
°' N
. Temp . % % % r-.µ OF Exx Eyy Ezz ta :::::
0.0 100.0 ........ Cl. 60.0 .......... 0:: (.!)
Vl 70.0 100.0 100.0 ., 0
°' 105.0 .......... 400.0 105.0 LO
""" +1 500.0 106.0 106.0 ' 0
L-.1
550.0 50.0
0.0 103.0 103.0 103.0
-0 100.0 98.0 98.0 98.0 c: OM 200.0 90.0 .90.0 90.0 .Q,.... r-,.... .µ,.... QJ 300.0 68.0 68.0 68.0 :::
350.0 50.0 50.0 50.0
TABLE C.2 Cont.
% % % Gyz Gxz Gxy
100.0 100.0
100.0
93.0 93.0 105.0
75 .0 75.0 l 06. 0
103.0 103. 0 103.0
98.0 98.0 98.0
90.0 90.0 90.0
68.0 68.0 68.0
50.0 50.0 50.0
al (in/in/°F)
a2 ( in/in/°F)
1. 7x10-6 1. 7x10-6
l .04x10-6 1. 04x10-6
.864xlo-6 .864x10-6
15.5x10-6 15.5x10-6
16. 5x10-6 16.5x10-6
18. ox10-6 18.0x10-6
20.25x10-6 20.25x10-6
22. Ox10-6 22. ox10-6
a3 (in/in/°F)
14. 1 x10-6
14. 1 x10-6
15.5x10-6
16.5x10-6
18.0x10-6
20.25x10-6
22. ox10-6
_.
°' w
. % r- Temp. % % % +'
ttl OF Exx Eyy Ezz Gyz :a: ic 100.0 120.0 120.0 l.J..J 0.0 104.0 ........ co r- 100.0 100.0 92.0 92.0 97.0 ttl c: 0 200.0 100.0 83.0 •r-+' u Q)
2:JO.C ICO.O 60.0 s... ..... ""O .....
100.0 45.0 c: 300.0 45.a 45.0 =>
l.J..J a.a 1a2.o la3.a 12a.o 104.a ........ cc rt 100.0 1ao.o lOa.a 92.a 97.a 0 -......: 0 2aa.o 98.0 96.0 83.0 CJ) ........ 0 -........ 250.0 6a.o 0 O"I ........ 0 300.0 97.0 93.a 45.a 45.0 L-J
* Ref. [l].
TABLE C.2 Cont .
% % Gxz Gxy
104.0 104.0
97.0 97.0
83.0 83.0
60.0 60.0
45.a 45.0
104.a 104.0
97.0 97.a
83.0 83.0
60.0 60.a
45.0 45.0
al (in/in/°F)
a2 (in/in/°F)
2 .2x10-6 9.6x10-6
2. 32x10-6 1. 11 x10-5
2. 4x10-6 l. 34x10-5
2. 7x10-6 1.68xl0-5
2.9lx10-6 3.45x10-6 ----- --- ·----------
2. 99x10-6 3.5lx10-6
3.04x10-6 3. 55x10-6
3. 26x10-6 3.7lxl0-6
a3 ( in/in/°F)
9. 6x10-6
1. 11 x10-5
1. 34x10-5
l. 68x10-5
9. 6x10-6
l . 11 x10-5
l. 34x10-5
1. 68x10-5
_, CJ) ~
. Temp. % % % ,.....
+' OF Exx Eyy Ezz IO ::E:
0.0 l 01. 3 101.3 101 .3 -le E 70.0 100.0 100.0 100.0 :::S> .,.... ...i-S::: I IO r- 400.0 95.0 95.0 95.0 +' c::(
.,.... l.O I-
800.0
0.0 104.0 104.0 104.0
70.0 100.0 100.0 100.0
-le 200.0 99.0 99.0 99.0 E...i-:::i I-;;: I 250.0 •r- ...i-EN ::so r-N 300.0 94.3 94.3 94.3 c::(
400.0 88.7 88.7 88.7
500.0 80.2 80.2 80.2
* Ref. [23].
-TABLE C.2 Cont .
% % % Gyz Gxz Gxy
l 01. 3 101.3 101.3
100.0 100.0 100.0
95.0 95.0 95.0
104.0 104.0 104.0
100.0 100.0 100.0
99.0 99.0 99.0
94.3 94.3 94.3
88.7 88.7 88.7
80.2 80.2 80.2
al ( in/in/°F)
a2 (in/in/°F)
4.7x10-6 4. 7x10-6
5.2x10-6 5. 2x10-6
5.6xl0-6 5.6x10-6
12.lxl0-6 12. lx10-6
12.9x10-6 12. 9xl0-6
13.0xl0-6 l3.0x10-6
13. lxl0-6 13.lx10-6
13.2x10-6 13.2x10-6
13.5xlQ-6 13.5xl0-6
a3 (in/in/°F)
4.7x10-6
5. 2x10-6
5. 6x10-6
12. lxl0-6
12.9x10-6
13.0xl0-6
13. lxrn-6
13.2x10-6
13.5x10-6
__, O"I <.J'l
APPENDIX D
Computer Program NONCOMl
166
Cards 1-5 (20A4)
Column
1-80
Card 6 (6I6)
Column
1-6 NE
7-12 NOS
13-18 ND I FM
19-24 NANG
25-30 IELET
31-36 I ELEM
Card 7 (616)
Column
1-6 NLOADS
7-12 NPSS (1)
13-18 NPSS(2)
etc. Repeated
NPSS(J}
167
NONCOMl FORTRAN USERS GUIDE
Contents
Title Cards
Contents
= Number of elements
= Number of nodes
= Number of different materials
= Number of different angles
= Operating temperature indicator
0 for 70°
> 0 for any other temperature
= Operating moisture content indicator
0 for 0% moisture
> 0 for elevated moisture content
Contents
= Number of load cases
= Load type number 1
= Load type number 2
NLOADS times
= 1 for axial strain
= 2 for thermal
Card 8 (516)
Column
1-6
7-12
NlNCRT(l)
NlNCRT(2)
168
= 3 for axial force
= 4 for hygroscopic
Contents
= Number of load increments for load case l
= Number of load increments for load case 2
etc. Repeated NLOADS times
Ca rd. 9 { 5 l 6 )
Column
1-6
7-12
13-18
19-24
KEY(l)
KEY(2)
KEY(3)
KEY(4)
25-30 KEY{5)
Card 10 (1016)
Column
1-6 L1NCPR(1 , I)
Contents
= Print indicator for grid
= Print indicator for strains
= Print indicator for stresses
= Print indicator for equivalent stresses
= Print indicator for displacements
KEY (I) = 0 for printing
l=l, NLOADS
Contents
= First increment of load case l to print
stresses, strains, and displacements
7-12 LlNCPR(2,l) = Last increment of load case l to print
etc. Repeated NLOADS times
Card 11 (2Fl2.6)
Column
1-12 SMY
Contents
= Scale factor for Y-coordinates
169
13-24 SMZ = Scale factor for Z-coordinates
The following card is repeated NLOADS times N=l,NLOADS
Card 12 (2Fl2.6)
Column
1-12
13-24
ALOADS(l,N)
ALOADS(2,N)
Contents
= Load increment for first load case
= Initial load state before applying increment
Card 13 is omitted if IELET = 0
Ca rd 13 (Fl 2. 6)
Column Contents
1-12 DEL TOT = Constant temperature for non-thennal loading
Card 14 is omitted if IELEM = 0
Card 14 (F12.6)
Column
1-12 DELMOT
Contents
= Constant moisture content for non-hygro
scopic loading
The following cards are repeated NDIFM times
K=l ,NDIFM
Card 15 (5El2.6)
Column
(Cards 15-28)
1-12 EKll(K,1)
13-24 EKll(K,2)
25-36 EK22(K, 1)
37-48 EK22(K,2)
49-60 EK33(K,1)
61-72 EK33(K,2)
Contents
= E11 tension modulus
= E11 compression modulus
= E22 tension modulus
= E22 compression modulus
= E33 tension modulus
= E33 compression modulus
170
Card 16 3El2.6
Column Contents
1-12 GK23 ( K) = G23 modulus
13-24 GK13(K) = G13 modulus
25-36 GK12(K) = G12 modulus
Card 17 (6El2.6)
Column Contents
1-1~ SPl(K,1) = Elastic limit stress for cr1 - £1 tension
13-24 Nl(l,K,l) = Ramberg-Osgood coefficient n1 for cr1 - £1 tension
25-36 Kl(l,K,l) = Ramberg-Osgood coefficient K1 for cr1 - £1 tension
37-48 SPil(K,l) = Bilinear intersect stress for cr1 - El
tension
49-60 Nl(2,K,l) = Ramberg-Osgood coefficient n2 for cr1 - £1 tension
61-72 Kl ( 2, K, 1 ) = Ramberg-Osgood coefficient K2 for cr1 - £1 tension
Card 18 (6El2.6)
Column Contents
1-12 SPl ( K,2) = 13-24 Nl (l ,K,2) =
25-36 Kl (l ,K,2) = Same as Card 17 but for cr1 - £1 37-48 SPl(K,2) = compression
49-60 Nl(2,K,2) = 61-72 Kl (2,K,2) =
171
Card 19 (6El2.6)
Column Contents
1-12 SP2{K, 1) = 13-24 N2{1,K,l) = 25-36 K2 ( l, K, 1) = Same as Card 17 but for o2 - £2 37-48 SPI2(K, l) = tension
49-60 N2{2,K,l) = 61-72 K2{2,K, 1) = Card 20 (6El2.6)
Column Contents
1-12 SP2{K,2) = 13-24 N2{1,K,2) = 25-36 K2{1,K,2) = Same as Card 17 but for o2 - £ 2 37-48 SPI2(K,2) = compression
49-60 N2{2,K,2) = 61-72 K2{2,K,2) = Card 21 (6El2.6)
Column Contents
1-12 SP33(K, 1) = 13-24 N33(1,K,l) = 25-36 K33(1,K, 1) = Same as Card 17 but for o3 - £3 37-48 SPI33{K,l) = tension
49-60 N33(2,k,l) = 61-72 K33{2,K,l) =
172
Card 22 (6El2.6)
Column Contents
1-12 SP33(K,2) = 13-24 N33(1,K,2) = 25-36 K33(1,K,2) = Same as Card 17 but for o3 - E3 37-48 SPI33(K,2) = compression
49-60 N33(2,K,2) = 61-72 K33(2,K,2) = Card 23 (6El2.6)
Column Contents
1-12 SP23(K) = 13-24 N23(1,K) = 25-36 K23(1,K) = Same as Card 17 but for , 23 - Y23 37-48 SPI23(K) = shear
49-60 N23(2,K) = 61-72 K23(2,K) = Card 24 (6El2.6)
Column Contents
1-12 SP13(K) = 13-24 Nl3(1,K) = 25-36 Kl3(1,K) = Same as Card 17 but for , 13 - Y13 37-48 SPI13(K) = shear
49-60 Nl3(2,K) = 61-72 Kl3(2,K) =
Card 25 (6El2.6)
Column
1-12
13-24
25-36
37-48
49-60
SP3(K)
N3(1,K)
K3( l , K)
SPI3(K)
N3{2,K)
61-72 K3(2,K)
Card 26 (5El2.6)
Column
1-12
13-24
25-36
37-48
49-60
61-72
SLl(l,K)
SL1(2,K)
Sl2{1,K)
SL2(2,K)
SL33(1,K)
SL33(2,K)
Card 27 (3El2.6)
Column
1-12
13-24
25-36
SL23(K)
SL13(K)
SL3(1,K)
Card 28 (6El2.6)
Column
1-12
13-24
UK12(K, 1)
UK12(K,2)
= =
= =
= =
173
Contents
Same as Card 17 but for •12 - r12 shear
Contents
= Ultimate stress for o1 - El tension
= Ultimate stress for o1 - El compression
= Ultimate stress for o2 - £ 2 tension
= Ultimat~ stress for o2 - £2 compression
= Ultimate stress for o3 - £ 3 tension
= Ultimate stress for o3 - £ 3 compression
Contents
= Ultimate stress for , 23 - r23 = Ultimate stress for , 13 - r 13 = Ultimate stress for •12 - r12
Contents
= Poisson's ratio v12 in tension
= Poisson's ratio v12 in compression
174
25-36 UK23(K, 1) = Poisson's ratio v23 in tension
37-48 UK23(K,2) = Poisson's ratio v23 in compression
49-60 UK13(K,l) = Poisson's ratio v13 in tension
61-72 UK13(K,2) = Poisson's ratio v13 in compression
If no therma 1 analysis is required skip to card 40
The following cards are repeated NDIFM times
K=l,NDIFM (Cards 29-39)
Card 29 (6!12)
Column
1-12 NTl (K)
13-24 NT2(K)
25-36 NT33(K)
37-48 NT23( K)
49-60 NTl 3( K)
61-72 NT3(K)
Card 30 (3Il 2)
Column
1-12 NT4(K)
Contents
= Number of linear segmented points for
E11 modulus percent retention curve
= Number of linear segmented points for
E22 modulus percent retention curve
= Number of linear segmented points for
E33 modulus percent retention curve
= Number of linear segmented points for
G23 modulus percent retention curve
= Number of linear segmented points for
G13 modulus percent retention curve
= Number of linear segmented points for
G12 modulus percent retention curve
Contents
= Number of linear segmented points for
a1 thermal coefficient curve
13-24 NT5(K)
25-36 NTALP3(K)
175
= Number of linear segmented points for
~2 thermal coefficient curve
= Number of linear segmented points for
~3 thermal coefficient curve
Card 31 (6Fl2.0) I=l,NTl(K)
Column Contents
1-12 PERMRl (I, K)
13-24 TEMPl(I,K)
= Percent retention of E11 modulus
at point I
= Temperature at point I
etc. Repeated NTl(K) times
Card 32 (6Fl2.0) I=l,NT2(K)
Column Contents
1-12 PERMR2(1 ,K) = Same as card 31 but for E22 13-24 TEMP2(1,K) = modulus
Card 33 (6Fl2.0) I=l,NT33(K)
Column Contents
1-12 PMR33(I,K) = Same as card 31 but for E33 13-24 TMR33(1 ,K) = modulus
Card 34 (6Fl2.0) I=l,NT23(K)
Column Contents
1-12 PMR23(I,K) = Same as card 31 but for G23 13-24 TRM23(1 ,K) = modulus
Card 35 (6Fl2.0) I=l ,NT13(K)
Column Contents
1-12 PMR12(I,K) = Same as card 31 but for G13 13-24 TMR13(1,K) = modulus
176
Card 36 (6Fl2.0) I=l,NT3(K)
Column
1-12 PERMR3 (I, K)
TEMP3(I ,K)
Card 37 3(El2.5,Fl2.0)
Column
Contents
= Same as card 31 but for G12 = modulus
I=l,NT4(K)
Contents
1-12 ALPl{I,K) = a 1 thermal coefficient at point I
13-24 TEMP4(I,K) = Temperature at point I
etc. Repeated NT4(K) times
Card 38 3(El2.5, Fl2.0) I=l,NT5(K)
Column Contents
1-12 ALP2{I,K) = Same as card 37 but for a2 13-24 TEMP5(I,K} = coefficient
Card 39 3(El2.5.Fl2.0) I=l,NTALP3(K)
Column
1-12
13-24
ALP3(I,K)
TALP3{I,K)
=
=
Contents
Same as card 37 but for a 3 coefficient
If no moisture analysis is required skip to card 51
The following cards are repeated ND I FM times
K=l,NDIFM (Cards 40-50)
Card 40 (6!12)
Column
1-12 NMl ( K)
13-24 NM2(K)
Contents
= Number of linear ~egmented points for
E11 modulus percent retention curve
= Number of linear segmented points for
25-36 NM33(K)
37-48 NM23(K)
49-60 NM13(K)
61-72 NM3(K)
Ca rd 41 ( 3I12)
Column
1-12 NM4(K)
13-24 NM5(K)
25-26 NBETA3(K)
Card 42 (6Fl2.0)
Column
177
E22 modulus percent retention curve
= Number of linear segmented points for E33 modulus percent retention curve
= Number of linear segmented points for
G23 modulus percent retention curve
= Number of linear segmented points for
G13 modulus percent retention curve
= Number of linear segmented points for
G12 modulus percent retention curve
Contents
= Number of linear segmented points for
s1 coefficient
= Number of linear segmented points for
82 coefficient
= Number of linear segmented points for
83 coefficient
I=l ,NMl ( K)
Contents
1-12 PERMR4( I ,K) = Percent retention of E11 modulus
at point I
13-24 TEMMl(I,K) = Moisture content at point I
etc. Repeated NMl(K) times
178
Card 43 (6Fl2.0) I=l,NM2(K)
Column Contents
1-12 PERMRS(I,K)
13-24 TEMM2(I,K)
=
=
Same as card 42 but for E22 modulus
Card 44 (6Fl2.0) I=l,NM33(K)
Column Contents
1-12 PMMR33(I,K) = Same as card 42 but for E33 13-24 TMMR33(I,K) = modulus
Card 45 (6F12.0) I=l,NM23(K)
Column Contents
1-12 PMMR23(I,K) = Same as card 42 but for G23 13-24 TMMR23(I,K) = modulus
Card 46 (6Fl2.0) I=1,NM13(K)
Column Contents
1-12 Pt1MR13(I,K) = Same as card 42 but for G13 13-24 TMMR13(I,K) = modulus
Card 47 (6F12.0) I=l,NM3(K)
1-12 PERMR6(I,K)
13-24 TEMM3(I,K)
Card 48 3(El2.5, Fl2.0)
Column
1-12 BETAl(I,K)
=
=
Same as card 42 but for G12 modulus
I=l,NM4(K)
Contents
= s1 hygroscopic coefficient at
point I
13-24 TEMM4(I,K) =Moisture content at point I
etc. Repeated NM4(K) times
Card 49 3(El2.5, Fl2.0)
Column
1-12 BETA2(I,K)
13-24 TEMM5(I,K)
Card 50 3(El2,5,Fl2.0)
Column
1-12
13-24
Card 51
Column
1-12
13-24
BETA3( I ,K)
TBETA3(I,K)
(6Fl2.6)
THE(l)
THE(2)
=
=
=
=
179
I=l,NM5(K)
Contents
Same as card 48 but for 82 coefficient
I=l,NBETA3(K)
Contents
Same as card 48 but for 83 coefficient
Contents
= Angle number 1 in degrees
= Angle number 2 in degrees
etc. Repeated NANG times
The following card is repeated ·NOS times
I=l,NDS
Card 52 (413,2Fl2.0)
Column
1-3 INODED( I)
4-6 !NODE( I, 1)
7-9 INODE(I,2)
10-12 !NODE( I ,3)
Contents
= I = U - displacement code
= V - displacement code
= W - displacement code
= 1 for force or non-zero displacement
boundary condition
= 2 for prescribed zero-displacement
13-24 YY(I)
25-36 ZZ(I)
180
= Y coordinate of node I before being scaled
by SMY
= Z coordinate of node I before being scaled
by SMZ
The following card is repeated NE times
I=l,NE
Card 53 (6X,5I6)
Col(.!mn Contents
1-6 = Blank
7-12 ND (I, l} = Node number 1 of element I
13-18 ND(I,2} = Node number 2 of element I
19-24 ND(I,3) = Node number 3 of element I
25-30 IMAT( I) = Material numcer of element I
31-36 !THETA( I) = Angle number of element I
Card 54 (2112)
Column Contents
1-12 NDCST = Number of nori-zero displacement constraints
13-24 NF CST = Number of no11-zero force constraints
If NDCST = 0 skip to card 56
The following card is repeated NDCST times
I=l,NDCST
Card 55 {2Il2,Fl2.0}
Column
1-12
13-24
~lODED( I)
MODE
Contents
= Node number 1)f constrained node
= Code for con'.;traint
181
= 2 for V constraint
= 3 for W constraint
25-36 DCST(I) = Displacement constraint increment
If NFCST = 0 no more input is required
The following card is repeated NFCST times
I=l,NFCST
Card 56 {2Il2,Fl2.0)
Column
1-12
13-24
25-36
Notes:
GENERAL:
Card 6
Card 7
Card 10
Card 11
Card 12
NODEF( I)
MODE
FCST(I)
= =
=
Contents
Same as card 55 but for force
constraints
Input units need only be consistent except for thermal
properties which must be.degrees F.
NE~400, NDS~400, NDIFM~lO, NANG<lO
NLOADS< 5
The first and last load increments are always printed
The scale factors are multiplied by the Y and Z coordinates
to obtain the final Y and Z coordinates
ALOADS(2,N) is applicable to thermal and hygroscopic
loading only. ALOADS(l,N) must be input as 0.0 for inplane
loadings.
Card 32 Node numbers must be given in counter-clockwise order
c c c c c c c c c c c c c
1'1A IN P K'JGRJ\M **-::'**>i< *-:::' ** A1' I i\l >:n;:*:O::;'<•:"~ ~":< * -t->::* ** * =<'* **** *
WORK DISKS 01,02,03,04
FASOOO'JO FASOul\)Q FASO:JllO FASOQ12u FAStJ0130
******************~~*****¥****~*******************************~***FAS0014U
RA1-1~3ERG-os:;c:o0 TYP~ l);!T A FCR STR.ESS-ST!~A I'J CURVES
*fASJJ153 *FASOQ160 *FASO'.Jl70 *fASOJ15l1 *FAS0019J
**~**~***~******~**~*****~*~***************~**********************FASOJ2JJ FAS•JQ21')
REAL i-a,,'!2,'13,Kl,K2,K3 FASOJ220 R~AL ~J3,MlJ,~2~1~3J,Kl3,K2~ FASOJ2JJ REAL*d YCJ~O,Z~G~0 FAS~024J c iJ:; '! ~J ~~ I G f) ~; ii y c \} :{. I)( 't 0 0 ' '-t ) ' L c c R 0 ( 4 cu ' ~ ) ' M !) ( 4 0 0 ' 3 ) FA s 0 0 2 5 J C!JA·1J;~ /GJK.3/ f.··::H{ 'tJJ) ,,\,.~cid40J) FASOJ?6J c CH:-i J \j I G 0 K 7 I !) s T x ' y .:, L ,L s L ' F :J, u LT ' F c R. c E ' D EL T ' y D ' z J ' s u ML\ ' ;n E s T , Ts r RTF As 0 J 2 7 ::> u: .. I 'J:~ / 8 DR l ii T: ,; 1\ x '.'W s ' ! p s ' .'u I ;= n ':''1'\ ~ r;, ~] VJC ;:!, , L I'.~ c' ~.J E ~' l'J E (.)~'Np, A."l u 1 F ~ s J 0 2 ~ J
l. J E ·~ d, ti J LUC,'(, l C' /\ U,. J ::>SS ( 5 l , .J f UJH, DEL T CT,•'< PLUS, ·"!L •l ~).2 F 1-\S t.i :.i2 '7 J C !J.>lu,·J I GJ;<.2 0 I ·SP l ( l :J, 2 l , 5 P 2 ( 1 C, 2) , SP 3 ( 10 l , SP I U l 1J, 2) , SP I 2 ( l 0, 2> , SF AS 003 Q 0
l P I 3 ( l J ) , · 11 l ;~ , l J, ,>_ ) , \J.:: ( 2 , l 1 , 2 ) , ,\i3 ( 2 , l J ) , i<.. l C .~ , l ') , 2 l , :<. 2 ( 2 , l ::> , 2 ) , K 3 { 2 , F 1-\ SJ J 31 J 21 'J) , .:K 11 ( i J, 2 ) , F. i<. 2 2 ( 1 iJ, 2 ) , EK 3 3 ( l J , 2 ) , GK 12 ( l 0) , GK 13 ( l 0 ) , SK 2 3 ( l 0 l FAS Ju 3 2 J
C'.J,.Ji•'iO'~ I ·~D i~'l. 2.1 1\T l. ( l 1)) , iH 2 ( i 0) , NT 3 ( l U l , PE K ·HU ( l 0, 10) , PE R.'lR2 ( 10,l0) F .l\S JO 3 3 0 l , ,-,, = -<. ii' . .J ( l J 1 J. J J , T ,:: W l ( 1 ) , l J } , T : , : P .2 ( l J , 1 .J ) , T c ·H' 3 I l :J , l .J l FA S u U Yt 0 CO>i.~O\~ /GOF;24/ fJT<t(l.J),'\JT5( iOl,H::•lP4(10,l;1),lEi·'P5(10,10) FAS00350 CO·HT~ !GJ;:...Lc/ AC1C-t.JJ),.\C2('t0vl,.\C3('1-0J) fl\S:)JJ6) cu:'1'1J(.j /.:,J:~r!.7/ t~LPl{ i.O,lJ) ,1~LP2(l0,1J) ,ALP3tl0,1Q) FASOiJ37'.) CJHlJ:~ /GiJF .. ~J/ SLU2,10),SL2.(2,10),SL3(2,1J) FAS00380 C -..'·1 =:\; I G :) .·,".i j I t: 11 { 1+ )J ) , .: ~2 ( l,.) ,J } , ;:: 3 3 hJ J ) , YJl l ( 4 J J ) , D fJ 12 C 4 JO ) , DJ l 3 f AS 0 03 -10 1(40J),0Jlb(~JJ),JJlL(40J),0023(4JJ) 1 0026(4JQ},Ll033(400),0036(400),F~SQ04J0
CX> N
2 '.JD 4't Ut 0 0) ' J 0 4 5 UH.l 0) ':)0 5 5 ( 4 (j 0) 'j) 06 6 ( 4 00) FAS 0 (J 410 COMMG~ /GOR40/ Gl2(~00),Gl3(40J),G23{400),THETA(~00),ITH FAS00420
2ETA(4JJ),THf(lOJ FASJJ430 COMMa~ /GOK42/ UTKE~P,UKEEP FAS00440 COM~JN /SORJJ/ Sll(4JO),Sl2(4Q0),S33(400),S12(400},Sl3{400),S23(40FAS00450
2J),S44(4JJ),S55(4JJ),S66(~0J) FAS00460 COMMON /G0~95/ Ul2(400),U13(400),U23(40J) ,UK12(10,2),UK13(10,2),UKFAS0047J
123(10,2) FASJJ490 CUi'l;.:u;-~ /t:J.Hl/ l;'.~n:: ('~Jc, j) FASJQ!1 jJ CO;lMtlti /EAH5/ l1'lJUE!J{40J),YY(400),ZZ(400),ITITU:(20,5),JK(3500) FAS00300 CO~MON /EAYlJ/ LR,LW FAS0051J C Li .·V·l u;J I:: 4;11 5 I '1 L) C ST , [,; F CST, ;~ ·~ 1) >:: 1J { SS } ., •JC 5 T ( 9 q} , u;o t F ( cy; ) , F CST ( r~")) , F .1\ S () J 5 2 J
1FCSTT(2 1 2) FAS00530 Cll•'i'.1.i\I /'.:AHZJ/ i:Y(+JJ),EZ(4JO),~X(40J),;:YZ(400),EXZ(400),EXY(400},FASOU5.+0
l s L.; :\ ( :t ~' ) ) ' s r '~V ( (~ ) ) } ' ') I r:, z ( ~tr)'! ) ' s I G y z ( 4 0 Q ) ' s I GX z { 4 0 J) ' s r G x y { 4 :JO) FA s Ou::;_) 0 C 0 "1.'I J: J I E.-\ :-12 5 I D [ X J { 't J J ) , CJ E Y G Ut J J ) , D E Z 0 ( 4 J J ) , D E X Y il ( 4 J J ) FA S J 0 5 6 J CG~M~N /EAH30/ SWl(~JU),SQ2{40U) 1 S03(400) 7 SWl2{400),S~l3(400),S~23FAS00570
l(4JO) FAS005d0 CC>l-1 J!\J I!:. 1\ 1-1 ·~JI \ .·d ( i .) } , N ~ 2 { l J) , f'fl 3 { l J) , ;\1 ~14 ( l J) , N:·15 ( 10 ) FAS 0 J 5 9 0 c c 'I 111 J ~J I ;,: Milt 5 I r r: .'U l( l 0 ' 1 0 ) ' TE i'i ·12 ( l Q ' l 0) ' TE W~'.3 ( l 0 t 1 0 ) ' T F. ,1'1[>,14 (l 0 ,1 0 ) FA s 0 0 s a 0 l,T[~~5(10,1J) FASJJ61J C_,,rn.~ /'C.A'rbJ/ PEJi :KHlJ,lJ),Pcf;i,:KS(l0,10) ,Pt:R1,iR6(10,10) FAS0062() Cl l,-.lf! 1 ~ /£:Ail5S/ i~t~T ~l{ 10, l'.:), jcT,\2( 10, l~)), d1=T/\3( 10, lJ) FAS0063u C:.J. '. .j LJ i'i I F:. Ad" J I L I rj C P ·-~ { 2 , S ) , K !:: Y ( 1 J ) FA S () J 6 rtJ CUA:J:J /F:.AH7J/ UU{-tJJ,3) FASO·J650 CC '~·'·1 G,\ I t.=. Ah f:; I S? LL.i \ l J, 2 ) , SP t L 5 { l 1., j , :; :1 1 1.:: { l J) , ::; ., ~:.; ( 10, 2) 1: ! 3 3 ( 2, l 0 F ;l S 0 i)C "-, Q
1 , 2) t KJ 3 ( 2 ,1 ), 2) , SL 3 3 (2 , 1 J) , SP 2. 3 (1 J) , ~!Z 3 ( 2, 1 J ) , K2 3 ( 2, 1J),Sl2 3 l l 0) , SF :\S OJ6 7 J 2 P 1 3( 1 J) , ~a 3 ( 2, 1 J) , Kl 3 { Z , l 0) , SL 1 3 ( 1 0) F l\S OOld u CJ.i'-\01~./d\i-1uv/ .~f3:J(luj,:-' 1i,;,,J{,iJ,iJ/,T .,,_:;J(lU,10),~JT13(10),p, .. !l<l3(1F~S')JfiOJ
1 J, l J ) , T , ; f) l _; ( l 0 , l J ) , :H 2 3 ( l J) , P ;, i ;.:_ 2 3 ( l u, 1 J ) , T. iR2 3 ( l u ,1 0 ) , ;~TA LP 3 ( l 0 ) , Fi\ S 0 v 7 0 \J 2T.'LP3(ll'),1Jl FAS00710
CIJ V·IJ'.'J I E.1\h05/ f'i.·l:d ( 1 J),? HP 33 ( lJ, 1 J), Ti-1:-1[{33 ( lJ, lJ), "-J:H 3 ( l J), P'•HR 1F,\SOJ7 2J
_, 00 w
2)(10,lQ),TM~~lJ(lJ,10),~~23(10),PMM~23(10,10) rTM~R23(10,10},NBETA3FASCJ73J 3(1Jl,TB~TA3(1J,1J) FAS00740 CO~MGN /EAHlQO/ ATOT{320JO} FAS00750 DIMENSIU~ l~(l0),Nl.~CRT(5),ALCAOS{2,5) FAS00760
C FA500770 C *~~*~*****~****~**~******~**~*****~*****~~~~*****~****************FAS00780 t.: c c c c c c c c c c c c
l
!TITLE = TITL= CA~DS NE = ~UM~E~ OF CRIGi~AL EL~~E\TS NJS = NU~dtR CF ORIGINAL ~ODES YCORD, ZCDR.D = NJOE COORD I :\JATES NIJ = f'iODE ;JUM.JER. IHAT = M1HE:-J.IAL i~U·UE?. DESIGl~ATICN FDR EACH ELEMENT
* OATA R~AD IN *** ;.1 T J T = 3 2 0 0 J LOAD-=J
OELMJT=J.O IStTT=J I Sc T';= 0 NPLUS=Q NBA!':O=J LR= l ::>
Rf/\O{LR,:>d,E,lD=5:>:J) ({ITITU::t!,J),I=l,2Q),J=l,5) \~~~I TE ( 12 , j ;; ) ( ( I T £ f LE: ( I , J ) , I= l , 2 J ) , J= 1 , 5 )
*FAS00700 *FAS008JO *FASOJi.ilO *FAS:JJ82J *FASOOC!30 *FAS00340 *FAS0085'.J *FAS0::>360
FASOJ830 FAS00890 FASIJQQ;)J FAS00910 FAS 1)0920 FASJ091J FAS 0 J<J :..') FAS0095 0 FAS00960 FAS00970 FASOO':>SJ FAS 0099 :) FASOlOOO FASJlOlJ FAS·)l020 FAS01030 FASO l JltJ
645
642
64.3
UTKEEP=O.O WRITE(U~,60)
WRITE(LW,61) {{ITITLE(I,JJ,I=l,20),J=lr5l \oi R IT E C U~ , l.; tt 5 ) f 0 RM AT ti, Tl 0 1 (:19 ( ' * ' ) , I I } R2A0 (LR,~2) N:,~os,~OIF~,NANG,IELfT,IELEM R E A 0 ( l R , 6 2 ) NL < J A D ::i , ( \ P S S { I j , I = 1 , :'-J L Q .t\ IJ S ) IF CIELcT.GT.0) ISETT=! IF { IEU'.M. GT. 0 l I SET:"l= l hRlTf (Lh,6~2) NLCAJ~,{~~SS(I},I=l,NLOADS) F o ~ -., AT u 1 , T 2 J , • i ~ u; ·L-> E 1. u f L r "'.D c As ::: s 1 , 3 x ,i s ,1 , L.: o , • L o ;, '.1
1S',5(2X,I5)) ~c~~ (LR,52) (NI~CRT(I),I=l,NLCAOS)
~~AJ (LK,62) K~Y
REA;J (LR,62) ((LL\JCPR(J,I),J=l,2),I=l,NLCJADS) WRITE {U!,643) CNI\JCi<.T(I) ,I=l,i'ilGAOS) fL.h.'iAT (//,Tlu,•,.J, .• .;:_;~~ tjF L(.;gi) I:·~c~t::it:.·~rs Pt:x LOAOHJG
l,S(2X, IS)) RC:).J(LR.159} S<Y,S.H
FASOlOSO FASOlOoO FAS0107J FAS0103J FASO lO':JO FASOllJJ FASOlllO FAS01120 FAS01130 FASOll'i-'J
TYP [ r'iU.'id cK FAS :.n 15 u FASOll.JO. FASJ117J FASOllJ') FASOlF~O FJ\SJ12'.J•J
C~SE',//,T20f~S01210
FAS01220 FASJ1230
;'.·:.HJ (L,\t:..>Ji llr•LU-<1.)(i,jj,:-"'.q.:..:.i,J=l,,Jl1J,.:u;:i) F/'.S0124U IF (I~LfT.GT.U) ~~~J (l~,SJJ 0ELT01 FAS1125J IF { ft L ~ '-'. • GT. U) K:: .\ 1 l ( l i<. , 6 3 ) DEL ilO T FAS 012 6 ') 0J 5 t\=l,:·iJI,- FAS01270 R~AJ(Lq,7~) EK11{~,l),~~ll(K,2),~K22{K,l},~K22(K,~),EK33(K,l),EK33FAS012~J
l(K,2) FAS0129J RE.::if; (L.::,,71) •,_,i~L.;C.\l,SK15(K),GKldK) FASJ13:JJ R c A 0 ( L fZ , 7 4 ) SP l { K , l } , ,\J l ( 1 , K , l ) , I< l ( 1 , K , l ) , S '> I 1 ( V.. , l ) , ~; l ( 2 , K , l ) , I< l [ 2 , F AS ·J l 11 J
lK,i) FASOl320 RE A !J ( L 1\ , 7 4 ) SP l ( ,<., .?. J , . ~ l ( l , I< , 2 ) , !<. l ( 1 , K, 2 ) , SD I l ( K , 2 t , N 1 ( 2 , K , 2 ) , Kl ( 2 , FAS •J 1] 3 )
lK,2) F~S0134J RE~ D [ LR., 7 4) S '.J 2 ( K , l } , : ~ 2 ( l , l< , 1 I , K 2. ( l , K , l l , SP 12 ( K, 1 ) , I'll 2 ( 2, K , l l , K 2 ( 2 , F ''Su 13 j 0
lK,1) FAS01360
__, co (.11
qEAOlLR,74) S~2(K,2l,N2(1,K,2),K2(1,K,2),SPI2(K,2),N2{2,K,2l,K2(2,FAS01370 lK,2) . FAS013du
REAO (LR,74) SP33(K,l),N33( l,K,l),K33(1,K,l) 1 SPI33(K,l),N33(2,K,l)FAS013Y0 11K33(2,K,l) FAS01400
REAn (LR,7~) 5P~3(K,2),~33(1,K,2),K3J(l,K,2),SPI33(K,2),N33(2,K,2)FAS01410 l,K33(2,K,2) FAS01420
REAO(LR,74) SPi3(KJ,~23(1,K),K23{1,K),SPI23{K) 7 N23(2,K),K23{2,K} FASJ143J R~AD(Lk,74) S?l3(K),~!l3(1,K),Kl3(l,K),SPI13{K),~13(2,K),Kl3(2,K) FAS01440 RE A 0 ( L :<.. , 7 4) SP 3 { K ) , ·~ 3 ( 1 , ;<.) , l<. 3 ( 1 , :<. ) , SP I 3 { K ) , 1~ 3 { 2 , K) , K 3 { 2, K) FAS D 14 5 0 REAO(LR,74)Sll(l,K),SL1(2,K),SL2(1,K),SL2t2,K),SL33(1,K),SL33(2,KlFASJ1460 Ri:AI) (LR,?.;.} SL23(K),5Ll3(K),SL3(1,K) FAS01470 SL3(2,Kl=SL3(1,K) F~S01480 REAJ(L~,74) UKL2(K,1J,UK12(K,2),UK23(K,l),UK23(K,2l,UK13(K,l),UK13FASJ1490.
l(K,2) FASOl5)0 WRITE(LW,77) K FAS0151J WRITE(L~,73) EKll(K,1),EK22(K,l),EK33(K,l),fKll(K,2) 7 EK22(K 1 2},EK3FASUl520
l 3 (,( , 2 ) , G :< l 2 ( K ) t (; l<l .::. ( ;( ) t G :< 2 3 ( K ) , U ;( l 2 ( t<. , 1) , U :< 1 3 ( K , l ) , UK 2 3 ( K , l) , tJ K l 2 ( F /\ S 0 l 5 3 0 2K,2J,UK13(K,2),U~2~(K,2) FAS0154J ~RITE(l~,71) FAS01551 WKIT~(L~,~5) FAS015o0 11'! ~I TE ( L .. .,.. , 6 (, ) SP l ( '.< , l ) , l\i l ( l , K, l ) , K 1 ( l , K, 1 ) , S? I l ( K , 1 ) , N 1 ( 2 , K , l ) , Kl { 2 F A 5 J l 5 7 0
l,K,l) FAS01530 l.j =!.IT: ( L :! ' { 7 ) s ;> 1 ! ,',I : } ' .• l { 11 ;-:_ 1 2 ) 1 Kl ( l, !<.' 2 ) ' s 0 I l ( .\. 1 .: ) '~1 ! ( ? ' I(' 2) ';( l ( 2 c ;, s J 1 ::; ') ,)
l,K,2) FAS01600 WRITE(L~,63) SP2(~,l),N2{1,K,l),K2{1,K,1) 1 SPI2{K,l),N2(2,K,l) ,K2{2FASOlblO
1,K,l) FASJ162.J A lU Tc ( L 1-, 1 S ~) St> 2 ( i<. , 2) 1f.;2 { l, K, 2) , K2 ( l t K, 2) , SP I 2 ( r(, 2) , !'J2 t 2, K ,Z ) , K2 ( 2 FAS 0 16 3 '.)
l,K,2) FAS01640 .~ f:', IT t ( L·J , 11 J J ) S ;> 3 J ( ;\ 1 l ) 1 '! 3 3 ( 1 , I< , U , K 3 3 ( 1 , K , l ) , S P I J 3 ( K , l ) , I'< 3 3 ( 2 , K , 1 FA S 0 l o 5 :J
l),K3J(2,K,l) FASDlG00 ,.; R IT c ( u~ , .:; J l ) s p 3 .:.) ( K ' 2 l , : ~ 3 3 ( l ' ~ ' 2 l ' i( J 3 ( l ' K , 2 ) ' s p I 3 3 ( K ' 2 ) ' r·! 3 3 ( 2 , K 7 2 F t. s -) 1 6 7 J l),K33(2,~,2) FASJ163J
__, co 0\
WRITE(LW,902} SP23{K),N2J(l,K),K23(1,~),SPI23(K),N23(2,K) ,K23(2,K)FAS01690 ~RITE(Lh,303) S~lJ(K),Nl3(1,K),Kl3(l,Kl,SPil3(K)·,N13(2,K),Kl3(2,K)FAS~l70J r.l R I TE ( L h , 7 J ) Sf' 3 ( K } , ;'J 3 { l , K ) , K 3 l l , K ) , 5 P I3 ( K ) , N 3 t 2 , K ) , K 3 { 2 , K ) F AS 01 7 l 0 WRITE(LW,73) SL1(1 7 K),SL1(2,K),SL2(1,K),SL2(2,K),Sl33{1,K),SL33(2,FAS01720
1 K) , SL Z-H iO , SL 13 ( K ) , S L 3 .( l , K) FAS 0 1 7 3 J 3 CONTINUE FAS01740
DO 191 KK=l,NLU~OS FAS01750 IF (.lfP SS ( K lO • E .J • 2 ) I Sc TT= l FAS 0 l 7 f> 0
Lil IF ( n'SS(::,,) .;;., ,..tt) ISEH1=1 FAS0177•J
4
IF (ISETT.EQ.l) CALL THI~C FASJ178Q l(NT1,NT2,~T3,NT4,NT~,NT33,NT23,NT13,~TALP3,PERMR1,PERMR2,PERMR3 FASOl790 2 , .~\LP l , 1\ L P 2 , t> '.; ;.<3 3 , P i ~2 ::; , P ·'i =< 1.3 , ALP 3 , T [MP 1 , TE .''1? 2 , TE'~ P 3 , T F ~1P4, T E :-1? 5 FAS 01 8 J 0 3,T 1·,J.J,1 ·.f~L.:>,T.·;KL;,, f<\LP3 1 l) FAS0181J
IF ( ISf:T;V:.E:,~.l) C,1-\LL THL,!C FAS01820 1 ( , ~ d 1 , :\J :12 , : ~ "\ J , i\1'·!4 , N "1 ~ 1 i\l :113 J , r,J H 2 3 , i' l .'': 13 , i\i b :: T A 3 , PE R r·; R. ' ' , P E P, ":; R S , PE K ~1 Rb , 8 E F AS 01 8 3 0 2 TA l, 0 ET AL'? .. , -; 1J3 J' p ,; nz 3 'p .. i,"'1p13 ' l>:: T A.3' r:::r1 ;'"11 , T i:.'1i12 'T t: MM 3' T HL\14, T E:·l ~'15 FAS Ji 840 3,T~M~33,T~M~2J,T~~R13,T~~TA3,Z) FAS018~0
REAO(Lt<.,72) (THE(K) ,,<=l,1·j.\f\JG) FAS'.H860 ..,J ;,z I T c ( L~ , l S <t ;., ) \~RITF: (t_~;,194-:.d (I,TrlE{I),I=l,i\ANG)
1 :: -t 5 F ;J -.:. l ;..1 i ( I I , T 2 1 , ' :.!. 'I G L :: i\1 J • ' , T 4 J , ' A 'l G L E ' , I I )
fASGlu7J FAS01B80 FAS013~JO
FAS019J:) FAS'.H<JlJ FAS Ol 9l:J FAS0l9JO fASOlq~o
FASOlS'.:iO FAS01960 FAS01S70 FAS019Hv FAS019"1'.) FAS020JO
19 1to'.. F8iZ'-lt1T (T22, rs,r.::.~,.=1L.j)
c c c
99
RE:\ G { l :<. , 7 6 ) ( I :-; l ),) t: ci ( 1 ) , ( I ;,~CD E ( I , J ) , J = 1 , 3 ) , Y Y ( I ) , l Z ( I ) , I = l , ND S ) RE At) ( L R ' 7 d ) { ( i\ I D { I ' J ) ' J = l , 3 ) ' I f~ /\ T { I ) f 1 T ' - ; . ( T ) ) 1 = l , • 'i: ) · ... •~ .· .. . ,... '. -~- ...... , .. ~, .. '· .. , .. -,.
SET U-LH SPLAC[:·1E\JTS Tu z::RU FU;.( CR.(JSS PL y A;•JO Uf\!IOIF~ECT IOi\Al
OG '71 I=1,:·i;.'.'iG IF(THE( I) .:-~f:.CJJ.'.).,\;D. THE (I) .0i[.0.0} NOOF=3 IF (1~uiJF.::·~.::>) GC T.J 7
__, CX> ......
6 7
10
2JlJ
12
l!t
c c c c c c
16
DO 6 l=l,iJJS IWJD'.:{ I ,l )=2 CD~!T INUE DO 10 I=l,NDS YY( I )=YY( I )*-5,.1y Z Z ( I l = Z l ( I ) ,;, S ~': l IF (XEY{l).\~.J) G~ TQ 2JlJ W R. I f t ( l>i , d 0 ) wRITE(Ll 1 <l0) ( I~;~1DC::J{l) 7 YY(I),ZZ(I),1=1 7 NOS) LJG 14 1=1,iE: K=ITHETA{I) THETA( I )=TrlE{ Kl Du 12 J=l,3 K= .'iU ( I , J ) YCJ~O{I,J)=YY(K) ZC 1J~J{ I,J)=lZ(l<) Y C u :;: 0 l i , 't ) = Y l- '--' -, i_; ( I , l ) ZCJR0( I ,-!t)=ZC1.=::<U( I, l) CL.t:H Ii'lUE :..j['J!.)='J
sr:cr 1::; ~ I (;:\LC.:L-. I,:::) THr: A;-\.c/:i. '.JF EACH ELl:1'1E:NT A~O OUTPUfS ~L:~E~T INFUPMATION
co.':T I\iUE SU 1<\=J.O
AR::A{K)=O.O DU lJ I=l,3 r.>::•.1 •J=-(-YC;--,_~{:(, I) +Y1-:..::1<L)(K, l+l) )q zcc~~OC:<, I )+ZCCJR.O(K, I+l) )/2.
FAS02010 FAS02020 FAS02030 FAS020ttJ FAS02050 FAS02060 FAS0207v FAS020'30 FA S020•JO FAS02100 FAS02110 FJ\$0212:) FAS021-~ J . FAS02140 FASJ2150 FAS0216D FAS02170 FASv213J FAS02190 FAS .J22 J•) FAS02210
*FAS:J223J *FAS0224'1
fAS.)22:,o FAS02270 FASJ22 •:W FAS02290 FASu2;j0J FAS0231J F4S02321
__, CP CP
18 AREA(K)=AR=~(K)+P~UD 19 SUi·\!\=S\J'iA+M'.EA( K)
I F ( KE Y ( l ) • :·~ E • 0) G 0 T iJ 5 0 riR IT E ( U·J, d2 ) DJ 2-PJ K=l ,N[;•J\X OG 2J3•J I= l, 3
2 U30 WR I TE ( L ~~, cl 3 ) K , 1W ( K, I ) , yr, no ~l ( !'~, I ) , l CJR D ( K, I ) 2 0 2 J '.'I ;.:;_i r::: ( L ;,; , J 4 ) f1 : '. c A ( K ) , I M .!\ T( K ) , TH E T A ( K ) 39 WRITE(Lw,64) SUMA 50 CALL CENTfS
i\IP s=:~p SS { U N P~CR=N I i'lCR. T { l) CALL DJITIL Cull c1U'.\U"i CALL Cjl~STi'.
LOAD=LJt'\O+l
C SfT ~RRAY SIZES ANO LOAOI~G TYPES
9jl GU TJ (l0J,ilu11~J,~LlJ, ~P~ lZJ LJSTX=J.)
OELT=~LJAJSCl,LUlO) TS rnT=>\LJ>4:)S { 2, LU.4J) FLJr{C'.::=J.J IOI':.2=Fl GC T.J 110
1 0 J F J ;.;. C ~:= J • v D~LT=·J.J
DSTX=,'.\,LO~iJS!l,LOAiJ) 1L)P~2='J
GO TrJ 110 l3J DcLT=J.O
FAS02330 FAS023ltJ FAS02350 FAS02360 FASJ2370 FASQ2380 FAS·J23<JO FAS02400 FAS02 1+ l 0 FAS02420 FASiJ2430 FASJ244J FAS0243') FAS J2't60 FASJ247J FAS02430 FASJ24:JJ FAS025UJ Ff\502510 F 1\. S ) 2 5 Z ·:! FAS02530 FAS02540 FAS0255·J FAS 02560 FAS02570 FASJ258J FAS02590 FAS02l.OJ F1-\SJ~til'J
FAS02620 FASJ263.) FASJ2640
__, 00 l.O
FURC := 1\LO ADS { 1, L.}A lJ l FA S026 50 DSTX=J.O FAS026~0 101M2=10 FAS02070 GO TO 110 FAS02680
121 IDIM2=10 FAS02690 FORCE=0.0 FAS02700 DELT=~LUADS(l,LO~D) FAS0271J TSf~T=~LC~JS(2,LOAD} FAS02720 DSTX=O.O FAS02730
llJ NGANJ=O FAS02740 49 Ll~C=l FAS02750
I F ( l L! ~ D • GT • l ) CA l L ~ QU ~ d ·.1 FA S 0 2 7 :J J WRIT~ (L~,45J) FASJ2770 W~IT~ (L~1250) 05TX,FORC~ FAS02780 I F ( ,, PS • f :.) • 2.) ;H I T E ( UI , 5 5 0 ) DE LT FAS J 2 7 9 J IF (1'li=>S.EJ.Lr) ~IKITf (U;,551) OELT FAS028JQ
5jJ FO{M~T (/,Tl0,'T~~PE~ATU~E INCkE~ENT 1 ,El5.7) FAS02810 .:>51 F:~'n,H (/,TlJ,';iOISTJK.c P;l.:-c:-:::1JT ·,~15.7) FAS02320
If (~PS.NE.2) ~RIT2 (L~,451) OELTOT FAS02830 451 FOx,:1AT (!,Tl0, 1 C1JNSL\:\IT TE'1PEC.:.ATUP.E FUR THIS LC/\DH:G = 1 ,Fl2.6) F.!\SJ28/+J
Ii:(··,;:..;.:;::.';) :·c:'.IT:::: (Ln,-+S2l J~L'~JT FAS02651 452 Fei.z:-L.\T (/,T10,•c:J.-~SL\,H ,•;i]!STUF~E CL'lTENT FOR Tl-'IS LCAIJING = 1 ,Fl2.FASJ2C:6J
16) FASJ2R7) IF t-':PS.E:Q.2.n:<,.1\w~.;:·:).Lt) GC TC 20 FASOZB80 IRST=J FAS02890 If-' ( lPS.:-,E.l) !?ST"-=l Fl\5029,)'.J I F ( ,-~ JC ST • : (~ • 0 ) G lJ T -.; 2 9 FA S 0 2 9 l 0 WRITE (LW,49~) FASU2920 DJ 640 I=l,~JCST FASOZ930 KKK= 1\CIJcl) { I) F~S02'; .;.o IF (~KX.GE.~~QF) KKK=KYK+IkST FASJ2950
64b i'il-<IT: (:_ ,_:~J) .:.~,_:~ST(I) FAS02960
_, 0.0 0
29 IF (NFCST.EQ.0) GO TO 20 FAS02970 WRITE (LW,599) FASJ29BO 00 647 I=l,NFCST FAS02990 KKK=NDDEf(l) FAS03000 IF (KKK.GE.NEQF) K~K=KKK+IRST FAS03010
647 W~ITE {Lh,500) KKX,FCST{i) FAS03020 4jC FO~MAT (//,20X,'GOUNDARY CONDITIONS',/) FASJ303J
499 FJf<MAT (2JX,•:.:L''i-ZE'{'J OISPLACC4ENT CUt-JSTRAli'.JTS',//,lOX,• E:'QIJ NU~'1BfF.~SU3040 lR', 2X, 'CU,·JSTR./\ FlT' ) FAS 03050
5 u CJ Fu~:; i •H Cl. J >~ , I l J , d :> • I) FAS i.) 3::) S 0 5 9 ~ FOk,'1A T ( / / 20X.' IF ']/{Ct: c Ol'IS H~A I l'iT s' 'I' lOX '' E:~u NUWH:R' '2 x' 'Cm!S TRJ.\I~ FAS0307')
1 T') FAS03Gd0 z s J F n ~. ·1 A T < / , T l J , • , " u K 1'i A L s Ti<. A I i': • , E l 5 • 1 , / , T i J , • No R , .. , AL F c R c E • , E 1 s • 1 > F A s o 3 e> 9 o
20 ~l=l FAS03100 N~S=O FAS0311J ~MS=J FAS0312J IF {\JfJS.E~.2.Dk.:)tlTUT.GT.75.CJ) ~hS=2 FAS03130 I F (; \ 1J S • E Q • 4 • l : .' • ~) :: L -l:J T • GT • 0 0 • J ) 'J ·l S = S F A S J 3 l 4 J IF ( .W;S.;:Q.2) CALL P::::~:;•jJ FAS0315·J
1 ( :\JT l,;. T 2, ;\JT 3, ~:T i-, "T ; : . :T 3 J, t-., T 23, :";T 13, ~IT ALP 3, P FP J'.Jf.'. 1, P ~R \j ~2, PU' \l? J FAS 0316 0 2 , Al P 1 , .~LP 2 , ;) cj r< :~ J , :J ; • .2;:; , P +n 3 , ALP J , TE:,·'? l, TE' l i=> 2, TC: 'H> 3, T ;=r.• P 1-, T £',, P 5 r J\ S 031 71) 3 ' T ·lP. ~ 3 ' T '. -~ ;<,. 2 3 ' T : :·u 3 , P. L p 3 ) FA s 0 3 1 a 0
If {N~S.~Q.5) C~LL P~~~JJ FASJ319J 1 ( •'i !l , .'! 12 , iJ '13, N ;.1 'q '; L) , ~lil 3 3, ;-; i'l 2 J , ~~ •ll 3 , \i ~ f.T ,.'\J , P '.:: t~ .'·Ji{-i-, PE: R :.~ R5 , PE'-~. MR6, ~J F. FI\ S 0 3 2 J tJ 2 r ;.\ l ' u ~ r ~ L ' p' 1. i'-\3 J ' p I i >,2 3 ' p ·l'·l ~ l 3 ' s ET;\ J , T f:V1; 11 ' l r: '·1!·:2 ' T ': :,. >13 ' Ti::~:, '>1l; ' T::: "F•l 5 F .I\ 5 'Li 2 l. 0 3,T:<:,~33,T-1:1,:;23,T,i,;U3,TdEfA3} FASJ322J C,~LL SIJU?D FAS03230
51 CALL i{;~;\T (i:.HJf(.H),L··1,IJU12) FAS032't0 IF (Lii~C.c;).1) C.~Ll ~!U'1l:L:t<. (MTOT) FASu3250 '·i 2 = 2 ~' I ·J I '2 ':' I J I . 2 + -,: 1 :·13=2~' I iJ I.·!.:'.+ ·!l-f.14=2 "•'1'13/\;,JJ~;-. C:Go2 +-i·1 J
FAS 032 60 FASJ327J FASU32<lJ
..... l.O .....
CAL L AS S ··l r:\ L ( Al u T ( d l ) , AT iJ T Hl 2 ) , AT 0 T ( M 3 ) , AT 0 T (, l Ld , U·1 , I D I M 2 ) NAV=~EJB~(~GANJ+l) NA V V = i\l t, ~ u :\'( 2 t- { :'J .5 !\ r U- i ) /:JC LJ B ) IF (NAVV.LT.NAV> N~VV=NAV MI= i'J3AND t-N c ,)6-1 iH=l N2=2>:qAV+>il M3=2*;-.i4VV+i·l2 C1\ LL Si:: Sr; L ( 1'. T :~ T ( ,.;t ) , .t\T 'lT ( 'ft:2 ) , J\ T Ll T ( . 13) , r l E 1, :-~ :1 M,; [), 1, I\' Bl CC K, ~ff QR,
1 i\l AV , ;.JI , 1 , 2 , 3 , 4 ) Ml=l '12=:\l[,;L\X+'.1-ll .1j=·~·-. :;.;A+·"IL. M4= :'-JC: :'lA X +r13 :"'15= ~Jf ''1.\X + 14 .·16= .'E ,·i.:\ A+.·i 5 M1= •\IE:,lAX +~'16 ~-13= i'l ':: .''1,l\ X +;·l 7 19=~~E,·l'\X+·l3
i~ i J= :'4E= 1lA X +.·l<-J ··111 = \][. J;\X +,'.;t J
FASJ3290 FASu3300 FAS03310 FAS0332.) FAS03330 FAS03340 FASJ335'J FAS03360 FAS03370 FASJ3330 FAS033?0 FAS03400 FAS0341J · FASD342J FAS J343 '.) FAS0344Q FAS03ttS 0 FAS03460 FAS03470 FAS0348J FAS 03Lt9 0
/l_::=·::: '.'..:~- ~:::: FASJ3.:JJ,) ·"113=i'JE ··i~X+'H 2 FA SJ35 la :'114= 3~:ND S+i\13 FAS J3 52 J IS I Z t=i'J EU d'·' ·J .JL iJC:< FAS 03 53 O CAL l 0 I St-' ( .!\ T <; T ( . I l. .J l , ~ I fJ rt ,. i. 1d , I .) I Z. .: } FAS 0 3 S 4 0 CALL Sfkl'J (,-1TOT{,'l),'\TCT(.-iZ),i\TUT(,,,3},/\PHC~1't),ATOT(i·1':5),:'\TOT(1"i6) FAS0355'J
1, 1\TC!T ( ':7), /\TtH (. '.J J, :HUT [ :·i9), l\TOT ('HJ), AT!JT ( 'n 1), ATtJT{ 1..112), ,'\TOT (;H3FAS J35:>J 2)) FASJ3570 1'114<·~".::~AX+•HJ FASOJ5JJ :'H 5= 1~~.>1AX hi 14 FAS '13 5 'JO 1'1l6= :·~t1';AX+:H 5 FAS 0360J
Ml7=NE~AX+~l6 FAS03bl0 Mld=NEMAX+Ml7 FASJ362J IF <t~PLUS.f.(J.9:,) GP TO l FAS03630 c AL l E s T ~A f.l ( AT u T ( ,·-1 1 ) ' AT ~n ( f.12 ) ' AT D T P-13) ' .'\TUT ( '-14 ) ' ;\ T n T ( ;l; 5 ) ' ~ T 0 T ( M:) ) F A s 0 3 6 4 0
1, A TOT( "i7) , ;.\TO f ( . .J.3) , AT liT ( ,1')) , AT OT <:HJ ) , f\ TGT PH l) , ATC T t ·~ 12 ) , ~n OT ( '113 ;- .~ S 0 3.:., 5 :J 2)' ~ TJ1 LH 4) '..\ T 1J T { .H :5} 't\ T dT( :H6) 'ATOT un 7) 'ATOT( Ml d) ) FAS 03660
CALL LlUTPUT FASJ367J IF C·IPLUS.GT.9) GlJ TJ 1 FAS03c80 IF (LlNC.GT.~I~C~} GO TO 52 FAS036~J CALL SIJU~u FASJ37J) GO Tu Sl FAS03710
52 L :::11'.\IJ=L GAO+ 1 FAS0372;) FASOJ7:i0 FAS OJ JltO FASJ375J FAS03760 FAS03770 FASJ3780
5'.:i5
IF (LC~O.GT.NLG~OS) ~C TO ~55 NP S = ,'j? S 5 ( L c; ;\ 0 ) NINCR=NINCRT(LOADl GO TtJ 952 j:~ITE(l!l,oJ)
STtJP C FAS01790 5d FURMAT (2JA~J FAS03800 ':5'-J Fu.~·ii\T (2fl2.J) FAS03810 60 FJR~AT (l~ll FASJ332J 61 F~;-<;·.;AT (/Tl;J,2J . .:;<t) FASJ383Q 62 FOR~AT (12IG) FA$03D40 6 3 F 2 .'.\. ·1 f1 T ( ;) c 1 2 • 5 ) FAS 0 3 o ::> 0 64 FCR..·1c\T (!TlJ,23!lC{'.·SS-SECTE;N~L AP,f\ = ,El2.5) FASJ186J u5 Fr;,{ .·I~ T ( IT L J, l 1 H::: • l • ST:{ t=: SS , T 3 5 , 4H i\ { 1) , T 5 D , -'t 1 i '.<. ( 1 l , T 6 5 , 1 3 HJ. L • l • ST;= .4S )3 B 71
l R. c SS, I .) J 1 't :-< ·d 2) , T9 :J, 1~:11<. ( .d ) FAS 0 3 Sa D 66 FO~MAT (/T3,l!HT~~SIJN -1 ,T2J,El~.3,T35,~12.j,T50,El2.S,T65,El2.5FASJ3~~0
l,T00,Fl2.5,T75,~12.5) FASQ3~J8
67 fJ,<."1AT (/T3,11HCJ·~.) -1 ,T2J,!::12.5 7 T35,~l2.5,T5),El2.5,T65,t=l2.5F~SBS1:) l,TJJ,~l2.5,T~~,El2.3) FAS03920
68 fOR~AT (/TJ,13HTENSIJN -2 ,T2J,El2.5,T35,El2.5,T50,El2.5,T65,El2FAS03~3J
l.5,TJ0,~12.5,T95,El2.5) FAS03940 800 FORMAT (!T3,13HTENSION -3 ,T20,El2.5,T35,El2.5,T50,El2.5,T65,El2FAS03950
l.5,T80,El2.5,T~5,El2.5) FAS03960 801 F 0 R ~::A T ( IT 3 ,13 H C 0C'i r> - 3 , T 2. 0 , E 12 • 5 , T 3 5 , E l 2 • 5 , T 5 0 , E 12 • 5 , T 6 5 , E 12 FA S 0 3 9 7 0
l.5,TJO,El2..5,T95,El2.Sl FAS0393J 69 FORMAT (/T3,l3HCU~? -2 ,T20,El2.5,T35,E12.5,TSO,Fl2.5,T&5,El2FAS03990
l.5,Td0,fl2.5,T9S,~12.S) FAS040JO 70 h.ir<. ;::11 <IT3,15HSHEAK. -12 1 T20,E12.5,TJ5,':12.5,T50,t=l2.5,T:.S5,EF~SJ401J
112.5,TGO,Cl2.5,T95,El2.5) FAS04020 802 FORM\T {/T3,l5HSHEAR -23 ,T20,El2.S,T35,Fl2.5,T50,~l2.5,T65,EFAS04030
ll2.5,TJO,El2.5,TY5,El2.5l FASJ4J40 11 FcR,'lAT c1rs,341-r-100IFE!l 0 .i\•\ileERG-GSG001) P.l\R1'i•lETt::RSl FAS0405J. 80::.> FOR\1AT (/Tj,15tiSi-lEt\;~ -13 ,T20.,El2.5,T35,E:l2.'.5,T50,El2.5,T65,EFAS04060
ll2.5,T8J,El2.5,T95,El2.5) FAS04070 72 F·.Ji~MH ( 1:JF12.6) FASJ40dQ 73 F~~~~T (/TlJ,2~HULTI~~TE ST~ESS VALUi:S,/T5,9HUIRECTIGN,T25,7HT~NSIFASJ4J9J
l iJ!\ , T 4 5 , l 1 H C C 1 P ,-~ E S S I U , ·J , I T 3 , l H 1 , 1 2 5 , E:l 2 • 5 , T 4 5 , E 12 • 5 ,/ , T 3 ,1 H 2 , T 2 F AS 0 41 J 0 2 5 ' ':: 1 -~ • ') 'T 4 :> ' != t ' • 5 ' IT .j' 1H3, T 2. ? ' E 12 • 5 'T 4 5 ' f. l 2. 5 IT 2 .) ' f, f-l s I~ f: f,. R 2 ~. ' T J 3 ' ;: f\ s J !~ l l 0 J = i .:'. • 5 ,; r 2 J , 1 .:> ;- ; ._ .... i-z i ::.; 1 , l .:s 3 , E i 2 • '.) , / 1 2 J , • ~ H::: A~ i 2 • , r :n , i: i 2 • s > FA s o !tl 2 J
74 FuK~~T {c~l2.5) FAS04130 15 FG~:-11\T (TlJ,'hHC:JULI !: ,T30,6!-iEllT =,El4.7,T55,oHt22T =,'?14.7,TdJ,6FASYtl4J
U i '~ .::, .) T = , ~- 1 ~, .• 7 , I , I :J :; , 0; : :": l 1 C = , ~ l '; • 7 7 T c; :5 1 S H :::: 7 ? C = , C l ~ • 7 , T ;1 0 , ·:'! ~; ~ J JC = FAS 0 4 1 5 0 2 7 tl'r.7 1 //TlCJ,l4riS:-J~A), i·\·JOULII:,T.JU,6HG12 = ,El".-.7,T55,6HG13 = ,El4FAS04160 3.7,TJ0,61l~2J = ,El4.7 7 //Tl0,l~HPQ!SSD~ PATIJS:,T3J,6HU12T =,El4.7,FAS041TJ 4T5~,~HJ13T =,fl4.7,TJC,6HU23T =,~l4.7,/TJJ,6HU12C =,El4.7,T55,6HU1FAS04180 53C =,tl~.7,f30,oHJ25C =,=14.7) FAS04l~a
7 6 F Ci' z ·H r h 1 j , 2 r 1 2 • ) J FA s J 4 2 J 0 71 FU~MAT (///Tl0,3JH~L\STIC CCNSTANT5 FOR MAT~~ItL,13//) FAS0;210
7L FURMAT (6X,3I6) FAS04220 8 J F C.J ;:, '1;\T ( I IT l J , 2. iii I ".I f l f\ L ·~ l.'i- /1 L C :: ;, :·:JI"' fl T != S , I IT 1 J 1 'J ;.1'1·-:1 '.) f. PT • , 5 X, 7H Y FAS 04 2 3 J
l C~L1;..,11Jx,711z CIJU1~./) F1\$1)-f.240
82
83 d4 06
c c c c
l
FO;{~·lAT (/T::,,79HELE. NIJ. NOOE NO. y cnm~. z COOR. AREFAS04250 lA MATERIAL ANGLE) FAS042GO
FURriAT <TlJ,I3,TUl,15,T26,2Fl2.6) FAS04270 FOR~AT (T53,fl4.7,T72,I3,TJ3,F6.2/) FAS0428J FORi'lAT (luX, I5,'·)X,FlJ.9,7x,F10.6) FASOl.t290 END FAS04300 SU W~ GUT I 1 ; !:: C .: i\JT i.: ,-{ FA$ Q .+ 3 1 Q KEAL*& Y,Z FAS04320 C;J:J\1J;'i /G'.JRl/ 'f(40J,ii-) ,Z{4v0,4) 7 1\J0(40'.),3) FASQ433J C0'·1"1Cf\i /GfJ'< 71 ns T >\' YSL ,z SL' F'·1Ul T 'F Qf?C F 'DEL T, Ye, lD, SU'-1/\, l\JTEST' TS n~ T FAS Jlt31t0 ,_,_ '1.J·i /GDRll/ .:.: :~\;(,:\!0S,·\JPS,i\DIF,'1,NANG,NINCR,Lli\JC,t-!ElJ,i,!EQF,f'lBAND,F,'.\SJ4350
l :'-l E ~ 3 , :·Jn L 0 CK , L [) 1\ 0 , i ~? S S { 5 ) , ~1 l: L ~.::::; T , D c Lr;::; T , i .; PLUS , ;.; f Q 3 2 FAS •Yt 3 6 :J CQ\l1lT'l If.Ahl?/ YCT(4.JVJ,ZCT<4UJ),JK(40JO) FAS04370 CU.·l ;,_J"l /rMHO/ L·~·.,L·! FAS04380 CD'i·lD~·I /EAH6J/ ~JISP;H2,5),KEY(lJ) FASalt-39.J
FAS044QO CALLEO F,\:JM MAPl FASJ't41J TrlIS R:JUTI1~;:. t-I:.i.J.::> TdE Ct1'!TE~, OF C:ACH ELE.•1Ei'!T FAS'J442J
'Y_· 1 I =l, 1'!': ,,~ X YC T ( l ) = ( Y ( I , i } t- Y { I , l.) + Y ( I , 3 } ) / 3. Q ZC.f( ll= {l( I, ll+Z( I,L>+Z( I,J) )/.3.0 c.j.;J ~:,0.:: IF (Kt::Y(l).!'.;!::.J) r<:::Tu?,:\i •·! ~ I T '.: ( L v·j , 3 ) rl R. I T...: ( l ;J , :'.) )
NE.L=':c ·.-\X/3 N.~:: L=·\ c L 0:,3 f\Jt xr:~:\=;,[,·iAX-~; .. ; ~L IF C;CL.EW.fJ) ·::.•i L~ 11)
0 u 2 I = 1 , ~ -~: L d Ill=I+l
FAS04430 FAS J4ft4'.1 FASJ44.JQ FAS04lt6J FAS0~47·J FAS'.)44SJ FAS.J4490 FAS04500 FAS J'i-!5 l J FAS0't~2J F-AS:J4530 FASO<t5 1t0 FAS04-550 FAS045~-J
112=!+2 FAS04570 2 WKITE(LW,4) I,YCTlI),LCT(IJ,IIl,YCT(I+l),ZCT(I+l),II2,YCT(I+2),ZCTFAS04580
1(1+2) FAS04590 10 IF (N~XT~~.EQ.J} RETUKN FAS04600
NS=~~~L+l FAS0461J .~1UTE(l1J, 1tl (! ,YCT( U,ZCT<I) ,I='.'JS,i'JE>1AX) FAS0462J RETU~N FAS0463J
C FAS04640 3 FfJl-\:'1AT (////,Tl5,3!tHCCt.RDii'JATES OF THE Elt'l.1ENT CENTERS/I) FASJ465J 4
5
c c
FOK~AT (l4,I6,f12,~ll.6,T25,El2.6,T42,I6,TjJ,El2.6,Tb3,~12.6,TJO,IFASJ4660
lo,To3,E12.u,T1Jl,El2.6) FAS0~67u F 1~1;~ '.·1 AT ( T 2 , :HIE L ': • ~ iJ • , T 1 -2 , 3 HY CT , T 2 5 , 3 1-l Z CT , T 4 0 , tl·l ~ u: • N 0 • , T 5 0 ,3 HY CT FA S G 4 6 3 0 l,T~3,JrlZCT,T78,JH~LE. NJ.,T68,3HYCT,TlOl,3rlZCT//) FAS04690.
END FASJ47JJ SUJRJJTIN~ K~AT (A~,LM,IOIM2) FASJ4710
FA$0l~ 72 0 CALLEJ FR8~ MAI~ FASJ473J
R L: 1\ L '" ~ ;\ ~ .. J , b:.) i': , C r:: 1\! , )Ji'! , !'.: t: ~ , G G '4 , Y , l , C , 5 , DC 0 S , D S I i J , T Hr< A 0 F AS 0 4 7 't ') l , A A, 1 i3, ...; C , :J•), >;- t. , .:_, ,, , ;,:-.. ·,, ~ l i. , 01 !. , •J l j, 0 1:-' 1 I) 2 .::'., u L _:i, J 2 ~, Q 3 J,) 36 , J-74, '} 't SF A$ '.)--t 7 5 •J 2 , 0 5 5 , D 6 6 , ;'\ :< , A r<. 2 , S T.-~ , f H , iJ E XX X , tJ E Y Y Y , 0 E l Z l , lJ F X Y X Y FA S 0 4 7 6 0
C0;-1."iOi\J /GOi;:!/ Y(4JJ,!t} ,Z{4CJ,4) ,"JJ{400,3} FAS04770 c c . : ' : I ~; '.Y .:) I l '.\ T { ': ] · .. ) I :~ ~. ( '; :; ::.: } F As J 4 7 3 0 C J '•~ ,.Jty i I G JD, 1 I :.i ST X , Y S L , Z S l , h 1 U l T , f(; RC t , J EL T , Y 0 , Z 1J , S UM A , ~H E ST , TS TR T F AS 04 7 9 0 CG: \.·F~:~ I GL)! 11 / ··l i::i·i 4 X, ;·~o S, i\P S, 1W Ii:: 1-1 1 1\1 A.'JG, ;'-.! P!CP , LI •\IC, NE Q, hi( CF,!\! BA\JO, FAS O!.r S ') u
L'ii::,)c1,1\JL;JC,.,,_, L~t1;;., .'>.1 .,;::; (;,) ,:JcL:'.;T ,:"11=LT':.1T, '-'.PLUS,i<t=C:FP FASJ~::-lJ
C0:1:·1\Jt\ /GJP_l_;;,/ ACl<'t)J),;J.CZ(4JJ),AC3(..;..;o) FAS04820 c 0 :'•l f•1 J: .; I u I) I"<. 2 7 I AL p l { i. tJ , 1 ) ) ' ~ L p 2 { l J ' l 0 ) ' A Ul 3 ( 1 0 ' 1 J ) F A s 0 4 B 3 i) C -: ' ' ~ : /;.~. :J <J I '.; l 2 ( '.t ) J ) , ~ 1 3 ( 4 J J ) , G 2 J { 1+ J J ) , T H C T 1U 't J J ) , IT : l F A S J 4 :.3 !t 0
2ET4(4~J),TH~(l0) FAS0~850 C Ol·1mJ:J I G J ''.J -:JI t i l ( 4 •J J l , :: -2 ::' ( 4 ) J ) , ':: ? 3 ( 4 J 0 ) , '1D11 { ... ) ) l , i) J 1 ~ { 4 J ·J ) , DD l 3 FI\ S J!+ ~; 6 J
i < 4 CJ rJ > , 1 o i o < 'to ·) > , . ; • 1 z 2 < 't .J :J > , ) u 2 .:> < <t .J J > , ,; u L 6 < ·t.J J 1 , ~. J J -_;, c :, J J l , J ,~ 3 6 c 4 J ·J > , FA s ~ 4 s 1 a 2 '.);) 4'.f. ( 4 0 iJ ) , D 0 4 '.;>( -1- ) 0 ) , iJ 0::; :> ( 't 0 :) l , 0 D [; o ( 4 0 J} FAS D4 8 a 0
1
2
3
4 c c c c c c
CGr-l,'1iJf\j /EAHl/ UJ·JOE(400, 3) . FAS0't69J CU~MON /EAH5/AA~(40J),BdPJ(4JO},CCN(40U),ODN(400) ,EEN(400),GGN(4JO)FAS04900 C8AMON /EAHlJ/ LR,L~ FAS049l0 CO\L10f\i /tAH25/ OEXJ( 'tOJ) ,IJEY0(4JJ) ,DEZ0(4JJ) tDEXYLl(4J')) FAS0 1i-920 0 I i·lE >JS I L"Ji\l l\K ( I 1 >I '.·i2 , I JI .-1l) , L; H I 0 I "•12 ) , S TtU 9 ) , TH ( 10} FAS 049 3 0 REAIMD 4 FAS04940 IF {;PS.t:~.l) GO Til lJ DD 4 I=l,:'l':i'·iJ\X K= L~1\T (I) TH'\AJ=THtTA( I )/'i7.2 h7'b C=t..:CJS!T i.\,J.LJ) S=QS I;\l (TH:->.AO) IF CTHcTA{ I) .:::rj.90.ll) C=J.O IF (;JPS.EJ.3) :..:u Tu 2. ALC l=AC l{ I> ALCL=AC2(1) AL CJ=.". C 3 ( I ) GO TJ 3 ALCl=J.;) ALC2=J.J ALC3=J.J OEXJ(l)=C~C*ALCl~JELT+S~S*ALC2~ocLT OEYO(l)=S*S~ALCl~G~LT+C*C*~LC2*UELT ;~.::-.<'.'.:.:;( '. )=2:< 0:''i''' 1 )i:_LT 0"L~LC ~-t.LCl)
DElU(I)=AL~J*DELT CO!'JT Ii iU~
TrlIS PO:<TLJ'>l Is Gf:.TTI1,!G THt Eu::tEl\1T STIFF:\JcSS .. JATRIX
FASU4S5J FAS04960 FASJ497J FAS 049JCl FAS049'JO FAS05JJJ FAS05010 FASJ.S020 FAS~5030 FAS05040 FAS J5u'>0 FA5050">0 FAS'J'.507'.l FAS0'.50EJ'.J FASOSC·JJ FASO::>lOO FA505110 FAS05120 FAS.Jjlj J FAS'J'.:>140 FAS05150
*FAS i)5 l 70
FASJ5190 FAS 052•) 'J
10 OG 12 l=l, 1Jti"1AX AA=Ai\N ( I) l:i8=33N ( I) CC=CCN(I) O:J=OO'.'-d l) E E = C E •'i ( I ) GG=GG>~ (I ) Dll=DDll{I) lHZ=DfH2 (I l Dl3=0Dl3{ l) Dl6=D0l;':;{ !) 022=0022(1) 023=1JJ23{l) D26=TJD2 ~ ( I) DJ3=JiJ33 (I) 036=J036{I) 044=0044( I} [J4.)=J04J ( I) 0:5::)=\JJS:)( II 'VJ.:1= )00 1'l {I)
' . ..-t • •.• .= •• ~ " :.. '
At\ ( 1, l} = ( .J::; j '·'c) .J ,;, l1:.Jt-j,_,J 0.·;-\A'· >-\f\.) / AfU~ AK ( l , 2 ) = ( .) 5 S >:< b J >;< U 1) + J 0 6 >!'Ai\>!< CC ) IM~ c~ M<. ( 1 1 .:> ) = ( 'J :J 5 ':• >J, '·' G :; .._ J ..; :; '~ ,.x ). ,:, r= c ) I MHZ A~(l,4l=!026~A~*AA+045~3d*hB)/~RR A~(l,::))={J2~*CC*~A+D4~~J~~00)/AkR
AK ( l , b ) = ( 0 2 ;:i ~' E r: ':<A::\+ J 't- :> ~' tH ·:<CG ) I A ,{ :'. AK{l,7}=(~3~*3H*A~+J45~d~~AA)/Ak~
AK ( l , s ) = ( \J .i .) ,:, 0 J ~' I\ .~. + ) 4 j >:: ;::, Y" c c ) I ,\ 'J ":>
AK(l,9l=(DJo~GG*~A..-J~5*~3*EEl/AR~ AK(2,2)={0jj*JJ*JJ+0~6~Cc~cc)/ARP
AK{2,3l=(J5j*00*GG•Do6~CC*~El/AP~
FAS0521J FAS05220 FASJ523 J FAS05241J FAS05250 FASJ526J FAS05?70 FAS052:30 FAS 0529J FAS 053C>O FAS05310 FAS 0532 J FAS05330 . FAS053ltJ FASJ53:>J FAS05360 FAS~537!1 FASJ5360 FAS05390 FAS0.34tJO F.l\505410 FAS0542.) FAS 05430 FAS05440 FASu545J FAS 05tr6 '.) FAS054 71) fAS0'54d0 FAS0549fJ FASll55JJ FAS0551J FAS05520
AK(Z,~)={02J~AA*CC+J45*D0*DB)/ARR AK(2,5)=(J26*CC*CC+J45*UD*OD)/ARR AK(2,6)=(026*~ECCC+C45*DD*GC)/AR~ AK(2,7)=(D3o*Ll8*CC+J.+5*DJ*AA)/AR~ AK { 2, 6) = ( u 3 f:)':::;r.""CC + ;)'i-:J >'.<Q:J '::( C} I ARR AK(L,9)=(0J~*GG*CC+D~S*DO*FE)/ARR AK(3,3)=(D5~*GGtGG+J56*~E*FE)/A~R AK(3,4)={026~Af"[E+D4~*GG*3b)/ARR
AK(3,5J={02c*CC~~E+D45*GG*DDJ/ARR AK(3,6)=(020*:E*Cf+J~S*G~~GG)/AR~ AK(3,7J={OJ~~il~~~E+D~5*GG*AA)/ARR
AK ( 3 , 8 l = ( JJ 3 :'::>,,. DJ >:- t E + '.J .+ 5 * G G ':' C C ) I A R R. AK(3,~)=(036~GG*~E+J45~G~~El)/ARR
AK(4,4)=(022*AA*AA+Q~4*dJ*P3)/AR~ AK ( 4 , 5 ) = ( •) 2 2 :i;: /\A t.: CC + ~Ft'• ':' f:LJ =::i H) } I A ~ i{ Al\. ( 4, o) = ( 1)22 >::;\.~"':: t: + Y+ 't>:<[) '3:::(; G) I AK~ AK(4 1 7)={Q44*bG*~A+D23•A~*B~)/AR~ AK(4,3}=(J44*BG*C~+J23~A\~QD)/~~~
AK(4,9l=(O~~~J~*:E+J2J*AA~GG)/A~R
AK(5,5)=(Q22*CC*CC~0~4*00*CD)/ARR AK ( S , (: ) = ( ·J '.? 2 ,:, Cl: >:: [: ~ +- J 'i + '" J :1 ""CG ) I A '-I 1::..
AK{j,7)=(J4~*DJ*~A+J23~CC~bDJ/A~~ AK ( 5 , J ) = ( J 't 4;.: 0 :) ':: C. C +.J 2 3 ,;, CC -'.: L' J ) I A!{ R 1\ '<. ( '.) , 'i ) = ( ')Lr .'.r '' ) '::: .: := + ~: _:, '' ·= '.~ :': ,-. : ) I \ • '."' AK(6,6}=(J22~EC*cE+D~4~CS*GGJ/A~R AK ( 6 , 7 } = ( LJ + ~ :;: G ~:; ::: t\ A + J 2 J ~' E I: "t J ) I tU. h: 14 K ( :_, , r) ) = ( l.J'.; 't "' ;~ :~ '·' C C + J ~ y: r: :: ;, D fJ l I A < !~ A;\! 6, ..; J = C J.:.,.. ,=•- ::~ ;..; ..:; \=:_~ct d.:!.. _) ) .. (: ~ ::·-: C ~ I I 1\ j·~ .. ~~
AK { 7, l l = ( J 3 3 t: b _»<:, '.:.1 + J 't !t~·'r'\ \'-'A:\ ) J A. r,_ !{ A K ( 7 ' 3 l = ( iJ J 3 * d [j * I )i.)+ ,;.-. 't ~~ A ~ >'.: c c ) I A ft II, AK( 7,9}=(JJi*~J*~~+J+4~AA~L~)/ARR ~K(d,5)={033*DJ*JO+J~4*CC*CC)/A~R
FAS0553J FAS05:Vi-O FAS0555() FAS0556'J FAS05570 FAS05580 FASIJ5590 FAS056JO FAS'1,610 FJ\505620 FAS0563J FASO~S!tO
FAS0565'J FASJ56~J
FAS0567u FAS 05680 F..\$0569•) FAS0570 1)
FAS0.5710 FAS05720 FAS05730 F/\S0574J fAS05750 FAS05700 FAS0577J F4S0573'J FASJ5790 FAS058JO FAS 'J513 l 0 FAS0532J F~'\SiJ:JC30
FAS05E40
c c c ZO!J
34 1)
33J
AK(3,9)=(DJ3*0D*GG+044*CC•EE}/ARR AK(9,9)=(033*GG*GG+044*EE*EEl/ARR DO 7 J=l,3 NDU=:-JD ( I, J) LM(J)=l~OO~(~DO,l)
L M C J + 3 ) = I.~ 1 1 J c Pi 0 D , 2 ) L M ( J +6) = I'~ C IJ L: ( :~ iJ 0, J )
7 CC'H I~lUf IF (;~PS.ilE.ll CQ TJ 30J
ST~Cl)=-Dl5*AA~J~TX S T ..\ ( 2 ) =-:.n S ~'CC ~'i) S L< s T'"' ( j) =- Ul. • .::>'·'.:: L. '·-..; .J T ·~ Sl.~(4)=-DlZ*AA~DSTX
STq(5)=-Dl2*CC*DSTX ST,:~ ( o ) =-'.) l ~>CE'" JS 1 >< ST~(7)=-~1;nBJ~1STX
~TKl0)=-GlJ~GJ~JST~
s Tr~< 9 >=-Cl 3,;, uG"'1) s r .< IF (LI'!C.E 1J.ll C.t\LL JA:'-~J·1i (~·!DIF,I1JI l2~L,·1)
JJ=i l..Jl.i JO:>~l I I=l, IDU~2 00 34J J=JJ,I~I~2 '''<.CJ, !U=-\~! Il,Jl JJ=JJ+l WRrTc (4) L'1,AK,STtZ GU 1 u 1.::
FAS05850 FAS058&0 FASJ5870 FAS056HO FAS05890 FAS059:JO FAS0591Q FAS0542J FAS05930
FAS0597J FAS059f30 FAS 059':1) FASJ60J.) FAS06010 FAS 06020 FASJ6J30 FAS06040 /..'..4SJL.ui0 FAS06060 FASJ6070 FASJ&OJJ FAS J6090 FAS~blOJ
FAS0611J FAS0617.0 FASOol30
C **~******t~**~**~*~***~~******~**********¥********~************~*~****FASJ614J C T H~i\; 114L ;_ji<, HY'.:;,w T rl'.;F, ;''.Al C [!;·W ;:::,,\ff :\T) FAS 06 l 5J
N 0 0
3JJ
45;) 4 -'r ,~!
DE XXX=OF. Xtl ( l) OEYYY= De YO ( I) OEZZZ=DEZ:J( I) DEXYXY=OEXYJ(I) TH( ll= ( Dlci>'.~iJ'.::Y:XX+02 ;,'"'Jt:YYY+036::.'<l)EZZZ+!J66i.:f)i:;:XYXY) ~<t\A TH ( 2) = ( Dl 6>: D .-: xx:< t-'.)l 6 t<l) f:YYY +D3 c,>::Jf l Zl +066 ';<!)EXY XY) *CC TH(3)={Jlo*O~XXX+ClJ*OfYYY+D36*0EZZZ+Ob6*DEXYXY)*~F TH(~}=(012~JEXXX+D2l~OFYYY+U2J*O~ZZZ+D2~*JEXYXY)~AA TH{5)=(012~DEXXX+022~GEYYY+Ll23*DEZZZ+D26*nEXYXY)*CC TH(6)~(012*DEXXX+OZ2~0EYYY+Ol3*0EZZZ+02b*O~XYXY)*EE TH(7)=(J13~UEXXX+D23~DEYYY+D33*DEZZZ+036~~~XYXY)*JH T H ( 3 ) = { lJl 3 ~' ._, ::: A '~ ;..: T..) L .i '•' l.J ;.: y '1"i' + i.; .:U ~- G i::: l l l + i) 3 [,, t,; ;) E x y x y ) >:: .) i) TH(9J=(013MO~xxx~D2J~DEYYY+D33*DEZZL+J36*DEXYXY)MGG TH ( 1 0 ) = ( 0 l l ;" i H:: ;, XX+ '1 1 2 ~: D Ct Y Y + 01 3 >:< IJ E Z Z l + 'J 1 6 * ~1 EX Y X Y ) *-A P R AK( l 7 1J)=JU,t:;1., AK ( 2, 1 J J =Cl G •:<CC AK { 3 , 1 J) = 0 lo;, EI: AK ( '~ , 1 Q } = u 1 2 -,;, ; . ~ AK{5 1 10)=Jl.2:*CC Ai<. ( 6, l J l = ') 12 ~' ~ t A,< { 7, l -i l = H 3 ,,<, '-', .I\;... ( 0 7 i. J) = J l_;:;c i .. J AK{ -71 lJ)=:)l"J~:G;
AK(lJ,lO)=Dll~A~R L' ( l ))=':':: )F I F ( L L'i C • t J • l} C AL L t: ~ f-1 J ~v c;,; L) 1 F 1 I J 1"12 1 L 1 ) JJ=l OJ 4•d I I=l. r ~·r '·1.2 Ju 4'...iO J=JJ, IJ I ·12 AK{J,1 I )=AK( I I ,J) JJ=JJ~l
~·J K I f t: ( 1t) L ,-! , J\ i<, TH
FAS06170 FAS0618J FAS06190 FA 506200 FAS06210 FAS0622'.) FAS062.30 FAS06240 FAS06250 FAS06260 FASJ627J FA506230 FASJ\':.290 FAS J63JJ. FAS0631Q FAS06320 FASJ633J FAS 1Ju3Lt0 FAS~635J
FASOt>360 FAS06370 FASJ630'J FAS\16YJJ FAS064Cl0 FAS06410 FASJ,.Az J FAS0641J FASOtA4J f:,S ') 64 '5 iJ FAS06t+60 FAS0647J FAS 06'+3 0
N 0 _.
12 cn;n rnuE RETUo\N END SUORUUTIN~ DISP (U,50LN,ISIZE)
FAS06490 FAS06500 FAS0651J FAS06520
C ***~******~~***~*~**~***~****~'********~*************~******~********* FAS06530 C SUdk.;JLJTliJc TU A~Rf\-.JGE CJISPLACi:.MENTS IN i\JUDAL OR.)'.::R FAS06540 C **********************''***************•*~*************************~***FAS06551
c
10
17
C CL·1J'1Q,·J / (~W: 11 I •Ji:=:·l '\;(, N LlS 1 '.\:PS, i'!O I F.'·1, NANG, :'J F~CP., l VJC, N EQ, i'i Em=, ~-l 81\:-iD, F.~ S J6 S :) :J l 1': i:: ,/ c, ''-i L,U.;C K 'L JI-\ I)'~; t> ') '::i ( j) '') ~ l; ;u T 'l) i: LT CT';~? LLJS 'f']f w :32 FAS Oo5 7 0
cu:i.·rnN /GOR42/ UTt<. E:::P,Ui<.Ef.P FASO&SJO COr·i,"!C~l /EAHl/ I\JJOE{ '1-JJ, J) FASJ659J cu.·1i'IGN /c/::..d7J/ U'J( -'tJJ,3} FAS066:)J REAL*3 SiJUHISIZE) FAS06610 D f ·It: ;·-1 S HH U ( \JU S , 3 ) FA S 0 6 6 2 0
REW PJD 3 LL 2= ~! ~ :J3>:< ,, :i L LlC:<. LL=LL2:-·'lt\,)cH-l 0,J lu I=l,~;JLC:C<
k,;::,:,J (J) {~~L •. {J),J=LL,LL~) LL2=LL-l
K.·~=l
Du 2 J i = l , , ., J S Dt".J 20 J=l, J U( I,J)=J.J \!J= PJJuE ( I, J) I F ( K ~ • f <J • :.i E (~ F ) K ~~ = K ; J + l If <du. :::r.,;. J > ·.:;~1 TC 2 J U ( I , J ) = S C! L i·J <:<. , l )
FAS0663 Q FAS 1)664) FAS06650 FAS066:JJ FAS 066 7 J FASJ6G30 FASIJ6C:H> FASJ67JJ F t.. S 'J :'.l 7 l 'J FAS06720 FAS0673J FAS06740 FASJ675J FAS0b76u FASOu770 FASJ678J fASJ679J FAS 06ti0d
N 0 N
K:J=l\N+ 1 2J Cu~•l fINUt
OIJ 30 I= 1, 3 DO 3J J=l,\JDS
FAS06Rl t) FAS0682J FAS06830
30 UU(J,I)=U(J,I)+UUtJ,I) FAS068'+~ FAS06850 FAS06860 FAS0687J
c c
REfU;{I~
EN::> s u _: :'. .; u r L, ~ s r , ·
lDSSXZ,DSGXi,U) t ci :: x , ,-, r y , 11 -= 1 , -! E 't z , c ;:: ~u , · J r x y , o s c .< , ., c; s Y , o s r, ~ • n i:; G Y z , F A s J ,::, ~: '3 "1
FAS06890 FAS06900 FAS06910
RE AL ;;q3 A!\;~ ' 3 H \j ' c o~ ' f)'.) ;J ' :: ·: : ' r' r: I ::.i ;\ J s ' y ' z FA s J 0 9 2 J CU-ti'!D:'I /GO':~l/ YUtJJ,4) ,Z{-t00,4),;\d(4'.J0d) FAS0:'.>930 v- : :·• /vT~.3/ I>l.H(-+.J::l) ,1-'d(400) FAS06940 c Cl .·L·.1J:~ I GD 1-, 7 / D s T x 'y s L' z ~Lt f ;,1u LT t F (]RC c' '.)EL T' y (J' l [)' s u :Jj ~ t :-n Es T ' Ts TR T FAS 06 9 51) Ci:J.··:·'lCh I GlJf<.11/ ~4 U;:\ ;(, :'LJ 5, >:PS, i~O IF .-i, :'JM...;G, N L\lCk, LI ~JC, l\;fQ, ~IE QF, N BA~JO, FAS 069 6:>
P:::::·~3, ULlCK,L2AfJ,h1PSS(5),JELMOT,:JELT8T,NPLJS,l':tQiJ2 FASJ697J c LH ·i ;j \I I G iJ 1:>. j 'j I ::: l. l ( -t J Q } I .: L 2 ( .;. 0 J ) ' i: 3 3 ( 4 'J J ) ' u.) l 1 ( 4 J J ) ' u;) 12 ( 4 '.) u ) ' JCH 3 F ;\ s 0 6 (> '.:l Q
l { 4J J) , )016 ( 11"'J J) , ,)J 2 ~ ( 't00 ) , DD2 3 ( 4J ,1) , JD2 6 ( 400 l , OD 33 ( 4 00) , IJD 36 ( 4-0J) , F il.S O,'l 9 9 0 2 IJ I).;. 't ( :1-) •) ) 1 • 1 ·Yt" ! '; :; _; ' 1 ).) s 5 ( !t J .J l 1 D 06 f;, { 4 }') ) Fi:\ s J 7 'J J J
C0.·1.·1ilN /G;J-:<.:+J/ Gl2( 'i-JJ) tl;l3(;.Jv) ,.:;L.::i(-'t'JJ) ,Tt-icTA(-tJJ), IT~-i FAS07010 2ET:~(4JJ),THtllJ) FAS07020
C ~'!'1•·1 J'·J /GIY; (,.z / UT.<.:::.:·), u:<. <:: ;-: ° FA S-370 30 UJ''i .; J: J It A ;t) /t, ~\,'.J ( 4 •J .J) , f:- ~LJ ( '• 0 0) , CC 1 l ( "t (~ ; ) , cJ .J ·J ( .,. 1) u) , ~ c ,·!( -'1 J ·) ) , ·_; C ;.J ( i-0 '.)) F.: S :J F>'t :J CC;·l.·L,~ /ri.l.:;LJ/ L~,l·~ FAS0705J C u;.; i 1 ·J, J I := t, ri 1 5 I N JC ') T , • ~ f C ST , i\ lJJ [ ;J ( 9 ':· ) , ~;CST ( g? ) , :\ J fJ ;: F { 9 9 ) , F C <) T ( c; 9 ) , FA S D T a 6 '.)
lfCSTT(2,2) FAS07070 cu:•;,"1 J.\l I~ AH 2 JI t '{ { 'r )) ) ' = z h ) J } ' '= x ( 4') J) , EY l { 4 J J ) ' i.: xz ( 1t'1 ,) ) , Ex y ( 4.) a) ' F \ s J 7 Jn J
l s l c x ( 4 J J) t s l '..} Y{ ·i- j J ) ' s I GZ ( 4(, l) ) 's I G y z ( 'tv u) '3 r :~:< l ( ~. :n) ' s I G x y ( 4 co ) F ,:\ s 0 7 ;y1..., Ci:i··;;:u··j /[i\!->.2j/ :l'.:'.,.{Li('t00) ,rJEYLH40JJ,0cZJ(4JJ),D[XY.)(lt0u) FASJ71JJ C O' ·H 0 :•J I ".:. A: l 6 J I L I 1 ~ ;) :Z { 2 , :5 ) , K c Y ( 1 J } F A S J 7 l 1 0 0 I 1F. NS I Cli'I 0 ~ X L; ~ '. ~ X) , ~) [. Y ( E:: t-l A Y. l , :JC. .i ( : , ~: :;~ X ) , J E ~~y { ~:-=. t1 X) , ~)EX Z. ( '-1 E .'~ 4 X) F ~ S 0 7 l 2 0
N 0 w
28
30
l , iJ E Y l { \ff MAX ) , D S l; X ( 1' JC ."1 AX) , JS G Y ( N E ~1 AX ) , DSG l HJ C: ~1 A X ) , 1) S G X Y ( N EM J\ X ) , FA S J 7 13 J 20SGYZ(NEMAX),DSGXZ(N~MAX) FAS07140 DI~ENSION UCNOS,3) FAS0715J DO 32 I=l,NE~AX FAS07160 NDl=~D(l,l) FAS07170 ND2=ND{l,2) FAS071~0 ND3=NJ(l,3) FAS07L9J K= P'lU (I) FAS07200 ARR=A~{I) FAS07210 AA=A~\.'l{ I) FAS07220 Bi3=GC.:d I) FAS0723J CC=CCH I) FAS07240 00=001\J{l) FAS07250 EE=E':i'J( I l FAS0726J GG=GCN{ I) FAS0l27.J 0 E Y ( I) = ( t, .\ "' U ( i JO l , 2 ) +CC :i< U ( '-! L12 , 2 ) + F E * U ( M 03 , 2 ) ) I ARR FAS 0 7 2 3 J f) El ( I ) = { J ~) t,: u ( :·~ G l t 3 ) .. )I) >!< 'J ( ~w 2 '3 ) +GG >:: u PD 3 ' 3 ) ) I Ar P.. FA s J 7 2 q J JEX( [)=JSTX FAS073'J-'._l IF ('II? 5. 1'i E. l ) l) EX { l ) =UK f c P FAS 0 7 .:H 0 0 F. y l ( I ) = ( i3 ~; '-'u ( ·~ J 1 , 2 ) + J t) '~u ( \j U2' 2 } +G G *U (., ~)3' z ) +A A ::':LJ (.\i 0 l ' 3 ) + FA s 0 7 3 2 ')
l Ct..>U (;•1J2, 3) +-c:!: '::u ( iOJ, 3)) I Ai~f-'. FAS 'J7J~ J D E X l ( I ) = { G :> '*' U ( ,·.J '.'J l , l ) t- D ! ) ':.: U ( ': U L: , l ) + v G ,;, ;J {.'!O 3 , U ) I J..\ :<. ~ F A S J 7 Vt 0 D c X Y ( I ) = ( A/\ "< U { , J iJ i , :;, J i- L.. L. "' .J { ;\i lJ 2 , l ) + !::: E ~' IJ ( r ~ D 3 , l ) ) I ARR F AS 0 7 3 5 v IF {'JPS.E:(J.l) G1J Td 30 FASG73SJ D F. Y ( I) = 0 c Y ( I )-0 EY :J ( I ) FAS 0 7 3 7 J DEL{ I l=u:=l( r )-)::lo{ I) FAS·J7380 J E X Y i ! ) = U t A 'r ( .i. J - ,_; '-- ,, Y _; ( i ) F A S 0 7 .::. 9 0 D c X ( I ) = Q !: X ( 1 l - J c X] { I ) FA S O J l~ ;) Q D S G X ( I ) = 0 0 11 ( I ) ;: ,) E X ( I ) + 0 :Jl 2 ( I ) ""' 0 i::: Y ( I ) + J 0 l 3 { I> >:: 0 E Z ( I ) + 0 D l 6 ( I ) >'.: C [ X Y ( F- ,ij, S :J 7 -+ l .-)
11) FASJ742J iJ SuY ( I ) =JD 12 { I I ,:,!) ~ X ( I ) +J .J2 2 ( I> :-::ij ~ Y { I ) i-~l02 3 ( l) ''DEZ ( I ) + D C2 0 ( I ) >::t) EX Y ( FAS J 7 43 0
11) FAS074~J
N 0 ~
c 31
32
0 S G l ( I ) = D 01 3 ( I) >:<!)EX ( I } +DD 23 ( I) :;'DEY ( I ) +DO 3 3 ( I J >:<[) E Z l I ) +!) D 36 ( I ) >::DEX Y { FAS J 7 't5 0 11) . FAS07460
DSG Y Z ( I} = D U1t 4- ( 1l ~'DEY l { I ) + 0 04 5 ( I> >:< 0 EX l ( I ) FAS 0 7 4 7 0 D S G X Z ( I ) = :J i i 'i- S ( I ) t.: 0 ,;; Y Z ( I ) + J D 5 5 ( I )':< ) c X l ( I ) F A S 0 7 4 J 0 u S G X Y { I } = 0D16 ( I ) '' '.J t ;(( I ) + 1J J 2 t,{ I ) *OE Y ( I ) + :J :13 o { I ) ,,, OE l ( I ) +;) i) 6 6 ( I ) *D [ X Y FAS Q 7 4 9 O
1(1} FASJ750J
f:X( I )=CX{ I)+CEXC I) EY (I>= EY { I) +:)tY (I) EZ(l)=cZ{ I )+CEZ{ I) E Y L( I ) = EY Z { I ) + :~ '.:: 'r' Z { : ) EXZ( I>=EXZ( I )+JEXZ( I) EXY{ I J=U\Y( I )+JEXYC I) s I GX ( I ) = s I ,~x ( i) +JS ~;x ( I ) SIG Y ( I ) = S 11_; Y ( I ) +iJ S G Y ( I ) s I GZ ( I ) = s I Gl ( i ) +w s ;_;z ( I ) SIGYZ{ I>=SluYl( I )+!JS}YZ( I) sr:;xz:<Il-=SF~X?< u+.JS'.;xz\ I>
CGfiT INUE R.cTU ,-<,-.J ;_: ·:)
FAS07510 FAS07520 FASJ753;) FAS07540 FAS07550 FAS0755J FAS0757J FAS07530 FAS075:1U FAS0760v FASJ761J FAS 0762 ·) FAS076JO FASJ764J FAS07650 FAS07650
s J .~ :>. Ju T r " c: :: .s T • ~ 1\ "' c u -~" , J :.::: y , D r: z , ; :: Y L , 0 ~ x z , : 1 ix Y , D s J l , D s Q 2 , n 5 Q 3 , cs G l 2 FA s :.> 1 6 1 J l,OSG13,DSG23,0~l,OE2,0E3,CE12,0f2J,DE13) FAS07680
C F4S0769J c THIS SUGr~GUTI1:E U~Tt.'.f\i,;Ji';cs THE EJUIV4LEf.JT STkESSES FAS077v·J C A·~o OEf~R:' Ii'iE:::, TH:: '-JC:~i I•JC~f"\D'.T·~.L SLOPE VALUt.:S FAS07710 C FAS0772J
RE~L*a ocus,os1~,T~~~u,c,sN,c2,s2 FAS07731 REAL ~J3,K3~ 7 Nl3,Kl~,N2J,~2j FAS07740 RE~l ~,N FAS0775U R'.:.:\L ~il,iJ2,N3, 1<.1,K2,K3 FAS07760
N 0 01
ca~~UN /GO~J/ I~AT(4JJ),AR(4JO) FAS07770 COMMON /G0~7/ JSTX,YSL,ZSL,~MULT,FCRCE,OELT,YO,~O,SUMA,NTEST,TSTRTFAS07780 C01'1MON /GIJRll/ ~ltf"l·\X,i'JDS,NPS,r.HHFM,i\!ANG,NI:\lCR.,Lil\lC,NEQ,NEQF,NBAND,FASJ779J
lN tQB, NB LOCK, L 0/\8, \JPS S ( ~) , OEL MOT, DEL TOT ,NPLUS, i'iEQB2 FAS 07800 C 0 ·FA Ci,\! I GD i< 21) I SP l ( 1 0, 2 ) , SP 2 ( 10, 2 ) , SP 3 ( l 0} , S? I l( l 'J, 2) , SP I 2 ( 10 , 2 ) , SF 1\ S 0 7 Bl 0
1P[3(10) ,~l(l,lJ,2},~2(2,lJ,2),N3(2,lJ),Kl(2,lJ,2},K2(2,lJ,2),K3{2,rAsJ782J 210) ,tKll{ li.J,2) ,::K22( 10,2) ,f-K33( 10,2) ,GK12 ( lU) ,GK13( 10) ,GK23 (lQ) FAS07330
CU M1'11Jf\J I GD ~:2. 21 NT l ( l:)) , 'H 2 ( l 0) , 1'! T 3 ( l J} , PER '1R 1( l 0, 10) , PF R ;v; R 2 { l 0, l 0) FAS 0 7 B'-t J l,P~R~R3(1J,lJ},T:~Pl(iJ,lil),T~~P2(10,10),TE~P3(10,10) FAS078~Q Cll·~··lON /GJ;i.24/ t'~T-i-(10) ,MT5{ 11)) ,TE'1P 1tC!0,11) 1 Tti':P5(10 7 10) FAS07ti60 CO'i:·!U\ /GDi"i.2b/ ,\Cl{40JJ,AC2(4JJ),AC3(4!JJ) FAS0787J CU~iJ~ /GD~27/ ALP1(10 7 1J),ALP2(1J,10),ALP3llO,l0) FAS07830 co~~UN /GD~28/ SL1(2,lO),SL2(2,lOJ,SL3(2,10) FAS07H90. CO:"ii,Jc;;,J /Gl)i<.35/ Ell( 1,J0) ,E22UtJJJ,EJ3(40J} ,OOllUtJ0},0Dl2UtJO) ,OD13FAS079JO
1 ( 't •)) ) , J D 1 .J ( '.,- ..., :J ) , ~)I) 2 2 { 4 U 0 ) , 0 0 r:H 4 Q J ) , D 0 2 6 { 4 J 0 ) , iJ 1) 3 3 { 4 0 0 ) , 0 D 3 6 ( 4 0 J ) , FA S 0 7 9 1 ;) 2DD~4(40J),D~4~{~0J),J055(~0Jl,D066(400} FAS0792J
COT1!JN /CD!-\4J/ Gl2(<tJJ) ,Gl3l400),G23{40J) ,THt::TA(ldO), ITH FAS07930 ZtT!\fltJJ),THi:(l,)) FAS07940
CC;,, ·i J ~; I GD r'. J JI S 11 l .+ . .J 0 ) , S 2 2 ( "1 0 J J , S ?. 3 ( 4 ) J l , S 12 ( 4 .J J ) , S 13 ( 4 J 0 J , S 2 3 C."t Jfi1 S .)7 1~ 5 J 2 .. .d,S.+tt(40J) ,ss.:;(Lj-C.J) ,S66(;.C:u) FAS0796~)
Cu ;,·,UN /GJk95/ Jl2(!t)Q) ,1.JlJ( 1tGJ),U23{40J) ,Ul<:i.dlt),2),UK13(10,2),JKFAS07Si7J 123(1J,21 FAS07980 c_~ ::-Ji;i\ /E::.:-llJ/ L~,L,; FAS079SO cc~~J~ /~AH30/ S~l{~)0),SQ2{400),SQ3(480),S~12(40Ll) ,SQ13(4CO),SQ23FAS030QJ
1(4Jv) FASOdOlu c:-1.n '~~ /F:M!4.J/ \ H( l J) ,\J:4.Z( 10) ,1\1(:3(10) ,:'Ji'l4{1'.J) ,:-'..·i)( 10) FAS03020 C ;_: ·1. 1 ui; I CA 1 i + :.:i I T -~." . .;i ( 1 G , 10 } , T [: 1· 1 ·~2 ( l J , 10 ) , TE \;:, 3 ( l 0 , 1 0) , T '=: 1 \~ 't( 1 J , 1 ) ) f 1\ S :H ) :. } l,TE~~5(1J,10) FASJ304Q CU·F·~c:i /EJ\H5C/ ?Erl-:W.·+{ lo, 10) ,PER'1 :~5( l;),l'J) ,PERr:R6( l'J, 10) FAS0.3050 C!YHJ,\J /Et\H55/ 0t_f41(1J,11l 7 2'.:TA2(1J,l.)) 1 tH::TAJ(lJ,lJ) FASC>d060 C0•"1.·1Jl'l /t:AH65/ f;'\IL(.'t00) FAS0"3070 CU: 1;•; G:~ I [ f;. iH 5 I SP IJ 3 ( 10, 2 ) , SP I 2 3 ( l u) , SP I 1 .J ( 1 Q t , SP 3 3 ( l J, 2 ) , \; 3 3 ( 2 , 10FASaJ0 SJ
N 0 en
2
l , 2 ) , 10 3 ( 2 , l 0 , 2 ) , S L.3 J ( 2 , l J ) , SP 2 3 ( 1 J ) , N 2 3 ( 2 , 1 a I , K2 3 ( 2 , 1 0 ) , SL 2 3 ( l J ) , SF AS 0 8 0 9 0 2Pl3{10),~13(2,1J),Kl3(2,LO),SL13(10) FAS08100
CD !'id u 1 ~ I EA H du I !'J T j 3 ( 1 J ) , P '-iR 3 3 ( 1 0, l 0 ) , T ~\ R J 3 ( l 0 ,10 ) , I\'. T 13 ( 1 0) , PM R 1 3 ( 1 FAS 0 811 0 1 J , l Q ) , l ."1R 13 ( l J , 1 C> ) , "l T 2 3 ( l J ) , P 1'-1R2 3 t l 0 , 1 0 ) , HIRZ 3 ( l 0 , 1 0 ) , NT A L P 3 ( l 0 ) , F AS 0 8 l 2 0 2TALP3(10,10) FAS08130 co~~UN /EAH55/ N~33(1J),P~MRJ3(1J,lJ),T~~R33(1J,LJ),NM13(1J),PM~~lFASJU14J
2 3 ( l ') ,1 0 ) , T I'. K 1 3 ( 1 0 , l J ) , :\! i'12 3 ( 1 u} , P :-.~MR 21 ( 1 0 , 1 0 ) , T i"l ~r-<2 3 ( 1 0 , l 0 ) , ~! 8 c T A 3 FAS 0 8 l 5 Q 3<l•J) ,TdETA3C 10, l'.J) FASOdloO
DIMENSION SA1(4JJ),KA01(1J),REDlflO),FACl(lJ) FAS08170 0 I;-..; E; ~ S I iJN !J '.:: l ( "J [ :-1 AX ) , 0 E 2 ( ~-lh'·~ AX ) , 0 E 3 ( NE ~-1 t\ X) , iJ E 12 ( bl i: MAX ) , 0 E 2 3 ( N ['Ii\ AX ) FAS 0 8 18 IJ l,D=lJCNE~AX} FAS08l9J
0 I :·\ENS I Ul !JS Q 1 UH: M c\X) , l:r S Q2 ( ,\ [: i'-~ AX ) , D S C.J 3 ( I\! E \'.AX) , u S G 12 UH= 1•1 AX ) , OS G l 3 ( r'.l Fl\ S 03 2 0 0 lF>1AX),DSG21(NE:·lAXl FAS032lJ
D I · IE l S I u :"-J 0 :: X (;-,; t . 1 AX ) , D t Y PE ;.1 AX ) , J -:: L ( ' . '- . L'\ X ) , 0 ~ X Y ( · .. , .:; , ; •'· X ) , Y X Z li·; r;: ·:: .'\ .>() F .-'\ S J 3 2 2 J l t Oc:Y L\ ·~l.·iA~() FAS18230
DO 2 I=l,NA~G FAS03240 KAJl(l)=O FASOB250 r : :~ l ( I } = ·1 • ' FA S 0 S 2 6 0 CO~Tl~Uf FAS0B27J KTJT=O FAS08280 KM=O FAS082~J Kl=J FAS083Ju DO 36 I=l,NiMAX FAS08310 S ~ l ( I l = ."\ o S { S 1Jl ( I ) ) FAS 0 8 3 ~ .) ~A1Lll)=J.J FAS0d330 TH~AD=TH~T~(I)/~7.20j7d FAS0334J C=DCJSlTHRAD) FASJ835J ~ = .-; ; f '• f T :~ i-?. /H)) IF ( I ri t:T A { I ) • E 1~ • 90. J) C= J. 0 C2=C'-'C S.2=S.i'.:S\ 8[1( Il=C2~:.JcX( l)+SZ''':JcY( I>-c:::s;~,:-o~Xf(I}
F4S083o0 FAS0037J FAS0833J FAS08390 F.AS0'34'JO
N 0 -....i
4
5
6
OE2( IJ=Sl*J;:X( I>+c2:::uf.Y( I)+C>:'SN~DEXY(I) DE3 ( I) =D[Z ( I l DE23(l)=C*OiYZ(IJ+SN*DEXZ{I) OE13(1}=-S~*JEYZ{l)+C*DEXZ(l)
D E 1 2 ( I ) = 2 • o;, { C ''.: S i i ::< D ~ X ( I ) -V'S : ~ >:: G E Y ( I ) ) + 0 EX Y ( I ) * ( C 2 - S 2 ) D S: H ( I ) = 1) c: U I ) I ~ 1 1 ( I ) OSQ2(I)=uZ:L( I)/522(!) DS '.J3 ( I } =D ~ J { I JI S 3 J l I ) OSG23( Il=Di.:2J{ I l/S'1.lt{ I) OS Gl 3 ( I ) = 0 E 13 ( I ) IS :J '.J ( I ) DSG12( Il=DE12! I )/S,.J6{I) S\ll( I} =OSJl (I) +5~1 {I) 5,:12 (I l =:Js::)2 ( I) +S !2 {I) 5 1.JJ( I)=tj5)"J3{ Il+S'.~3( Il SQ23(l)=JSG23( l)+S~Z3(1} S Q 13 l I l = S S ,-; l 3 ( I l + S J l J ( I ) SQ12( I l=JS,_;12< I l +S·H2 {I) K=IMAT(I)
KS=l lF (..l-d(I}.L~.3.J} '<C=2 F :\ I l ( I ) = f ~ 1 U I ) + ,'.\ o ~ { S :,) i ( I ) ) I S L 1( K G , K ) Ul.d l)=:..Ji<.1~(;~.·:r,) Ul~(IJ=U~l3(K 7 K~l IF (AG'S(S(Jl(l)).<:,E.SLl(K.J,K)) GU TC 9 lF (A3S(SJ1(1)).LC.S'->UK,KG)) GC: TC S I F ( :\ c S ( S '.J 1 ( I ) l • Lt: • S ~ I l ( ,, , KG l l G :-; T J 6 KK=2
Ki<=l
FASOd410 FASU8420 FAS08430 FAS08ttttO FASJ945J FAS08't60 FAS08470 FASG94BJ FAS08490 FAS03500 FAS•Jd5lu FAS0852Q FAS J~53;) . FAS0:35tt0 FAS08551 FASOd'56J FAS08::>70 FAS0353·) FAS035';;0 FAS Od6•JO !=ASJ~.'.:-tJ
FAS08620 FASJ3630 FASJ36'tJ FAS0:365Q FAS036:JJ FASJ867J FASJd680 FAS03f,-)0 FAS03700
7 E 11{ l ) =Kl ( K K, K , K ·:; ) ':' 'J l( ,<.:<, K, r( G) >:<( ,"~ c S ( Sr~ l ( I ) ) H' "" ( I'll ( K K, K, 1( G ) -1 • J ) + l • FAS 08 7 l ') 10/cl\ll(K,l\u) FASJH2J
N 0 00
8
9
10
Ell(I)=l.J/Ell{l) GO TU 10 Ell(I)=EKll(K,KGl GU TO 10 IA=ITHETA(I) KAOl(I~)=KAOl(IAl+l
FACUIA)=SAl( I)/~,:1S{Si.ll( !))+FACl(IA) :\AiJO=i"..AfJO+l lf (THiTA(l).~Q.0.0) KZ=~Z+l KG=l IF (SQ2(Il.Lt.·J.J) :<..:i=2 FAILIIl=FAILCIJ+AbS{S~2(!))/SL2{KG,K) U2:J (I l =U:~::_:_:. { ·~, i<.~;) I F ( i.i.. d .S ( .S J 2 ( I J ) • '_; :: •. :> L !. { r( .; , /, ) } ;; LJ T C l tt I F L~ tJ s ( s (} 2 ( I ) ) • L :: • $ p 2 u~ ' KG ) ) ,, G r 0 1 3 IF {AGS{S'.:l2(I)}.L~.s;-,~2(i'-.,:<.G)) GiJ T'.J 11 KK=2 GO TU 12
FAS03730 FASOH74Q FAS0875J FAS 08 760 FAS0d770 FAS 037 .10 FAS087J•J FAS08BJJ FAS08Sl0 FASODP.20 FASJ833J FAS0884G FASJ835·'J FAS Oc3660 FAS 088 7,1 FAS 0~'~3 J FASOJ8'70 FAS08q0J
11 KK=l FASJ~91) 12 E 2 2 ( I ) = .< 2 ( Ki~, !<., i< G) "''!2 ( K K, K, KG} '·' ( .,\ JS ( S Q2 ( I ) ) );:<>:<( .\J2 ( I\ K 1 K, KG ) -1. 0) + 1 • FAS J :3 SI 2 ·)
1 •
14
15
2.'=11~ .. -"-\ :,:, ~) FAS0393J E22( I )=l.J/E22( I) FASQ3940 GU TO 15 FAS08950 '= 2 2{ : } = : j' 2 ~ ( ;<. ' :·~ --» GO T LJ 1::) K:\ L>J=K ADD +l I r ( T. ! ': T /-. ( ! ) • ::= j • , J • J } i( ; = ~~:: + l KG= 1 IF ( S '../ j ( I ) • L '-:. ') • u ) Kt;= 2 Ft IL ( l ) =FA I L ( I ) +:\Li S ( S '~ 3 l I ) ) ISL 3 3 l ;\ C , K) I r ( 1~ .:i S ( S (.{3 ( I )} • G:: • S l 3 3 ( i\. G , K )) G Q T '.J l 9 I F { tb S ( S (J 3 ( I ) ) • L f: • S P 3 3 (.<_ , I~ i_; ) ) u 'J T 0 1 d
F "· S IJ ·:: '~ ~: J FASOd97'.) fAS0398,) FAS•JJ9'1J FASQ;JOJO FAS09010 FAS:J9020 FAS0::I030 FAS 09•JltJ
N 0 l.O
16 17
13
l ~ 20
21 22
23
24 25
IF (A85(SQ3(l)).LE.SPI33(K,KG)) GO TO 16 KK=2 GO TO 17 KK=l
FAS09J50 FAS09060 FAS0907J FAS09030
E33(Il=K;3(K~,K,KG)~N3J(KK,K,KG>*lAGS{SQ3(l)))**C~33(~K,K,KG)-l.JJFAS0909J l+l.0/~K33(K,KG) FAS0910J E3J(l)=l.0/E33{IJ FAS0911J GO TO 2J FAS09llJ c 3 .3 ( I ) = E K 3 3 ( K , lC3 ) FA S 0 9 13 O Gl TJ 20 FAsog140 KA0U=KAOU+l FAS09150 KG=l FASJ9150 IF (SQ 2 3 ( I) • L c. j. ,; ) i\. C = 2 FAS J9 l 7 J . FAIL(l)=FAIL(I}+~ascs123(l))/SL23(K) FAS09l80 IF ( A 8 S ( SQ 2 J ( I )) • ,; c • SL 2 3 ( K l ) '; C T fJ 2 4 FA SJ 9 l 9 0 IF (ABS(SQ2J(!}).L~.SP23{K)) GO TO 23 FASJ92JJ IF (ABS(SQ2J1IJ).LE.SPJ2j(K)} GO TO 21 FAS09210 KK=2 FAS09220 G..J TO 22 KK=l
FASO(i23J FAS0924')
G 2 ::> { I ) = ;'(z ."1 ( :<_:!,, ,( } 0< '<: ~ -, ( ,<. r._ : < ~ 0' { ;: .·::; { S •l? 3 ( I ) ) ) '·• >::( :' l 2:, ( :<.K, K )- l • 0} + 1. 0 I _;1<, 2 FAS J9 2 5 J 13(K) FASJJ26J
G23( I )=l .J/G2J( I) FAS09270 ~O TJ L5 FASQ928J G2}fl)=GK23(KJ FAS09290 J'. ~~, 2~ FAS09300 K .\tJ i)= K t\i} 1J + l KG= l I F ( :) cH 3 ( I ) • L F • •J • 0 ) :<.1.3= 2 FAIL!IJ=F~IL(I)~'i5[5~1~Cill/3L13(K)
IF {:\SS{S',/13(Il}.G,:.SLLHK)) GCJ ftJ 29 I F ( A J S ( S Q l J [ I } J • L c • S P l 3 ( K ) ) G C T •J 2 8
Fi\SJ::i3Li FAS09320 FAS0933J FAS09 3't0 FAS09J.30 FAS '.)93~J
N __, 0
IF (AbS(S.H3(1)).L'.:.5PI13(K)) GO TO 26 FAS09370 KK=2 FAS09380 GO TO 27 FASJ9390
26 KK=l F~S09400 27 Gl3{i)=Kl3(K~,K)~~lJ(KK,K)*{AGS(SQ13(!)))**(Nl3(KK,K)-l.O)+l.O/GKlFASJ9410
29 30
31
13(K) FAS09420 Gl 3( I)= 1. O/Gl3 ( I} FAS0943!J GO TO 30 ·:;1.1( I)=G!\l.JC'.O GU Tu 30 KADJ=Kt\OD+ l KG=l IF (S~l~( Il .L;:: • ..;.J) .<>Z FAIL(Il=FAIL{l)+AilS(SQl2(1})/Sl~(KG,K)
IF (A3S(~Q12(1)).G~.SL3(KG 1 K)) GO TO j4 IF (AoS(SQl~{I)}.L:.SPJ(;~J.1 GJ TC 33 If (A3S(S~l2(!)).LE.SPl3{KJ) GO TO 31 K <.=2 G.J TiJ 3.2 KK=l
FAS 09'i4J FAS09450 FAS09460 FAS J94 7 ,) FAS0~430
FAS0~4JO FAS09500 FAS095l0 FASJ952J FAS0953J FAS09540 FASJ9550 FASoq5,:,a
-, ) -:: 1 2 ( r ) = •( .) ( ;( ;< '/ ) >P 13 ( rZ ,<' K ) J' ( .'\ 3 s ( s CH 2 ( I ) } ) ~< ~'( ~ ~ ( !~}'' K ) -1 • 0 ) + l • 0/GK12 ( K FI\ s ) 9 5 l J l) fASJ95JJ
G12{ Il=l.J/Gl2( I} FAS09590 G 1.i T :J 3 '.) FA S J 9 6 J J
~3 Gl2(Il~~~l2(K) FAS09610 Gu TJ 35 FAS0902J
J~ K~JC=KA~J+l FASJ963J 35 IF (KAJJ.G~.l} KTOT=KfLIT•l FAS09640
CC.i\4TI:JUc C ·-' ': T I . : U ::: ff ( KT ~JT • [ '~ • 1\; :>. ·.:. ~() >.l PL lJ S = 1 ') IF (ttPLUS.E }.lQ) .,;·~ITF:(L'·i,ttl I
FAS0''.1650 FASOi66J FAS"J9670 FAS )'Jf d J
N ..... .....
39 40
c c c
IF (~PLUS.EQ.10) GO TO 4J DO 38 I =l, iJ.C\,.'llG IF lFACl( 1) .EQ.J.O) KADl( l)=l.O IF ( FA C 1 l I ) • t=Q • 0 • 0 l FA C 1( I l = l . 0 R E :Jl ( I ) = F ,•\ C l ( I 1 I:< .'\ 0 l ( I l CtJ1H IiWE 00 39 I=l,il!C:."it.X li4= ITi-ICTA( I) Ell( I )=RtJU L::. H'El l{ I>
C :J::T I 'Wt: IF (Li·'H ... ..":.~.,~l.'JCR.) S,ETURN IF (UELTJT.GT.75.J) CALL PER~OO
l (;. T 1, -lT 2, \]T5 t ,\' T 't , ". T 5, ~; T 3 3 , :'. T 2 .:'., i; T l ?. , 1>1Tii.LP3, Pd: 1 ;,U, PE R\1 R2 , PEP MR 3 2 , AL P 1 , A LP 2 , P ,'i R .:U , P ·H ~ 3 , P; lR 1 3 , J\ L P 3 , 1 !:>: P 1 , T Hi P 2 , T Hl P 3 , T E 1': f> 4 , T EM P 5 3,TAR33,T~R23,T~~l3 1 TALP3)
IF (Jf:L'1CT.GT.0.GJl CALL PEf"'MOD
FAS0969J FAS J97 .,) ,J FAS09710 FAS09720 FAS09730 FASQ9740 FAS09750 F.l\S O~ 76 J FAS09770 FASJ97~U
F6.S097<JJ FASJ980.J FASOJb 10 . FAS09'-~20 FAS0983D FAS0984J FAS093::>0
1 ( \1·11, -.;''.~ ,~J:-13, i'J 14, "1 ~"i,: !<::U, r-.~"2°3, :: ,i 13, :.ir~q 1\3, P~·) : 1 ~4,? u:-,;.:K.5, P:=P .":RS, 3 EF ~S ~»? S~) J 2 T k l , £.it: TA 2 , P: < >l ~ 3 3 , ;) . i" L J , P · 1: '. K 13 , :) E T 113 , TE; L'1 l , T f.'·h·iL , TE \: :'~ 3 , TE ~1 M4, T E :'1;115FAS0 9 8 7 0 3,L·lf!,!U3,T 1·1,'h2.=:,T :,'lP,lJ,T::)fTA3) FAS09880
SUJ:\:J\JT.L.:.: TS UP..J;\lc VALUES OF SIJ
R c Al* d DC JS , J ~ I ~ t TH.-'.;. C , C , Si , t C. t: , .) 2 t C 4- , S '~ REAL :·I, N ~~~L \~,~2,~J,Ki,Kl,~3
C ;J:" '-1 :J .'I I G YU I Li /1 T ( '+ J J ) , :\ ~ E A ( L1- (; 0 j
FAS'.Vi890 FAS09<1J0 FA50991J FA509920 FAS099:JO FASJ994J FAS09950 FAS099o0 FASJ9970 FAS09980 FAS09990 FASlOODO
N __, N
c 2
3
CO:'·iMi..l.-.i /G0'}.7/ tJSfX,YSL,ZSL,F1\1ULT,fGRCE,OELT,YJ,Z.D 1 SUt'i1\,NTEST,T.SfRTFASlOOlD CO;•i:·101\l I GDR 11 / 'lEi-i ~x 'NGS, NP s, '.\JD I Fr·i' N Ai·J G' NI NCR' L Ii'JC' ~!EQ, l'>JC:QF 'N B.1\NO' FAS l JJ1-0 1NE~B,NGLGCK,LOAD,NPSS(5),DELMOT,OELTCT,NPLUS,NEQB2 FAS10030
C 0 '1 ; .. HJ~ I G 0 ;-'.. 2 JI SP 1 ( 1 :), 2 ) , SP 2 ( l 0 , 2 } , SP 3 ( l 0 } , Sr 11 ( 1 0 , 2 ) , S P I 2 ( 10 , 2 ) , S FAS 1 0 0 ·t 0 l P I 3 ( 1 0 ) , : J l ( 2 , l ·J , 2 ) , \J 2 ( 2 , l J , 2 ) , ,\J 3 C 2 , l J ) , K l { 2 , 1 0 , 2 ) , K 2 ( 2 , 1 J , 2 ) , K 3 ( 2 , F A S 1 J J 5 J 210) ,t:Kll( 10,2) 1=K22{ lJ,2) ,FK33( 10,2) ,'.;Kl2(10) ,GK13 ( 10) ,GK23 (10) F~Sl'J060
c u i··: :-h.F~ / G o F 2 z. 1 1 n l < 1 :) > , , n 2 < i o > , l'i T 3 < l o > , P c: R : w l< 1 o , l o > , P F R ;.; R 2 < l o , l o > F As l o o 1 J l,P;:R·~f<.3(10,lJ) rTciAPl(l.J,lJ) ,T~:-1P2(10 ,10) ,Tf.'-W3(lu,10) fASl003u c 0 ~if·l C 1~ I G J R.2 4 I ~.a 4 ( l t)) ' f\H 5 { l c ) , TE ;'1p4 ( l 0 ' 1 0 ) ' T HiP 5 { 1 0' l 0) FA s l 00 9 0 C:J.,~l',1Ulll /GtY<..20/ AC1<4JJ) , 1:'.\C2(liJQ) ,AC3(4JO) FASlJlJJ C u ,.: . ·i G .\l I G 0 P. 2 7 I ALP l ( l 0 , 1 0 ) , A L P 2 ( 1 0 , l J ) , ALP 3 ( 1 G , 10 ) FAS 1 0 11 0 COMM8~ /GD~J5/ Ell(~JJ),E22(400),E33(400),9Dll(40J),0912(40C) ,OD13FAS1Jl20
1(40J),J016(4JJI ,0J22(4JJ),G023(40J),JJ26(4JJ),DD33(4JJ),0036(4JJ),FAS1Jl3J 2 i) iJ !+4 ( lt DJ ) , 0 0 4 ':J ( .;. '.) 1) } , : I Li 5 5 ( :,. 0 0 ) , 0 C 6 .) ( 4 0 0 ) FAS l 0 l 4 0 co~~LlN /GO~~J/ Gl2(4JJ),Gl3(4JO),G23(~00),THETA(4JO),ITH FAS1Jl5J
2ETA(4JJ),T~C(l0) FAS10160 co~~j~ /GD~Sci/ Sll(40J),S22(400),S33(40J),Sl2(400),Sl3(40J),$23(40FAS1017J
20),S44(~J0),S55{4JJ),S56{~JJ) FASlJlCJ COM~O\ /GOkJj/ Jll{~jJ),U13(4UQ),U23(400),UK12{10r2l,UK13(10,2),UKFAS10190
123(10,2) FAS10200 c~,,;.,._'I /c;.;...nlJ/ Li'..:.,Lil FASl;J210 OI:·1E1·JSIC1~ S{36) FAS10220
FAS1J23J J ~ '.) I -: l I 'I ~ l ,\ y FA s l \) 2 4 3 Ou 3 K=l,36 S(K)=O.O Kt<,= I ·:.~TC I) SCll=l/E:ll( Tl S(Z)=-Ul2CI)/~ll(I)
S l 3) ::o- Jl ::S C I > It: l U I ) $(7)=$(2) S(6)=1/EZ2Cll
FAS10.2':i0 FAS1026J F~Sl0270
Fi\Sl)2JJ FASl02~0 FAS103UO FAS1J31J FAS1J320
N __, w
S(<J)=-U2J( I )/EL.d I) ${13)=5{3) S(l4)=S(9) S(l5)=1/E33(I) S{22}=1/G23(Il S(Z·n=l/Gl3( I) S(36)=1/G12(I) Sll( I>=S( 1) S22(Il=S(d) 533{ I l=S( 15) Sl.:'.£ IJ=S{2} Sl3(IJ=S(J) S 2 5 ( I ) = S { en S44tl)=S(2~)
S55(Il=S(2~l
S&o{ I)=S(3;)) SA=S(l)*S{8)¥S(l5)-S(l)~S(9)*S(9)-S(8)~S(3)*S(J)-S(l5)*Sl2)*S(2)
1+2 • * s { -2 } ,., s ( ;, ) ,,, 3 ( J) C 1 1 = C SC a ) "-' S I 1 5 ) - S ( ) ) ,, S ( ~ ) ) I S to. Cl2={S(3)*S(S)-S(2)~5{15))/SA C22=(S(l~)*S(l)-5(~J*S(JJJ/SA Cl3=(S(2)*S(9)-S(JJ~~(8} )/SA CJ3=(S{1JnS(3)-S(2)~S(2))/SA
C Li= { S { 2 }':< S ( 3 )- S ( 9 ) * S { U } IS A C41~=1. /S ( 22) C5::i=l./5[2;1 C6CJ=l./S{3.J) TH {AD= TH c TA ( I ) / 5 7. 2 ')? 7.;, C=iJCUS { H1;<,:,.01 S N= l) S I \j ( T HK 1'4 iJ) I F ( r H ': T A ( I ) • E :-i • 9 J • JJ C = J • :J CZ= C ':'c
FAS10330 FAS1034J FAS10350 FASlJ36J FL\Sl03"l0 FAS103d0 FAS l JJ ·jJ FAS1Cl400 FL\510410 FAS10420 FAS10430 FAS1J440 FAS l J!+5 J FAS104.SO FASliJ47C FAS10430 FASl·J4-.iJ FAS105JO N _, F1-'\SlJ510 ~
FAS1052J F<\S 105 J,) FAS1054u FASl0550 FAS1J5t.O FAS 105 7 ,) FAS105:j) FAS1059J FASlJ6J') FAS10610 FAS 1 ,; ;·, :~ :.; FAS 1 ~>C3 J r AS l ,J 6 1-t·1
5
c c c (.
C4=C2>;'C2 S2= Si'~* SN S4=S2*S2 0Dll(!)=Cll*C4+2.*C2*S2*(Cl2+2.*C66)+C22*S4 J012(l}=C2~S2*(Cll+C22-~.*C66)+Cl2*(S4+C4) D013lll=Cl3*C2+C23~Sl
UD16{1)=-C*SN*CC11*C2-C22*S2-(Cl2+2.*C66)*(C2-S2)) 0;) 2 2 ( I ) = C il ~= S 4 + 2 • * C 2 ~, S 2 * ( C 1 2 + 2 »:< C 6 6 } + C 2 2 >:: C 4 OD23(I)=CL3~Sl+C23~Cl
0026 (I )=-C*SI~"* (Cl l ';'S2-C22*C2+( C 12 +2. ~'Coo)* CC2-S2 l) IJ•J33 (I l-==C33 OOJ0(Il=C*SN*(C23-ClJJ OJ44(l)=C44~C2+C55*SZ DD 4-? ( i ) = C :;' S: l =:: ( C 4 ·'t-C 5 :5 ) DD5j(l)=C53~C2~C44*SZ
DD66(l)={Cll+C22-2.~c121~c2*S2+C66*(C2-S2)*~2 CUNT INUE
fi'~O
SUBsOUTI'-lE Pt:2>·:~o
1 ( 'iT 1 , rH 2 , ;J T 3 , \JT 4 , i'! T 5 , NT 3 i , "IT 2 ~ 1 :\ T l 3 , ~TA l P 3 , P ;: R i·l K i , P ER : 1R 2 , Pt: k >1R 3 2 , AL p l ' ~~ L p 2 ' p 1
'', 3 3 ' p "i :~z j ' p l ~ 1 3 ' l\ l p J , T F ', r l ' TE>l p 2 , TE :-1 Q 3 ' T ~ 11p4 , T FM p 5 3, T -1R3J, f.-IR2.3, T ·liUJ, T ~LP3)
SU3~t)UTI•'k TiJ RE1JUSE MOtJULI fDR- TilEk'-1~L >lR HYGR,JTHF~i·1 AL LOADINGS
FASlv650 FAS l tJ£6 0 FAS10670 FAS1068J FAS10690 FAS10700 FAS1J71J FAS10720 FAS10730 FAS107ttO FAS107::50 FAS1J76J. FAS10770 FAS107d0 FAS1J7~J FASlOBOO FAS1J810 FAS1U82J FAS 1083!J FAS1JJ4J FASlOS:>O FAS10B60 fi,SlJ;)7J FAS10850 FAS1J'3-JO Fi\ S 1 JS.J 'J FAS1091J
CCl:·lr·i:YJ I Gr)f' J/ L11\T { -:tJu) , M{E i\ Utoo) FAS 1J92 0 co,,;-~ CJi-: I G Jr, 7 I :J s T x 'y s L, z s L , F .\' u LT 'F CR.Cc' D t:L r 'y !) ' l D' s W! A' :'.JT ls T' Ts T n FA s l 09 3 \) CC ' 1 ': f ',,I I G 0.::.. l 1 I 1 r: ' .\ '< , , ! ;, <) , ") : , ,. l I F , ., , ~ i 1\ •'-: G , ~-; I r'i CR , L I 'l r: , !\ E;,; , 1\ E CF , :-.~ GA l\J Q , Ft\ S l 0 9 4 0
1 N -: o d , :-. ;:; L c c ,., , L c; A D , N P s s < 5 > , 1) t u~ u T , 11 1: L T o T , !') ? Lu :; , r j c: o , 3 2 F As l J ~ s J D I '~ F. • ~ s r u ;'J Lj r l ( l u ) ' 'lT 2 { l J } , f\; T 3 ( 1 \) } ' p E 11 M iU ( 1 J , 1 tJ } , pf R ~-1R2 ( 1 J ' 1 0 ) F I\ s l ~ 9 6 0
l,PER~RJ(l0,10),TEMPl(lC,10),TEMP2(1J,lO>,T~~P3(lJ,lJJ FAS1J970 Dl>lEf~SION NT·Hl0),'.'if5(lU),Tf1W4(l0,10),Ti::'~P5(10,JO) FAS10980 IJii~E;~SHl.\i 1\;T33{1J),p··;;z33(10,10),T'H;.33(10,l0),NT13(10),PMR13(1 FASlu'.NO
1 u , l J l , T:-1 R 13 ( l J , l J ) , l'H2 3 { l J l , Pi l k.2 3 l 10 , 1 J ) , H1R2 3 ( 1 J ,1 0 } , NT AL P 3 ( l J ) , FAS l l J 0 0 2TALP3(10,l0) FAS11010 COM~JN /GJRZ6/ ACl{~JO),hC2(400),AC3C4JO) FAS11020 JI;~.t'·J3IU.~ :~L,">l(lJ,lJ) ,1\L?2( il),l.J),ALP3{10,H)) FASll030 co~~ON /CJR35/ ~11(40J),E22(4QO),E33(400),DD11(4J1),0012(400) ,DD13FAS11040
1(400),0016(%JUJ,0J22(40Jl,OD23(4JJ),D026(4JJ),UOJ3(40J),0036(4JJ),FAS11050 2(Li',"t(L1,JJ) 1;_; r'.:! 1JJ) ,OG.5.5{ '.-88) ,J0!:6{40'J) FASllOf->O co~~J~ /G0~40/ Gl2(4JO),Sl3(~00J,S23(400},TYETA(4JO),ITH FAS11070
2ETA(4JJ},TH~(10) FASll050 CT1>;Gi'J /E:dilO/ LR,L\·i FAS1109'.) WRITE (6,lJlO) FASlllJJ
lJlJ FC ~il1\T (T2J, ,,.,,.,,;::<.n:'~":' PER.'1GO u; =:c>:n;'"'*:<'~""*"-' 1 ) fASllll0 XLI~C=LlhC FAS11120 DT=TSTRT+(XLI~C-0.Sl*DELT FAS11130 DO 11 I=lr''!c,'1AX FAS1114() N K=I~AT(I) FAS11150
__.
°' fH rl 1 ='.~ r i ( i<.l -1 FAS 1116 J NTH2=iJT2(K)-l FAS11170 ~Trl3=~T3{~)-l FASlllJJ >;T 1.:?>J=;.!T33(K>-l FASlll90 ;-.i T'.-l 2 3 =:. T :: 3 ( :z ) - l F AS 11 2 J 0 NTHlJ=HlJ(i\)-1 FASll.21J DJ l :<.i<.=l,:iTHl FAS1122J KZ=KK+l FAS112JU If (JT.Gt..ff=·lPl(i<.K 1 ;-d.s.\.~J.L1T.Lt.fEi·lPUK.2,:\.)) ~C T·J 2 F~Sll24J
1 CO:Hf\JUE FAS11250 ',1 ~ I T t ( u-; , 12 ) D T FAS 11 2 6 0 GU T•J -199 FAS11270
2 E: 1 l { I ) = ( P c 1,. ·1 i U ( ,<. !<. , K ) - ( ( .J T - T c ;'-: r' 1 { K K , K l ) I C T U1 P 1 ( K 2 , K ) - Ti:: ; 1 P l ( K K , K ) ) ) * F A S l1 2 d J
l(PC~~Rl(KK,K}-PERM~l(K2,K))}*Fll(l)/100.0 FAS11290 DO 3 KK=l,NTH2 FAS11300 K2=KK+l FAS11310 IF (DT.GE.TE·'·1P2(KK,r0.f\!'~i).OT.LE.T.E:\1P2(KZ,K)) GO TJ 4 FAS11320
3 CmlTUiUE FAS1133J WRITE (LW,12) OT FAS11340 GO TO 9~9 FAS1135J
4 .: 2 z < r > = < P :=: ~, 1~2 < Ki<.. , K > - < < u T - TE 1'·'. P 2 < i<. K , K > l / < T e:>1 PL t K2 , K > - TE i 1 P z < K K, K 1 > > ~,FA s 113 6 o 1 (P[Ri·!R2(KK,K)-Pt.<.0!K2(K2,K)) )'-~E22( I )/100.J FAS11370
DO 5 ~K=l,NTH3 FAS113qo K2=KK+l FAS11390 If (;)1 .v::.rt.1Pj(r\K,l\).i'..:J:J.UT.LE.rdP3\K2,i<.}) GG TU 6 FASll4JJ
5 CO~TINUE FAS11410 WRITi (LW,12) OT FASll420 GO TO q9q FAS11430
6 G 12 ( I l = ( P t: i·d : r.._ 3 ( K :< , K ) - { { 0 T -1 t: ~·1P3 ( K K , K ) ) I ( T E \1 P 3 ( K 2 , K ) - T c !-': P 3 ( K K , K ) ) } ~~ FA S 11-4 4 0 1 ( P E: '."-F· l :U ( K K , K ) - ? ~ R '1 R 3 { K 2 , ;{ ) ) ) '* G l 2 ( I ) I l 0 0 • 0 F AS 11 4 'J 0 N __,
DJ lJJ KK=l,.·!H:~J FASll460 ...... K2=KK+l FAS11470 I!= (:H.GE.T.!1'J3(KK,i<).ATJ.DT.LJ:.T lR33(!<.2,Kl) GC TS 101 FASll4'3J
iJJ l. "'~;'J: FASll490 h~ITE (LW,li} DT FAS1150J G1 TJ SYS FAS11511
l 01 L. J J ( I ) = { P i~, 5_~ ( K .<,rd - { { :..., T- f: ;,- 3::: ( ;c< , K) ) I ( T 1': :\ 3 3 ( K2, Kl - P.1?.3 3 ( :< K, !O ) ) tq FAS 11 5 ~ ') 1 ~ .. i ~ 3 3 OZ:< , :O - P i•i r', 3 .:.H K 2, K ) ) ) ':' = 3 J ( I ) I l 0 0. FAS 1153 J
!JJ 103 !'~K=l ,rJTY2J FAS115't•) ~2=KK+l FAS115~J IF (0T.GE.T-n2J(KK,~).A'EJ.'.H.u:.T.'ii<.23(K2,K}} GL' PJ 106 FASll56J
l J 5 C •.1 '-'. TI . ~ U f: FA S l l 5 7 u h.dTc ll1J,l,.~) •JT FASll5dJ ~G TJ 999 FASli59Q
l J LJ 1; 2 .:> ( I ) = ( P rl ~'. 2 3 ( K K t K ) - ( ( J T - T' ', R 2 3 { K K , K ) ) I ( T "! R ~ 3 ( K 2. , K ) - Hl R 2 3 ( K K , K ) ) ) ;~ F l\ S l l 6 CJ J
1( P. ·l R 2 3 ( Kr< , K ) - P •l R. .2 3 ( K 2 , K j ) ) ~' G 2 J ( IJ / l J J • G F A S 11 6 l 0 DO 110 KK=l,NTH13 FASll620 K2=KK+l FAS11630 IF (DT.GE.T~Rl3(KK,Kl.AND.DT.LE.TMRll[K2,K)) GO Tn 111 FAS11640
110 Cu1HliWE FASll050 WRITE (l~J,l2) JT FAS11060 GO TO 99S FAS11670
111 C 1 3 [ I ) = ( P '1fd 3 ( .-, .<., ;<. ) - ( ( LH-T' ~ ·"' 13 ( K K, K) ) I ( T \] '·d 3 { K 2 , k'. } - Ti1l:13 ( K K , K ) } ) ~: FAS 116 3 0 l(P'iiUJ(r<.l<.,K)-PtlRlJ[K2,K)} )i.'Gl3( Il/100.0 FAS1169J
11 CONTINUE FAS11700 IF (~PS.~E.Z.A~0.NPS.NE.4) RETURN FASll710 ~RITE (6,1020) FAS1172J
1020 FOK~AT (T20r'*****~ ~LPrl~ PE~~co ~********') FA51173Q DC 20 I=l,NE~AX FAS1174J K=I~AT(I) FASll75J ,.lfrl"t=.~T'}(K}-1 FAS1176) NTrl5;NTS(K)-l FAS11770 :n t. L =:~TA L p j ( K ) FA s 11 7 a 0 o.:: 7 :1..:<.=1, :T:-;:,. FASll 790 K2=K~+l FAS118JJ If {JT.Gf-.T:.:>l?4U<.K,K) •. \:·~1J.CT.Lt::.TEi'P4(K2,K}) GU T•J 8 FASlElJ
7 CG~TI~JE FASlld21 '' R I T :: { L \;! , l 2 ) D T FAS l 18 3 0 ~~ TJ ~~; FAS1184J
:J AC 1 ( I l = t" l P .L ( l\K, K) - ( { ,J T-T c" l P4 ( K ,(, ,<. l ) I ( TE ,.1? !t (i<.2, K ) -T;::: '-1 P !t [ !\t(, K ) ) ) q Al F :\ $ 118 5 J 1Pl{K~,K)-ALPl(K2,K}) FAS11150 1)J.; ,\\.=1, ·!P~.'.J FASllB70 K2=KK+l FASllc8J IF (JT.GE.TC~P5(KK,K).~NJ.DT.LE.TEMP5(K2,K)} GC TS 10 FASll81J
9 CO~TINUE FASll900 \.\f'ITE (L\•J,12) rn FAS11910 GJ TO 999 FASll92Q
10
140
2J
9'79 c 12
c c c c
AC 2 ( Il =ALP 2 ( K K , K ) - ( ( J T - T E i•\ P 5 ( K K , K ) ) /( l f; l P 5 ( K 2 , I\ ) .- TE HP 5 ( i< K , K } ) )':<( Al F AS 11 9 3 0 1P2(KK,Kl-ALP2{K2,K)) fASll94J
00 140 KK=l,NTAL FAS11950 K2=KK+l FAS11960 I F ( I H •• ,; :::: • T AL p 3 ( K K ' i< ) • ;.\ N 0 • l) T • L E • T A l p ] ( K 2 ' K ) ) Gu T a 1 4 1 FA s l l 9 7 a CONTINUE FAS11930 WRITf (LW,12) DT FASll99J Gu TQ 999 FAS1200Q AC3(1}=ALPJ(KK,K)-((OT-TALPJ(KK,Kl)/CTALP3(K2,K)-TALP3(KK,K)))*(ALFAS1201J
lP3{KK,K)-ALP3(K2,Kl) FAS12~20
CGNTINUE FAS120J0 RETU~N FAS12040
STOP FAS12C5J FAS120:.JG
FO~~~.'.\T (/////L:.J,:):'>rl ~;\XI:,,J·· VALUc c;: Tc;JPE~UU;<E CHl\i'--iGE HAS BEE'~ EFAS12Cl70 lXCEEO~u, ·r~~PE~ATU~~ IS •,[12.7}
END S-JS),'~lJTI:,~ ; ii,.:.
l ( rn 1, .'.JT 2, H 3 , :\ ! 4, !°'i TS,\ T 3 .:3, .\J T 23, ~ 1 T13, NT Al P3 , P ~;-.... :·lR l , PE k "lf~2 , PER MR3 2, ~\LP l , I\ LP 2, P. ~·-· ..J J, P. i 1\2 3, P .P 13, r1 l P 3, T F:r c P 1 7 T E>1? 2, T f :~I':,, T cY 04, T FM P':) ...> ' I . ' .. , _;.) ' r :·l r{ 2 3 ' T . '. ~ l 3 ' T .. L ,:. :.) ' ' ~ ,; 1\ )
SUd'<.'JUTPJE H.i K~,\D INF81-~?· 1\Tir-;;1 Fl!? I:'lCf'.E'ii:::1'!TAL THE;--'.,'1AL LOAOI:'!G m:. HYGf.1.CTrlE.k. JL\l L:J'\Oli'iG
FAS12080 FAS lZO'YJ FAS 12 l·JO f ;\S 121. l '1 FAS12120 Fl\SL213J F 1~S l 21 't:) FAS12l?J FAS12l60 FAS12ll0
C \.J1·i'i .J>1 I G ,J,; l I ::, ~ L'. 1 Y S '- ; 7 ) '- ~ i:- 'l 1 L T • ;= r :> C [ , ')FLT , Y L1, l f), SI.Fl A, 'lT E S l , TS TR T FAS 12 if,) CLJ;·.; ·lU\i /r:;l);t 11/ 1•l [\: 1~ X, ,'JDS, :\JPS, r·.;u H- ;"1 7 ,,Jt,:6, i': I >1C1', LI ·'-!C, 1'.c 'i, ;' t 1rJF-, :\ ;) :\,W, f t\S 121 9')
l :\J E 1d,.-1 i:J LCC1\, L C'A J, J\J PSS { 5) , JElr·lGT, De LT OT, !'F'LIJS, NEQ 02 FAS 122 .J J 0 Ll HJ s I Li~ 1H 1 ( l ) ) ' a 2 ( l ) ) 1 f·' T 3 ( l () ) ' ? ;:; k r·:,u ( l J '1 -J ) , p ER :,:K 2 ( l 0 ' l 0) FA 5 12 2 l 0
1, P:: 1-1. 1·'.~'3 ( 10, i .J l , T:: Y l { L , l ~ l , l L.. •. ? .2 ( 1 ) , l:;} , T 1: ~? .3. t l J , l C) f AS 122 2 0 J f ·if. :~ S I 0 :'i ;\IT It ( l J ) , . J 1 5 { 1 ) ) , T i:. I\? 4 ( l .J , l J ) , T ~ · IP 5 ( l J , l ,J ) FAS 12 2 3 .) Ol,·1i::•'~S!i_/J 1\LPl( lJ,10) ,ALPZ( 10,lJ} ,AU'3Cl0,l0) FAS12240
N __, l.D
DI~E~SION ~T33(1J),PMR33(10,10),T~R33(1J,11),NT13(1J),p~Rl3(1 FAS12250 l J, l 0) , TMR 1 3 ( l J, 1 J) , ~Jf 23 ( l ;) ) , P r1R23 ( l 0 ,10) , T MR2 3 { l.J, l 0) , f\JT ALP 3 ( l 0 ) , FAS 12 260 2TALP3(10,10) FAS12270
CGMMDN /G0~35/ Ell(~JJ),E22(4JO),E33(4JO),D011(4J)),OJ12(4JO},DD13F~Sl22BJ 1(4J0),00l~t4JJ),0022(400),D023(400),0026(400),0033(40J),D036(400),FAS12290 20J44(4JO),OJ~?(~00),DDS5(~00),CD66(40J) FAS12300
COMMON /G8~4J/ Gl2(4JJ) ,Gl3(4JO),G23(400),THETA(40U),IT~I FAS12310 2 E T A ( !~ 0 J ) , Hi :: ( Fl ) F ,\ S 1 2 3 2 f) CO~MON /EAHlO/ LR,L~ FAS12330 DO 6 K=l,NDIFM FAS12340 RE AD ( l ~ , 8 ) ~; T l t K I , · 'l T 2 ( K ) , :--n .3 3 ( K ) , "H 2 3 ( K ) , rH l 3 ( K ) , >JT 3 ( K ) , ~JT 4 ( K ) , i H 5 F A S 1 2 3 S ~)
l(K),NTALP3{K) FAS12360 ;H.-ll=~Hl(K) FAS1237:) f\ITH2=1H2 ( iO iH rl 3 = '\ T J { K) !H·-14=fH4(K) NTrb-=NTS ( iO NHi3 ::\=:iT 3 3 { :< l : ' I; 1 l J = "l T 1 3 ( ;<.) ;\jTH23=fH2J( K) i\J T !IL-= 1~ T C. L P 3 l K } 1': :_ ,~ J : L .~, , ~ ) { ? ~ :-, . ~~. l { I , K } J T ~)~Pl { I , i·"' ) , l = 1, ,·, r H l ) Rf A 0 (LR, 9} ( Pc P-. 1 ;._2 ( 1 , K) , T t: :·l P 2 ( I , K ) , I= 1 , r-n H 2 ) f<. f: !U ( Lr<. t 9 ) ( P ;.~,: 1 3 ( I , K ) , T '.H 3 3 ( I , K ) ,I = l , ~n H 3 3 ) RE J\ ~) ( L I'. , ; ) l P , i r: 2 3 ( I , ~ ) , T >: .=!_ 2:, ( I , K ) , I= l , f·~ T H 2 J ) RE~\ u ( U{ , 'J } ( P ·1 j; 13 ( I , !() , T "1'.U 3 ( I , K) , I= 1 , 'JT H l 3 ) R [ .-~ )( L .-. , -; ) ( f) 1: " . '. ~ ~ : ; '. l i r :: . · '.-· 3 ( I , K ) , I= 1, ~n 1-l 3) R. E 1\D ( L K, 1 5 } { i\ L P 1 l I , I\ J , Tc ·1 ?4 C I , K } , I= 1 , :'ff W+) i{ E ~H) ( Li~ , l 5 ) ( :.. L P 2 ( I , K ) , T t: :'·:f' 5 ( I , K ) , I = l , ':Tl' :J ) CZ;: A') ( L •-\ , 1 '.5 ) ( :, l P J { I , -<. } , T f\ L P 3 ( I , :<. ) , I = l , ;., T :. L ) IF CPt\:.':: 11.21 GU TJ 3J WR. IT ~ ( Ul , 7 )
FAS123d0 FAS12390 FAS124JO FAS1241J FAS1242rJ FAS12430 FAS124!tJ FAS 1245~) FAS12<1-00 FAS12470 FAS 1248') FAS 12'd.J FAS125vJ FAS12~10
FAS1252J FAS125"10 F4S 125-'tJ FA51255J FAS 12?u0
N N 0
DU 20 K=l, i'<D I F.'1 FAS1257u NTHl=NTl(K) FAS12580 NTH2=NT2(K) FAS12590 NTHJ=liT3 ( K) FAS1260J NTH't=;·JT4{ K) FAS12610 NTH5=,·H S ( K) FAS12620 i-..JTH33=(!T3 3 ( ,{) FAS1263u ilJE-1u=:-n13 < K > FAS 126'•0 NT ri 2 J = ,\ff 2 _j { K ) FASL2650 NTAL=NTJ\Lr>.3(K) FAS12660 tlRlTC(LW,10) K FAS12670 ri R I T E ( L 'ti ,11 ) FAS126<lJ lJ lJ l I=l,NTHl FAS12690
1 \/Rllc (LW, 14) P:::~;'•Rl( I,K) ,TUiPl( l,K) FAS12701.l ~l ~ I T E ( L .~ , 1 L: ) FAS1271 J DO 2 I= 1, :H rl2 F;.\512720
2 WR.ITt(L.,i,lLt) PE~~R2(1,K),T~~P2(!,K) FAS 1273·} N N 'tirU T 1= (L\.1,112) FAS12740 __.
OU 1111 I=lri'JTH:J3 FAS 12 750 1111 \•/QI T,:: (L-l,14) P r;~-u H I , K) , I:''' P 3 3 ( I , K) FAS127,:,)
,. :' I T ·~ {ui,113) FAS12.71J DJ 1112 I= l, \JT H2 J FAS127'.JO
1112 '!ii.~~- I T ~ (l.';1~:4) ? .,~ 2 3 ( I , K) , T1•i R2 3 { I,:<.) FAS l27"JJ ..JGITE { L 'ti , 1 l 1t) FAS12£JO DO 1113 I=1,:-n1.nJ FAS12610
1113 •'ff. I T f (L,v 7 l4) P~Rl3(!,K) 7 T~Kl3(1,K) FAS12JZO WP. ITC( L 'ti, 13) FAS12830 ')·j j I=l,:':T1-•3 FAS1284l1
3 fl:~. I T ._: ( l ',; , 14 ) P ;:= :\. 1 r'..-;; { I , K) , TE,'' P 3 { I , :< } FAS12B50 WRlTt=(LW,17) FAS1286J DtJ -t f=l,;.T:l.:i- FAS1227J
4 ..iRITc(L~J,16) i.L? l (I , 1') 1 T ~ ·i?4 (I, K) FAS128d0
WR IT E ( U~ , 1 8 ) DO 5 I= 1, 1\JT H 5
5 WRITE(LW,16) ALP2C I,K),TE:'1P5(I,K) wtUTE (LH,115) 00 1115 I=l,NTAL
1115 ~JfUT1: (Lt-4,ll)) !\LP3(I,K),TALP3(I,K) 2J CCiJTitlU':
KETUkN 30 WRITE (LW,70)
DO 2JO K=l,1'JOif-i1 NTlH=fiT 1 ( K) ~TH2=NT2(K)
NTH 3= >; T 3 ( K } NT rl'+-=; JT It ( K) NTll5=iH5(K) NT rl 3 3=; ~ T 3 3 ( K ) NTH 1 3 = iH 1 3 { K ) NT1i23=!H23(K) ;~T \L-= .L\LPJ{:O .. .JRITt:(l,.J,lJ) K 't! '.\ I T:: t Ui, 11 0 ) D Cl l 0 J') I = l , ·'H i l l
1 J J J ,,.; <.I T t ( L ~l, l 4- ) P ·~: .. '. :~ l ( I , K} , T F: :~ P 1 ( I , K ) ~.;R lTi::(Ui,120) DU 2JJJ I-=l, :HH2
2 '.) J) : . ~ I T .~ ( L 't; , 1 tt ) P : R. 1 ;.? 2 ( I , K) , T '.::MP 2 ( I , K ) ~~RITE (Lh,111(•) DO 199;) I=l,:-.JTHJ.3
1'~9d 1~RITc (L,~,11+) p .. J.<.3:_)( I,K} ,T.'1RJ3( I,~} ~·J ;;. I T t ( L ·i , l u '.• ·J )
\J _I 1 ·n .L I = l , . , f.-J 2. ~· 1991 .~"~IT::: {L~•,14) P''i;Z2J( [,K) ,TMEZJ(l,K)
FAS1289J FAS12900 FAS12910 FAS12920 FAS12930 FAS129 1+J FAS12950 FAS129.'>0 FAS1297) FAS12CJJO FAS12990 FAS130u:J FAS130l'J. FASl3020 FAS13G30 FAS13040 FAS1305u FAS13060 FAS1307() FAS130d0 FAS 130'10 FAS131JJ FASl.3110 FAS13120 FAS131->J FAS13140 FAS13150 FAS13J:,J FAS1317J FAS1:>130 FAS131YU FAS1320Q
N N N
19~2
3000
'.)J'.)Q
199 I 20J
c 7J llJ 12J .i. l l :~
13 >l J LJ'l l 130 14 15 16 1 7 )
1 '3 J
WR lT E ( LVJ, l B 8 ll DO l 992 I= l, ;HH 13 WK. I T t ( L ~-I , l 1+ ) P fi,k_ 1 J ( I , K ) , T t~ !-./, 13 ( I , K ) \~RITE ( L ~ , 13 0 ) DO 3JJJ I=l,'JT:-l3 WRITE(L~,14) PER~RJ(I,Kl,TEMP3(1,K) ~RITE(U~,17J)
OU 4JOO I=l,NTH4 WRIT~(L~ 7 l6) ALPl(I,Kl,TEMP4(I,K) W~ IT d U-J, l d 0 ) O:J 5·JJ J I=l, ''iT:--l> ti i{ i T;:: ( L.;, 16) 4 LP 2 ( I , K) , T ::: 1 PS { I , !<} tlR IT t ( L\'i, 1382) [J::l i ·n 3 I= 1 , :H...;. L ~i 'd T :: { L;. , 1 :1 ) • \ L 1) 3 ( I 1 K ) , T AL P 3 { I , K ) C01'JTJ,\hff RETU:.\;~
FAS13210 FAS13220 FAS1323D FAS13240 FAS13250 FAS13250 FASl3270 FAS1328J FAS1329J FAS133JO FAS13310 FAS1332C> FAS1333J FAS1331+u FAS13350 FAS13360 FAS1337J FAS 13 3 '3 •)
f = ~ f L\1 ( I I , T 1 U , '1- 'th l 1 :- u :--.. ,.\ T I r:,, F: 1 ~ p: CR [ H:: ''JT 1\ L ,'-; D I STU,;. f:: L 0:. D I \ C) F .l $ 13 :,..; J FGRMAl (/,T2J,15ii~ ~~TENTICN Ell,T4J,'~JSITU~E CO~TENT 1 ,/) FASl34JJ F J r;, ·.; 1\T ( I , T Z 'J , l ':l L' .-'.. ': T E i\ T I 0 I' '. E 2 2 , T 4 J , ' f.1 G S I TU i\. f C rJ ,\JT E 1' ! T ' , I ) F AS 1 3 4 l J f,_j.\;·,T u,r2Jy.j._,, ' .. Tc .. i~-·- :: _ _;~~,1·i:~, 1 .":J:)I-:-u::: c·_;T~·''T 1 ,/) r\SlS42.} F- C_I K >1.4 T (/ ' f 2. j ' l 5 H ~ ~ET E: J\J r I u r: G 2 3 'T 4 .) ' I ;-j ;] s I Tu~~ i: c y ~ T c NT I , I ) FA s l 3'd '.)
F•1K. iAf (/,T20,151L ,,t:T1.::~JTIG1~ Gl3,T-'t0,"~JSITU:-\E CUl\iTENT' ,!} FAS1344-J FC!l'),"iAT (!,T20,l.)'-1_; ?.f-:Tr \TIOt; Gl2,Tl1J, • .. : . .:::sIT'JP[ c:: lTE~JT',/) FAS1345) F G ;(.'•1.'\ T { T .~ U, !- l .J. , , T , J, ;- l :J. t,) FAS 1 .:Vi G 1) FJ~.ti:\T {j(fl~.5,FL~.7)) FAS13470 F G ;\ 1·1.4 1 ( T 2. J , C. l 2 • 5 , T !.;. J , F l J • 6 ) F AS l 3 '1· :;, J F : ;:'. 'l :\ T ( I I I , T L 1 2 J :-i ·Ll S I T J:~ t CC c FF I l I E ;n , T 41 t 1 r'·lf.i I$ T lJ RE C J~>1 TENT ' , I , Ti:: 1\S 1 3 49 u 125,'~cTA-i') FASlJ~OO
FtJ"\'';.n {///,T11,2J~-L'·iJISTu1:·r: co:::FflCIC..:<T,T'1-l, 1 .·lCISfU9.f (t)i•H!=:\T',/,TF/\Sl3'.:•l.J L~.Jt 'J.:T,~-2') FAS1.i.S20
N N w
1802 FORMAf (///,T19,2Jri~JlSTLl~E CGEFF1CIENT,T4lr'~CISTURE CONTE~T 1 ,/,TFAS1J53U
c 7 8 9 10 11 12 112 13 113 11 :1-17
la
115
c c
120,'3fTA-3') FAS13540 FAS13550
FO~~AT (//,Tl0,43rlI~FORMATION FOR INCRE~ENTAL THERMAL LOADI~G) FAS1356J FOK~AT (6112) FAS13570 FORM~T (6Fl2.7) FAS135~J
FO~~~T {/,Tl5,lSHN~TERIAL NUMJER,15) FAS135JO FOR:'-1AT (/,T20,l5il~ RETENTIOi" Ell,T40,11HTE'WERATU!<.E,/) FAS13600 FJ!'.MAT (f,T2J,15H'~ RETt=r'1TH11-.J f22,Tl;J,llHTE:1Pt:kATIJ~F.,/) FAS13610 F G ,<. '·'-\ T { I , T 2 J , l 5 :-r.~ :-:. ~ T f :'J TI CJ' . E 3 3 , T 1t 0 , l 1 YT :: '·1 i' HJ T U •'. ~ ,/ } F .!\ S l 3 S 2 J FOK~AT (/,T20,l5Hl KETENTICN Gl2,T40,11HTE~PE?ATU~E,/) FAS1J63J FLl~~AT t/,T20,15H~ ~ETENTION G23,T40,llHT~~PERATURE,/) FAS13640 fJ,-v•:Ar {/,T2),15H'.; :'.~T:::·:TIU..J GlJ,T4fJ,llHTE"1P[l~ATU)E-,/) FAS136'5') FG~~AT (///,TZJ,lJ~TrlERM~l CUEFFICIENTrT4J,llHTE1PERATU~f,/,T25,lHFAS1365J
1ALPH4-l) FAS13t:i7J Fu K ;,1 r~ T (/ I I , T 2 J , l <J h I!-;;;::-'. '·i.' L CG E FF I C I E i'1 T, T 4 Q , 11 il T F. M? E K. AT U i::.;: , I , T 2 0 , 7 Hf.\ S 13 Ci S 0
lALPHA-2) FAS13690 f 0 ~1'1 AT (I I I , TL Q tl 9 :H :-1 Ei~. i-1 ·\ L C Cf FF IC I c~·l T, T 40, 11 HT'.: >1? t: KA l U1'c , I, T 2 '.), 7H Ft\ S 13 7 J :)
lALPHA-3) FAS1371 J ~NO FAS1372J S U .j R ;JU T i. '.i E Ii'< I T I l FAS 1 3 7 3 cl
******~***u**~*~**~***~~*~********$**•*************************** FAS1374J
R. E Al * 8 A t\ :·h e. 3 r J , CC 1'-1 , Ll 'Y; , E ;: N , G G ~ , Y , Z COMMON /GOkl/ Y(4JJ,4),L(400 1 4),NJ(40J,3)
Fi\Sl37?C FASlJ760 FAS137!0 FAS13730
cu \·I u·i I '.JL.J•<.l. ii Ii-:::~;.;~ , \;~) s ' ·\: p s' f\1G I F :4. t\I A\: G' ;\j I nci-: t l I ~J:::' f\~f: 0' ~]:: QF '~J B;\:·w, r- t.;, s l 5 7 ') J 1 \J '.: 0 J , i 1 f, L 0 C K , L Ji\ ! J , , ~ r) S S ( 5 ) , D '.: l:-i C T , D t l T G T , ,;; P L U S , "l .::: ~ 3 -~ F A S 1 J :' J J
Cll'l''iOfJ /GDKZO/ SPt( l•),2) ,.:>?2( 10 1 2) ,SP3(10) ,SPiltlv,2) ,SPI2ll0,2) ,SFASlJdlJ .LP i:., l :.. ..., l , '; l { : , ::. ~ 1 2) , ! ., ( 2 , l .) , 2 ) , '. ~ ( 2 1 l ,) l ~ K l ( 2, l J , 2 ) , K 2 ( 2 , 1 J , 2) , K3 ( 2, FAS 13 b 2 J .2 1 \J ) , c K 11( 1 0 , ?. ) , ::: .< L.l. ( l \J , ..:.'. ) , EK 3 J { l () , 2 ) , 0 Kl 2 ( l J ) , G rd j { l :J ) , G '< 2 3 ( 1 !) ) FA $ l3 :.. ~5 ') c '.]1•::·1 Di~ I GOP j j I F 11 ( ·+Ju ) ' = 2 2 ( 'tu a } ' E 3 3 ( 4 u l) ) ' D f) 11 { 4 0 i)} ' DD l 2 Ut J 0 ) , ')I H j f As 13 c :t J
c
1 ( 4.J·J) , J 016 ( 4 J J) , DD 1-Z ( 'tJ J ) , '.JDZ 3 ( 40 () ) , DJ 2.:> ( 'i- JJ) , Du3 3 ( 40 ~) l , 003 6 ( 400) , FAS 13 8 5 J 20t.H!d 1t00), OD45( 400), 0055 ( 400) , DD66 ( 400) FAS 13860
COl·P·iO,'~ /GDR.ltO/ G1l{4'.Ju),Sl3('1JJ),G23( 1t00),THETld400),ITY FAS13870 2ETA{40JJ,THEC1J) FAS13080
C iJ: ·H CH I GDP. 9 5 / U 12 ( 4 U 0) , U l 3 ( 't J 0 ) , U 2 3 ( 1t 0 0 ) , UK l 2 ( 1 '.) , 2 ) , U i< l 3 ( l 0 , 2 ) , U :<.FAS l 3 fl q O 123(10,2) FAS13SJJ COMMU~/EArll/ I~OJE(4JJ,3) FAS13910 CQ,'1,·!UN IE AH3/ AMU 40 '.J) , di3;H 400) , CC iH 40J), C'D.\l ( 400) , i:: EN ( 400), GGi'l ( 400) FAS 139 2 O c 1 .• ·~ u •·: / ?- 'I 'H s / 1
-; 11 cs r , , : F c s T , ~;co E o H s > , o cs r c '-;' ·n , "o o E F < 9 9 > , F cs T< 9 9 > , FA s 1 3 9 3 J lFC5TT{2,Z) FAS13940 COM~UN /GDRJ/ I~\T(~JO),~q~A(40J) FAS13950 cu 1 'l-1 ;)'J I f '\>-! 2 ) I ;:: y ( + J J ) , E l (It I) J ) ' E: x ( I; J J ) , ~ y z ( 4 J 0 ) ' [ x l ( 4 0 J ) ' E x y ( 4 J J ) ;:: .Pi. s l 3 9 6 u l,~IG~(~v0),SIGY(4JJ),SIGZ(400),SIGYZ(4JO),SIGXZ(4JO),SIGXYC400) FAS13970
C 0 M ·101•J I f A rl 3 0 I SQ l ( 1+ J 0 ) , S q 2 ( 4 0 'J ) , S C 3 { 't 0 J l , S :~ l. 2 { ~ J J } , S •.; 13 ( 4 0 0 ) , S \.12 3 F AS l J 'i .i .J 1(4JJ) FAS13S~O C0•4.c!Of~ /EAH7'J/ U:J(4J'J,J)
F cs r T < '1 , 1 > =a • o l: c;:, l -1- ( J. ' ~ ) = ~ • 0 FCSTT(L,2}=C.u FCSTT{2,l)=J.J ,; lJ 1=1,,.::-,u\,,
Ki<= I>;A T { I ) E:ll( Il=EKll(KK, 1l '::2.?( I l-='= :.:2<-<:<, l l E3J(l)=EK33{KK,ll G23( I l=GKLHi<.K} Gl3( I )=Gl<.13( l<i<l
Ul2(I)=UK1L(K~,l)
Ul3( Il=UK13(:<:--., l) U2J(l)=UK2J(KK,ll
FAS140·JO FASl4JlJ FAS l ·'tJZ '.) FAS l't03 '.) FASUtO.;.Q FAS 14rJ50 FAS 1.;. J ': J FAS1487J FAS14'J30 FASl4090 FAS141JJ FAS1411J FAS14-12J FAS l!tl3•J F1\Sl414J FAS14150 FA$l'r160
N N <.11
H.1
20
EX(l)=O.O EY(l)==J.O EZ(I>==u.;) EXZ(l)=O.O EYl(l)==O.O EXV(l)=J.u SlGX( I )=J.O s r c Y < 1 > =u • ·'.) SIGZ( I )=O.O si:~:<lt I>=n.o SIGYZ(l)=J.fJ SIGXY( !)=0.0 SGl<Il=u.0 SQZ(l)=J.O SQ3 {I )=J .o SQl2(l)=J.O S.J l 3 ( I ) = J. J SQ23{!)==0.J A A>: ( I ) = ( Z ( I , 2 ) - l ( 1 , 3 ) ) / 2 t3 ti,~ ( I ) = ( Y ( i , 3 ) - Y ( I , 2) ) I 2 c c.·n I ) = ( z { ~ '3) -z ( I' l ) ) I 2 OU~(I)=(~{i,l)-1( 1,3))/2 EE,\!( I)=\Z( 1,1)-Z( i,2.) )/2 GG ,; ( T I= ( Y ( I , 2 ) -Y ( I , 1 ) ) I .2 CLJ.~l PJUE OJ ZJ J=l,3 iJu LJ I=l,i~uS
Uu ( I , J ) = '.). J
Ei'iU
FAS1417Q FAS14130 FAS1419J FAS14200 FAS1421J FASl422u FAS14231J FAS 1'~2 4J FAS 1-"tZ 5 0 FAS142Gu FAS1427J FAS142B:J FAS14290 . FA S l /t 3 •1 1.) FAS14Jl0 FAS14320 FAS1433J FAS143".tD FAS l't3 ".>J F~Sl43:)o
FAS l 't:., 70 FAS143:10 FAS 143 ie) FAS144JO FAS14'tl0 FAS 1'~420 FAS1443·J FASl't440 FASl't450 FAS144t,0
SU.iiZ.>.JfT'~t t':>JJ.·iU,; FAS1447G C***~~*~~*~*¥*******~***~****~~*~**~***~*~*~~**~*****~******¥********* FAS144~J
N N O'I
c SUtJi<OUTHJE TO GdJcRAL: f. 1~Ul\TION NU;<.1BERS FAS14490 C**•¥*~*~***********~~*********~**********************~******~******** FAS14500
COMMON /GDR7/ OSTX,YSL,Z5L,F~1ULT,FORCE,OELT,YD,Z0 1 SUMA,NTFST,TSTRTFAS1451J C m1 "1 JN I G 0 R l l/ ~ E :vl AX , iJ lj S , MP S , i'JO I F ;.1 , NANG , N I NCR , l I 'l C , NE~~ , N E Q F , '.'JO AND, F AS 1 't 5 2 'J
c
1 i\Jt JU, :fo LCCK, LC~ D, i'Ji> SS { 5) , DEL MCT, 1JE L TOT, >IPL US , 11!EQ B2 FAS 145 3 0 com-10;~ IE /~H l/ HJJDE UtOJ' 3) FAS 14540 CO~MON /EAHlO/ L~,LW FAS14550
I F ( UJ A 0 • G T • J ) NE 1..J=O NEWF=O 00 l iJ I = 1 , t·D S D'.1 l 'J J = 1, 3 I:~=PWJE(l,J)
GJ T•J 35
GO TJ (l,2t1),I'.J l NEQ=1\iECi+l
1.,,..;o::::< 1,J)='~:::~ GC! T J l 0
2 Hi'JOt(I,Jl=O lJ C1Y'1 TI\•u:::
20
30
~-i ru Tc < u,1 , 0 (; > IF (:~PS.,\::.11 Gt~ T:J 20 .-.· ~ I T : ( L ' ' , 1 :; ) } ( ~ , ( I -; _i1:. ·:: ( I , J } , J = l , 3 ) , I = 1 , N :1 S )
:~ E ~ F =; i E: ~I 2+1 OU 3J I=l, ;,iDS D:..! 3 J J = 1 , J II-' { li•uU~( I,J) .;,jE. ,;L:Jf) r··JfJO[( I,J)=HJ!!JE( I,JH-1 CO•H I \UE ;"iR In: ( L h d d J) { I ,( I\ JJ:: ( I, J) , J = l , 3) ,I= 1, ND S) ~'iR IT E ( Ll,i, LU 1) i·EOf
FAS145')0 FAS1457J FAS145SO FAS1459J FAS146:JI'} FAS 14(·10 FAS14620 FAS l't630 FAS14640 FAS14650 !=AS 14E,~') FAS l '+6 7 J FAS 1'1·6 J !J FAS 14,;dU FAS l't 7JO fASl471J F A.S l 1t 7 2 J FAS14730 FAS1474Q FAS1475J FAS147uJ FAS14770 FAS147d.) FAS147·)J FAS1430J
N N ......
C ~: l'.;-,:: l,'::: t.~.J!". ;;.::=;" ~-;~-: ~;: ;;:;... *;;:. ::.: ,_.:;;;, ~·.: ~' ~:.;:. ;.;.: ;: :~,~~~.::;::: * >::* >:: .:-,:: >;: :!;. :;:* ~c ~:: ;~1,:. ~:. 1';;; * * *~' :::.:;:: :;: >;--:*_~: :-~;:: * * * *:>~: t,:*~' * * ,~: * * ;': C RESET EQUATION ~U40EkS IF NECCESARY c *****************~~~****************************~~*************~~****
35 M~=NPSS(LOAD-l)
NN=NPSSCL01\J) ms r=0 I F ( :·.j -1i • E;) • 1 • ;.\ i.; :) • r~ :-~ • •'l :: • l ) If~ S l = l
41 IF ('-li-\.f'.;'.:.l.J\~JD.N\J.f:,).l) IRST=-1 N1=.~<~EC+ I KS T
c
40
IF (IRST.EQ.l) NEQF=~EQ/2+1
IF tI~ST.E~.0) RETURN DO 4.J I-=l,,·~DS
DO 4J J=l,3 IF ( Ir< •JU f { I , J ) • GE • :~ E ·.J F ) I 'JG t>t: ( I , J ) = !f,l .J IJ !: ( I , J ) + I R ST C rn~ T I i'J lJ E NE«JF=O IF (li<ST.E'o.).l) .JtGF=~c"1/2+1
~i P. I T E ( L \.\ , 9 9 ) "~ • -~ I T ': ( L '.·: , 1 0 IJ ) ( I , ( I '~ 0 •) >: { I , J ) , J-= 1 , 3 ) , I = 1 , \ :) S ) d:{iT:: (L•~.~,2Jl) :·._.c_~;=
RETUR'~
9 ') F~ i< >1 AT ( I I ' Li) x ' I = (~ u ,.\ T I ::-~ .'l t: u \ 0 [ Ks ' ' I I ' 5 ( I :\J JD t: ' ' 4 x ' ' u ' , 4 x ' • v • ' 1 :. x ' ' :; f , 4 x ) )
1 J J f J ,:, ; ... f ( :i { 4 I :.i , •t ;( ) J 201 FLJ~0AT(//,l0X,'~VEK~~E F0~CE ~QUATIO~ NU~BER 1 ,f5)
E~-Ju
S ·J~,f,'.JUT L·: [ r:L'l>i l~I (fl 0 IF, I 'l 11-'.2, u~)
FAS14810 FAS 1Lt320 FAS14B30 FAS14840 FAS1485J FAS14860 FAS14870 FAS14S8J FAS 14B'JJ FAS149J0 FAS1491Q FAS14920 FAS14930 FAS14940 FAS149:50 FAS1496J FAS1497'.J FAS14980 FAS14'190 FAS150:)0 FAS15010 FAS15020 FAS15-:'JJ FAS1504.J FAS15C50 F J.\S 150 ')0 FAS15u7J FAS15030 FAS15C90 FAS1.51J.) FAS15110 FliS15120
N N co
c
CJ,'1MDN /GDRll/ ;~E."1AX,NDS,~IPS,:\DIFi-1,:\JJ\0JG,NINCR,LI.r~C,f\JEQ,~JECF,N8A\JL>,FAS15130
1NEQ3,~BLCCK 1 LGAD,~PSS(51 ,DELMOT,OELTOT,NPLUS,NEQB2 FAS15140 CO~MUN /FAHl/ I~ODE{4J0,31 FAS1515J Dl:~El'JSiuN L''-H I•JI:'-12) FAS15161) M I \l= l 0 0 0 :) 0 F AS 1 5 l 7 0 MAX=Q FAS15ldv DO 1 0 I= 1 , I 0 I .'12 FAS l 51 9 0 IF (L:~(I}.E'}.0) SiJ T:J 10 FAS152JJ If= (L ,(!).-;C.·'1'.;;{) .~;.;\=L·~(!) FAS1521J IF (L1•1{ I) .Le.MI:-.!) >!IN=LM{ Il FAS15220
10 CONT J!'Jl.lt FAS 1523 J NLJ IF=.·lAX-MI!'Hl IF ( 1'JD IF .GE .l'rnANO) >l,3AT\1D='.'JD IF K.E TU'.\.\J EN 1)
SUB~'.UU r I •\JE cc.-~STP.
FAS15240. FAS152SO FAS152.JJ FAS15270 FAS15230
c .. i hlJ,, /E,\Hl/ p;.Jt)t( -tOC,3} FAS15320 c~:·•:'1'>: /:_--;;1-:1J/ t.·:, 71_, FAS1:53J!J c.:...1.1·:,·i /E,~:1lj/ ,,GC'.:>f,"ifC3T, .. _:_i~:){·J'0),JC.ST(-J')),i·T'.EF(00),FCST(99), F.L\Sl534J
l~CSTT(2,2) FAS1535'.J co~~ON /G0~7/ DSTX,YSL,lSL,FMULT,Fr~cE,DELT,YD,ZO,SU~A,NTFST,TST;~TF~Sl53SO
C .J. ; : 'I u ', I J J '. 11 / : : =.: ' '" ': , '~ ~ S , ''l? S , ~: 1 d F '. ·1 , i''. ,\ \! G , ~ J I \JC'~ , l I NC , i\l E G , 'l E Q F , N BM~ D, FA S l 5 3 7 J l NE J i1, ;'IB L 0 CK, LUA 0, ~JP .:i S ( 5 ) , :J:: U, L'J T , 0;:: L Tt.H , ;\PL US , ti E Q •J 2 FAS 15 3 a 0
C FAS153~J Rt:AJ (L\,lJJ) '.\li.JCST,'!FCSf FAS1:5l~JO
IF (\u~Sf.i:.;.O) ;q fJ 11 FAS154la o:~ lJ l=l,;·:DCSf FAS15420 R c /'.\;) ( LR d l 0) '\i ,-, :~ '.: :) ( I) 1 ,'1-JIJ E , OC 5 T ( I) FAS 15 4 '3 J i\l U) EI)( I ) = I;] CJ G ~ ( : J .:, 0 e i) ( I ) J '1 r JD f ) FAS 1 5 4 4 J
N N \0
to ccrn 1,.1uE 11 IF (~~CST.EQ.O) GU T~ 21
DO 15 1=1 1 '.·JFCST READ (LK.,110) ;.,jQEF( I) ,,"1t::.'OE ,FCSTI I) K= I~.::; DE ( ;"i 1J 0 t:: F ( I ) , 'l •J ;) ::: ) N 0 0 c F( I ) = I i\, ~ J t ( : ~ .J Ll ~ r ( I ) , "I JC' E ) ;"lGJc=>lUDE-1 IF CFCST(Il.GT.J.J) FCSTT(~UDE,lJ=FCSTT(~JnE,l)+FCST{I) I F ( f- C S T( I ) • l T • : ) • u ) F C S T TC i'I C 0 E , 2) = f C S T T ( M IJ 1) f, 2 ) + F C S T ( ! )
Li CJi\Jf LW f 21 IF {NDCST.EQ.J.1~!{.r·JFCST.i:::).J) GO TO 31
OJ 16 1-= 1 , :.1 F CST D~~ lb J=l, .·;.JCS T IF (~SDcLl(J}.N~.~aJEF(I)) GO TU 16 K = !-.; Cl 0 E 0 ( J ) GO T'.J 1500
lo CCl.iT PiU: 3 l L D; ·: T ;: i.'J E
If C-·~!:~F.E.1.J) .<.C:TUc{.'j IF (iJ~)CST.f!.,l.O) GJ TJ 2£. o:~ j;) I= l, i\'.DCS r
3 J I f ( 'WU := D ( I ) • GT • "l :: -i F ) ii 0 :3 c D { I ) = 0: C ') ED ( I ) - 1 2 2 I F ( : ~ F (, Sf • t < • 0 ) S .J T <J 2 4
IJG L5 1=1 7 1' !=-CST 23 IF U~1ltJi:F(l).GT.;J~>:1f) ~cJJl:F(IJ=~:JOEF{I)-1
l)JJ ,.;<.iTC: (L• ,<:i3)K 3JO S fJi'l
1 J ) f c LZ '·1 ~ T ( 2 I 12 ) liJ F,J::. ;1-1T{2f l.2 7 rL::. )) ')j Fw~:·~,\T{//, 1 ~'F01~Cc A'~I) DISPLACE::i:::-.n SPcCIFicO AT E(JU. l\!U;H)~R 'd3,
l ' EX EC J f I G 1 ~ H: 1~ "II f\iA T E J ' )
FAS 15't5•) FAS 15't60 FAS1547Q F4Sl5480 FAS154~J
FAS155J0 FAS15510 FAS1552J FAS15530 FAS l 5Sti-J FAS1555J FAS15500 F~Sl':>57J FASl'.>530 FAS155'JO FAS156JJ FAS150lO FAS15620 FAS156Ju FASl5640 Ft\Sl?6:JtJ FASl'>.'.;60 FAS15u70 FAS1~0RO FAS15r,)0 FAS157J') Ff,Sl57l:J FAS157ZO FAS15730 FAS157-t0 FAS15750 FASl57f:>J
N w 0
END SUG~OUTINE ASS~BL (AK,RY,~,FF,L~ 7 1DIM2)
FAS15770 FAS15780
c c c
*~**~***~****~**~~****~****~*******************~*******~****~********FAS15790
c c
l :)J c c
ASSEM3LE ~TIFF~ESS ~AT~IX TWO BLOCKS AT 4 TIME FAS15SOJ
RE AL* B AK , F:il , A , Fr- , X l A;{ G !: F !\ S 1 5 8 2 0 CO~Mo·~ /GJR7/ DSTX,Y5L,ZSL,F~ULT,FORCE,OELT,YD,l0,SUMA,~TEST,TSTRTFAS1583J cm1;~ o·~ I GDR.11/ 1'J[i1ii-\ x ,,·ms' :•:P s' i !U If ,\J, '\JA;·JG' i'l I \!Ct"~' LI ;-.:c '1'\EO' ~;EOF '~JBAr-.io, F ASl 5 o'i-0 lNEJB,NJLOCK,L~~D,NPSS(5) 1 0~l~QT,~ELTCT,~PLUS,NEWd2 FASlj35J COM~O~ /EAY15/ NuCST,~FCST,~COE0(99) ,OCST(99) ,NOOEF(99),FCST(9~), FAS158SO
1FC$TT(2 7 2) FAS1587J DLEUSIC:'i L 'i( 101'12) ,AK( IJFl2, IOir-12) ,RH( IOL-12) ,A{ i\i:=t..i?2,0:t.M,O) ,FFL"J!:i-'\Sl:>G80
lQti.l) FAS15L90 REWIND 1 FAS15SJ0 XL~RGE=l.010 FAs1sq10 I~ST=O FAS15~20
IF fi'I PS. 1'~ t: • l) IR.ST= l NEo.11=:·icJb+l i'JCST<<fCST l.J SH IF T=O
INITIALIZE ASS":1·l:JLE 1J t\RRAYS
0] lJJ I=l, .:EQ62 Ff(l)=J.O DO 100 J=l,~JA\J ,\(I, J) =J.-J
AVE:F~vt: FUF~CE Gk THER:·1AL r<.IGHT HA 1W TERM
FAS1593J F/<.S 159.!i- 0 FAS159:5J FASl'.>9~0
FAS15970 F /\S 1.:>:i 3 J FAS1599J FAS16Cl0
FAS16J3J FA51604:J F/\Slt.0:50 FAS l6J·.)0
FAS16080
N w __,
NEQFF= 1'lEC~F- -JS HI FT I F ( ·~ t .J F F • G T • J • M; :J • "·l f: ·) F f • L :: • !\ E Q B 2 ) f F { N ::: Q F F ) = F F ( N E w f F ) + F !J ~ C f.
FAS16100 FAS16110
c c c
****¥****~***********************************~*********************FAS16120
205
:>50 SJ-J 6tJO 7JJ
c c c
LQOP OVEK ALL ELE~ENTS
OJ 7 •)0 1·1= 1, f\Ui·i E Rt AD ( 4 ) l >i , AK , RH !) I~ 6 j J I= 1 , I I) I .. , 2 I I= UH I )-:Ii SH I FT IF (ll.LE.J.OR.II.GT.NEQB2) FF{ I I )=FF( I I )+;<.H( I) l)•J 5 C.J J =l, Lil i2 JJ=L.-1{ J)-L"l( I) +l IF (JJJ 5JJ,SOJ,550 A ( I I , J J ) = A ( I I , J J ) + •\ K ( I , J ) c·::iTI~:Jc
CU.JT l·uUf Cu \J TI i'JU E
GO TO 600
If- ( .-.; I-':::> • t:'J. 2. 1.1::., •. \JPS • t:= :,) • .;. ) \..~(_; T 0 7 (j 9 IF (i~C.Sf.fG.0) r;o TJ 7J5 Oll 21..J '<=t,,~CST
K.\.=- :'. _; :=; 1: F ( :< ) - \:SH IF T IF L~~J:Jt:f=(!\J .GE:.•~t;,)~) i<t<.=Kr\ t-!.~ST IF (KK.Lt:.U.LJR.KK.GT.1'Jh~·32) GO TG 2JO FF( Kr.)= FF(:( K ) + F CST (\.)
L.JJ ~,,;·,il.u~
7J? i,= ( ,,JCST.~·~.O) G1J T1J 799 OG /jJ J=l,~DCST
FAS1613J
FAS16150 FAS16160 FAS1617J FAS16180 FAS16l911 FAS1620'J FASl62lJ FAS1622J FASl6230 FAS1624J Fl\SlA250 FAS162o0 FAS1627J
FAS 162'1.J
FAS l :'.>310 FAS1632J FAS1633J FAS1634 U FAS163'JQ FAS 1636.1 FASlo370 FAS16330 FAS16390 FAS104JJ
N w N
75J c c
JJ=NOD ED ( J} -~JSH I FT IF (~-.JOOEO(J} .GE.0JEJF) JJ=JJ+IK.ST IF (JJ.LE.o.m~.JJ.·;T.NEQH2) GO TO 150 A(JJ,l)=A(JJ,ll~XLA~GE
Ff(JJ}=OCST(J)*A{JJ,l) CONT li'~Ut:
FASl6"t-10 FAS16420 FAS16430 FAS16440 FAS16450 FAS16460
c 799
*::~!l::t;t::.* ~.:~;: *,;:: ~: :::.;;~ :;:*~~:~: ;;~~:~:: ~~,!~ ;{-:....:::~: ~** * ~ ~:*~: *:s!' ~= :~ =::: ** :::: :~ * ~~_:.:: '** *** -1_.:.~r:# ::c *~,: ~::: ~~ * ~~ J.!~::: *:~*>::*FAS 16 4 :.:; J
lOOU
lJJ
c
\ff IT E (1 ) ( {A ( I , J ) , I= 1, 1'-J:: I) b ) , .J = 1, NB Ai"! f1) , ( F c ( I ) , I= l, NE OB) FAS 16 5 0 J If ( ·:.'.:lJ.l'j;JLl:Ci-,) G,; TG 1.))) FAS 1651 ~ ~JR I T E ( l ) { l ; \ { I , J ) , I = N HH , h E Q U 2 ) , J = 1 , N BA N 0 ) , ( f F ( I ) , I = r l E (11 , i'! E Q b 2 ) F .t\ S 1 6 5 2 0 ,~Si-l!FT=~O'~f-IIFT+:H:~J~12 FAS 16~ 30 RETU:{N FAS 165~0 G ! :J Ft'. S 1 6 5 '.J ') sual~QuTiilE iJU1·1;J[f~ (·HUT) FASl::,5::,J CO.MG>! /GDP 11/ f\j :::.~r~x 'I JCS' ·w s, ,\JD IF ; ... 1, i'V\f'JG 'I\! l"KK.' LI ;\JC' t.'f.Q, r-JEQF 'N ~·Ai\JD, FAS l 5 ~j 7 8
li'! E: -.J 'j, ;.,i j L ,-; CK , L'::• id, :\l ;> SS ( '.5 ) , ':'> ~ L '. -::; T , fY: L T ,_; T 1 : ; ? l JS , ,\1tQ;J2 FAS 1 6 5 >=l C CiJ'·11lLi"J /1'.:AHlO/ LR.,L·J FAS165~0 ~k 1~d= ( ( ·H:.:.<T-\: 0t\.'IJ+ 1 ) I { '~::'i'lMA i'~IJ+ 5) ) I 2 Ft\ S 16 A 0 ') Ir (.:t:i,:j.Gl.1·1:=."1i •• ::.;)=,.,~~ FAS16610 l'WLOCK-=li\Ew-1>/'lc·~:J + l FAS16o2J I F Pb L LJ CK>:<: 'l C '.J :5 • L T • , E •.J ) '~ ;:i UK K = N e L 0 CK + 1 FA S l 6 f, 3 0 .'~:T::: {L·:,l.J')) :::::;, •':(\J, '3l(C~,N? . .J..\!\'"') FAS16t:"i-)
f0K'•l.L\T (//,llG, 1 '>jlJ .. i...:K :jf cC,;tJ1\TIOi'lS',E>,/,Tld, 1 ~-JU:if)[K OF t=QUJ\TIO:~SFASlo650
t P i:: i<, b L u c. K • , I ~ , / , T i 0 , • •'l u , ; 0 E P o F e L o c K s • , I :::> , / , T 1 o , 1 s t.. 1\l o ~..; r J T 11 • , 1 s 1 F As I •":> o b o 1\JE(:c~2=:l[l,;C)>;,z FAS 106 7 J Rt.=TU!:<i'J E:-JD
SU tr.<JJ UT [!'..; i :) r: 5 : ~ l ( ,\ , ·3 , .\I 1\ ~.I\ , i'.: E Q , 'l I\ , "! V , f·! Q L r l CK , ~ ~ E '~ ':\ , ;v, f'. V , ;.; ! , r-.; S T I F , 1 i'-1.:.,t:u, ~L,~.k)
FAS 1S6,.J0 FASlGr-:,90 fAS16700 r-AS 16 710 FA$ll)7?0
N w w
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
THIS )U2.l<.u!JTI1';t: CALCULATES TH[ SOLGT!ON OF THE SI.·IULT/1i\jcfJUS EQUATIONS
WHEP. E K R x
= =
K X = R
PuSITIVE DEFINITE tlATldX SUPPLIED RIGHT HA1\JD SI'Jl.: VECTO~~S C~LCUL\TEU SOLUTIGN VECTOKS
L If~EARFAS 16 7 30 FAS1674J FAS16750 FAS16760 FAS1677J FAS167JO FAS167~u
FAS16S'JC> FAS16810
TH':: '1.:\Tt-UX K A:·iiJ fdt: K.IGl!l llAf\JJ $!fl(; VECTOi~S R olCJCKS FR'.J''I TAPE ~lSTIF • THE SOLUTION VECTORS
ARE READ IN X ARE STORED
FAS16820 FAS l 68'jJ FAS16840
I N o L .JC KS T '· 1 R ~ V E i'- S :: C fUE R 0 N T A P E NL •
* ;': * :!:: * * * I N P U T S :(: * * * 'i<
NEQ = Of EQUt1 TI 01115
MA = i-IAX L•lU~: H1\l F aANm; IDT H OF K {P\JCL. DIAG'JNAL)
NV = NU~J~R OF LC~O VECTORS
NAV = NEQJ*(MA + NV)
Ml = ~A + N~OB - 1
i\!STIF , N'.·'.~·1 , i'Jl , NP = TAPf NU,·!HcKS
FAS1685J FAS16f6J FAS 168 la FAS168.33 FAS1689'.) FAS 169,JO F-AS16910 FAS16920 FAS1693:J FAS16<740 FASlo950 FASlSt,S,') fASl6970 Fl\$169?.0 FAS109SU FAS17C>JQ FAS1701J FAS1702'J FAS17GJO FAS170ttJ
c C A , B , MAXA = STORAGE ARRAYS c * * * * * ~ * * * * * * * * * * * * * * * * * * * * * * * * * * * c
c
c
c c
l~PLICIT REAL*3 {A-H,U-Z} cu~~JN /EAHlJ/ LR,LW D I>lfi'~S I G.\4 -1 ( ,jp_ V) , r; {;~.4 V), :!.'\ XA U1 I J
Mi-1=1 MA2=MA - 2 IF (:l;\2.tfJ.G) l.:'..2=1 I NC= i\J :.: «J'1 - 1 j~ .. ~ A=i~ Ebi h ~ ... ,A . l T o : L·1 '\-2 l I .'l E ~ J i- 1 N'.: b= NT J;;<1'J t: ~ D
i\JEE.T=iH:B + l\;EJt3 ;·~ II 'J = f,; :::: i) ~ >:< f j V tL1 V V = ;-,i .:: b l >; • .., V
~H= ~L
l~Z=;'JK
R f ;-i! i\J .J !'; ST I F ~.;. := -~ I 4 J 0 0<) .. ~ 'J REA I l\i D ;\il
K-f:w lt~J N2
MAI1\J LDUP CVEi\ All tJLuCKS OG 6JJ ~.J=l,f·lolJC~ IF- (i!J.!,E.lJ Gd T'..J lJ ;-: 1.: .~ u { i'~ S T I r- ) :~
IF UEQ.GT.l) G~J TJ 100 r~ ,~ X A ( 1) = 1
FAS17050 FAS1706u
* * FAS1707J FA$1708Q FAS17090 FAS171JO FAS17110 FAS1712,1 FAS1713u FASl 7l1-t0 FAS1715J FAS17160 FAS17170 FAS17ldJ FAS17190 FAS172'JJ FAS17210 FAS17220 FAS1723J FAS17240 FAS17250 FAS17260 FAS17270 FAS172:JO FASl 72'i0 FAS17~J0 FAS1731J FAS 173 21) Ft1Sl'l~ll)
FAS1734-J FASl 7350 FAS 1 73<J0
WR.ITE ( f\JP .. Cu) A, r'1AX;\ FAS1737J IF CA ( U) l, l 7 4' .3 FAS17J30
l KK=l FAS1139J WH.ITC: ( Ul, 1 J l J) KK,\ll) FAS17400
3 DO 5 L=l,i\lV FASl 7'tl0 5 A(l+L)=A(l+L)/A(l) FAS1742tJ
KR=l + 1'lV FAS17430 Wl\ITE P-iU ( ;\{ i<K) , K K= Z, KR) FAS 1 7 L1:4'J RETUKi~ FAS17450
10 IF ( 1'J Tb • E 'J • l ) G'~ ._, TIJ 100 F;\517460 R ::1-J I 1'l'.:l ~n FAS1747J
R.bHiW ~l2 FAS1748J REAO ( :d) A FAS17490 c FHJD CJLLH.'J rlt:IGHTS FAS1750J
100 l<U= 1 FAS17510 KM=,~1l!W ( .·lA, r...;t:tH>) FASl 752 0 i'lt\XA ( 1) = 1 FAS1753J l)tJ llQ :~=2,:1! FAS l 75tt 0 N IF ( >i-i'·I .'4) 1.20' 120' 130 FAS1755J w
°' 120 KU=KU + N:::·J~ FAS17S~) KK=!\U FAS1751u · .. ·'..=-"iI I iJ ( i·!, \,\',) FAS l 75!JJ
G.:.; Td l ~tJ FAS17':J'1J l 3') KU=KU + 1 fASl-/600
r~ \= :,u FAS17610 IF (N-t;t:Q!)) 1 .;. J ' l !t J ' 13 ,J FAS17620
136 i'1d= '·li·1 - 1 FAS17630 ll+J Ot; l6J r\ = l , >L'-~ FAS176!t0
If (;,(i\K)) ll•J, 10'.J, ilO Ft,Sl76)0 l tJ (j KK=i\i\ - n~c FAS176'JJ 110 :1,.\ X.'\ ( •\!) =KK FAS17670 ,..
FASl76':10 ....
IF (A(l)) l 72, l 74, 1 76 FAS1769J 174 KK= ( NJ-1) *i•l cQLi + l FAS17700
IF ( KK .GT .:\J l=C.) G" ;J T·J 59J FAS17710 WUTE (U;l,lJJJ) Ki<. FAS1772J STOP FAS177JD
172 KK= ( ilJ-1} ';'\!:'.Cl' + l FAS17740 .~i<. I TE (L.~,lJlJ) K1<.,A(U FAS177'50 c Ft.Sl77c0 c FACTORIZE Lt:ADI f'iG dLOCK FAS1777J
1 Tb Oi) Z0 1J 1\l = L , i\ E \J ii FAS177GQ t~ll= ~lAX A (;ii) FAS177'10 ff {\ii-:.:) 2JJ,2..JJ,~1J F!\Sl7(;0J
210 KL=:~ + INC FASl.7810 K='~ FAS17J20 D=J. FAS17<330 OJ 220 KK=o\L t r'.H, F~C FAS17840 K=K - l FAS17t35J N C= 1\ ( K1(} I A ( :O FAS17360 w
-....J 0=1.) .... c;'1\ ( KK) FASl 7c-f J
22() 1\ ( KK) =C FAS170H'J ,\ < :n =.:. L·:l - D FAS! 7f»;'J
c FAS179JO IF {11(.\j)j 2 l_ 2.. , 2 2 '.t , L .1 J FAS1791J
224 K K = ( ; i J - l ) ·::,\ [ <..; ;.; + ;.J Fl\Sl 7920 If (:<.K • .;T.rJE•;) GU T'J 5 9;) FAS17930 ~iR IT E (Li,lOJJ) i<.K FAS 17S 1tu STJD FAS179'.50
222 Ki<.= { "iJ- l) ':' ,';[ '),) + ,,1 FAS1795) ~H 1 TC: (L11.,lJ1.J) i<. ,, , ·\ < :n FAS17970 c FAS 1798•)
230 I•.> ·J(: '~ ~ FAS l 7t;:;J OCJ 2 1tJ J = l' :-11\ 2 FAS1800J
280
3JJ
2 'tO c
430 c
2 \JJ c c
MJ=~AXA(N+J) - IC IF (MJ-i"l) 2 !tO, 240, 2 30 KiJ=;'-1!J\JO ( i''tJ, :~H) KN=N + IC C=O. OJ 300 KK=Kl,KU,I~C C=C + A(KK)*A(~K+IC) A(KN)=/.\(K;"~) - C IC-=IC
K=N + i'MA Ou 43J l=l,'~V KJ=K C=O. D:J 440 KK=KL,rlH,P.JC KJ=i\J - 1 C=C + A(KK}*AIKJJ A ( ;() = .-\ ( i<.l - C K=K + ·'iEQt:3
C ~ f~ ;.:. Y G \IL:?, I•, I :_, T" \ ~ LI . :·, u L CC :z::; 0 J 't'J J i! K = 1 , :,, T :::. If ((i':K+!-iJ).1;1.i•!C'.LXK) 1;'.J TC 400 i'l I= 1·~ l If- ((;~J.t::~.l) .GR.('JK.t:·.J.~HG)) 1\JI=i\iSfIF ;CA'.J (.'JI) 0
:·i'<.= ~;I~~'.) ( { ~~i:+l) ''~·J t;•J.J, '1 I} If (."1A.t:(~.1J ,·.L=,ir<. :·10= '.11 - 1·ll
FA5l~OlJ FAS18020 FAS1803J FAS18040 FAS18050 FASl8JAJ FAS1U070 FAS180JO FAS1'3Q90 FAS18100 FASlollO FAS1LJ120 FAS 1~313') FAS1814J FASl:Jl50 FAS18160 FAS1817J FA$18li0 FAS181CJJ Ft.Slc:;';)J fASlB21Q FAS1822J FAS 102. ·s0 FAS132!-t0 FASl82::iJ FAS13200 FASlv!.70 FASlti2JJ FAS162SO FAS l 63 :J:) FASld3lO FAS18320
N w CX>
c
51~
520
538
5:) J
5 7)
KL=i~cW8 + ( :·~K-1) ~,;,;Ejd"'':~E·;~2. N=l
OJ 500 i'1=:•;L, 1,m NH= :1A X.f!\ ( .'-'l) KL=i\.L + Ni.:·Jo IF (NH-KL) 50~ 1 jlJ,510 K='\E~J
D='J • 0 1.J 52 J i~l\.= KL, '~H, I 1~C C= A ( KK) //\{ K) D= D + C •:< /\ ( K K ) A('<K)=C i<.=K - 1 3 nn = J < ,,n - o IF (~0) 5SJ,5JJ,51J IC= ; c ~~0 DO ::>t>D J=l, \1L)
r1J=.'·!.'.\XAP1+JJ - IC IF ( lJ-~( L ) 5 ltO, 5 5 J, S 5 ;) KU= 'il rh){ ·1J, \JH) K:~=:J + IC C=O. DO j7j KK=KL,KU,I~C C=C + AC~K)~~(~~+ICJ 5(1\.'.·d='J{K~l) - C L= IC + ,\J!:::QJ
K:'~=:~ + i\I;'/,\ K=:~t.:·.~.J + i':'.~iJ\
0-J olJ L=l, :1J l<.J=K
FAS18330 FAS 133ttO FAS 18350 FAS18360 FAS1337J FAS133d0 FAS1839J FAS18't·10 FAS l8 1tl 0 FASlfi4-20 FAS 18Lt30 FAS1341+0 FASlo4:50 FAS lf14f:i,J FAS1847J FAS1?3480 FAS134'JJ FAS185JJ FAS18510 FAS1B520 FAS13:530 FAS 1854·'.) FAS 1::;53 J FAS135J0 FAS 1 B57~) FASld5'3J FAS135.;tO FAS 186·)0 FASl66lJ FAS13620 FASl!3630 FAS1864J
N w \0
C=O. FASl8650 DO 620 KK= i<L, f·lH, I. ~C FAS1866J C=C + A(KK)~'A(KJ) FAS18670
620 KJ=KJ - l FAS186d0 o(K".J>=f-HKN) - c FASld6)J Kil= KN + N:: Ob FAS1B7Cl0
610 K=K + t.Jf ~.Ji:; FASlf:l71J c FAS18720 505 MD=MO - 1 FAS187JO 500 N=N + l FAS1874J c FAS18750
IF (NTB.NE.l) GO TJ 560 FAS 13760 WRITE { ~LH.:U) A, ~'1A;\A FAS1>3T/O OJ ':i7J I= l, .'~AV FAS187aG
570 A.(1)=8(1) F-'\Sl07'J) GU TO o·JJ FAS13DvJ
56J 'rlRITE ( :~ 2) ~') FAS1881J c FAS18820 N 4J) cc;, Tl; :U'.: FAS13C3J ~
0 c FAS18f40 i'l=:\11 FAS1385J 1·11=.J2 F~Sli~St':>J ~J2= .'·1 F~.$108 7J
? ::)'J ·,;:-<I TE ( 1 :-<- cD) A, ~AXf\ FAS18B3'J c F.AS 13~390 6 Ju CJ;llT I.'JUE fASl:39QO c FAS18910
L VtCTU...; u~C~S0~STITUTIC~ FAS1d920 DU 7JO K=l, N~iVV FASl 5S1J
7JJ o!Kl=O. F.~$1894'.) ~'.::'ri I "JiJ 1'; L FAS13450 c FASlt>96'J
32J 310
650
8'>5
.J7J
DJ 8JJ NJ=l,.'JdLCC;< BACKSP.~CE NRED R >.:AD ( :V{ E 0) A, •'I .\X ;\ B AC K 5 PA C ~ 1-.J R :: L) K= ~~r: ;H DCJ 810 l=l,\JV DO 82.J I=l,N1:J EH J<.) = 13 ( K-.'1E'.:io) K=K - l K=i< + NEtH + l'iE5 K;-J=0 l'CK= .--.;;·/ ·\ NJif=.'Jc:Jn If (NJ.t:Q.ll i'DIF=:~'.:i~2. - (,~5LCCK*-NE•W - Nl:Q) 0 iJ J 5 5 L = l , 1-N Du 350 K=l,.\JD!f 3(K~+K)=A(KK+K)/~(~)
K:<=KK + ,·Jt=t)d K:,=Ki~ ~ N'::JT l f ( ,.~ 4 • C: J • l ) G .:; T.j Jl S :·1L=1,:E Qi3 + l KL=;'/c -.13 D'J 3bJ i'-i='-!L,;"1J :<. l = K L + "J :: ~::i :J KU= ·1ti ;\1\ ( ;·1) IF (KU-i<..L) 'J60,370,J70
K.·i= i OJ 88J L=l, :~V KJ=K U CJ J 9 J :<. r< = < L , '.-( 1J , I ': C :--, ( i< J ) -= ':\ ( :<. J ) - ~ ( :\. . .;. J '~ . ..i ( i'\ l l
FAS18970 FAS18Sd0 FASlS99J FAS19CJO FAS19010 FAS19U20 FAS19030 FASl904-0 FAS19J5u FAS190o0 FAS19070 FAS19030 FAS19J'JJ FAS19100 FAS19110 FAS19120 FAS1913J FAS1914J FAS19150 FAS 191 . .:,0 FASl ·Jl 7 J FAS19l·W FAS19l'JO FAS1920J FAS1921J FASlq22J FAS192D FAS1924J FAS1925u FAS1926J FAS19270 FAS l 92o'J
c
890
tl80 860
920
94') 9Ju 918
915
'J60 9 ;;J
c
dO.J c
KJ=KJ - l KH=KM + NEoT K=K + NEBT CO!H I NUE N=l'JEQ3 DO 91·) I=2,:·JE,._if3 KL="J +INC KU=:·1AXA ( i\J) 1F OW-KL) ·~1J, ~ZJ,920
K=:~
OU 930 L=l, :N KJ=K DO 940 KK=KL,KU,I~C KJ=KJ - l u{KJ)=b(KJ) - A(K{)*8(K) K=K + NEoT i\i='.\J - 1
o<.K=() K!l=J DJ -i S 0 L = l , .'N D :J -i ~ J ~( = l , :-.1 fl-'. J KK=Kl<. + 1 ,'d ;_1, :< ) = B ( K >i + !\. ) Ki,=;-<.f; t- NEuT
tlRITf PJU (l\(r<.),i<.=l,N':JV)
lJJJ F0~,·1•H (// 46H STOP *** ZERO 0IAGCNAL ENCGUNTEREU DU~l~G, l J H L~ .~ 'J • T IC; : .) .J L u TI L I~ ' I 1
2 1JX,18H fcJU/\TIG;\ ;-.,urH:.l::K =, 16 )
FAS19290 FASlgJQJ FAS1931J FAS19320 FAS1933J FAS1934u FAS19350 FAS1936'.) FAS1937J FAS 19330 FAS1939Q FAS19400 FAS19til0. FASl 9tt2 J FASl'J4JO FAS1944J Fa.s 19430 FAS1946J FAS1947J FAS1948J FAS1949J fAS195JJ FAS19510 Ff,S 1Cj520 FAS1'..l5Ju FAS1q540 FASl'J55J FAS195~,0
FAS19570 FJ\Sl95d0 FAS1959:J FAS 196'.)u
N ~ N
c
c c c c
lJlJ FJRi4AT (/ l
501~ v!J\R.l'-II:~G ~** f'if::.GATIVE DIAGClNAL ENCDUNTERED OU~ING, fASl9610 18H EQU~TION SOLUTION, I . FAS19t2J
2 ljX,loH EWUATICN ~JMDE~ =, 16, 5X, 7HVALUE =, ~2J.a ) FASl963J
RETU'.~:\
E:\\lD SUBRJ0TINE ~UTPUT
SUJ~OUTINC TCJ PRI~T STR~SSfS,STRAINS,EQUIVALENT STRESS~S A;'ll) TOT.~L r-.:oOAL !) ISPL/\CE:-lci'JTS
FAS19640 FAS19650 FAS19l60 FAS19670
FAS1:16C:O FAS19700
*******~~********¥******~****************************************FAS1971J REAL*8 Y,Z FAS19720 CO:·t1Ji'i /GO'i\l/ Y('tJ0,4J,Z(400,4),ND(~00,3) FAS19730 CJ l'1C>! /GY~'+2/ UT.<.~:;;: 0 ,uKr::.:P FAS197!t0 CG~~Ll~ /GDR7/ OSTX,YSL,ZSL,F~ULT,FC~C~,0iLT,Yu,zo,su~l,~TEST,TSTRTFlS197SO Ct.Ji'1i·i~,': /SJK.J/ I "'~T{ltJlJ), :\'\UtuO) FAS1976J C 0 +•I 0 :'l I G 0 ~ l l / , ~ c:: .A ~< , •\l t; S , NP S , f>..1 C I f 1 , 01 ANG , ~l I NC K , l L'K , ~i '.:Q , \J E Q F , N BA i'l 'J, FAS l 9 7 7 0
l.'E: ·~ :j, :~ o LC CK , L cir:, D , .'JP 5 S ( S ) , :J fl 1-;c.. T , DE l TOT , NP LU S , i'! F- Q 32 FAS l 97 B 0 c,-";.,_~ .. /.::,_:.:1lJ/ L :~, L_J FAS197'JJ CJ ·l · 1 C ,\l IC: A 1-i 2 .j I E Y ('+ 0 0 ) , f l Ut 0 0 ) , EX ( 4 0 J J , E Y l ( 4 ~)() ) , 1:: X l ( 4 ~) i:.d , i:: X Y ( 4- Ou ) , FAS 1 9 8 0 0
l S I G X h J J ) , S I GY ( tt ·) J ) , S I G l ( 't 0 '.) ) , S I G Y l ( 4 C 0) , S I G X Z ( 'i- •J )) , S I G X Y (t;- 0 '.) J F .~ S l 9 iH :) C ij;-;.·, _:;,.j I.:.:~ rU JI S ~ l l /~J .J) , ~ · 22 ( '~J -J ) , S C13 ( 4) J ) , S :H 2 { t:_l J l , S •H ~ ( '• :,~ 0) , .S '22 3 f AS l 'hl 2 J
l(~')J) FAS19830 C·l.·'.-l.-,-, /'::.f\':--l65/ Ft\IL(4J1)) FAS198'tJ CCH··.].·: h:Af-:60/ LiiJ'~P.-~(2,::)) ,~~Y(10) FAS19f35J COM~O~ /~AH70/ UU('tOJ,3) FAS19d60 OIM~~SIJN NYl2J) fASl987J
~ FAS1988J WRIT~(L~,30) FASl989Q GO TJ {10 7 Li,2J,27', !·!PS FAS199)0
lv ST.ZX=LI~C*U.STX FJ\Sl9910 i~!dTE (L.;,99) Lii'~C,CJSTX,STkX FAs1·1.:;2J
15
21
2J
26
20
33
c jo Lt2
1t 7 ') j
1 J't
lJl
l J .3
GO TO 26 DE l T 0 T = 0 E L T '~ L I ·l C tT S T n WR I TE ( L ~ ,! J 4 ) l Ji'K , ) El T , DE l T 0 T GO TO 26 OEL~OT=DFLT*LI~C+TST~T ~RiT~ (LW,103) LINC,JELT,JELMOT GO T'l 26 FRCTJT=FO~C~*LINC WRIT~ (LW,101) LINC,FORCE,FRCTOT If (~PS.EQ.l) GO TJ lJ UTKE '::P=UT\t ~P+' J~<.'":: co WRIT~(LW,4ll UTKEEP SU l=J. J DO .:D I=l,~it::--11\X SU 1=SU,i+SIGX( I) :;'IH~( I) C ~'"i T 1,·J Uf~ AVt:= SU' d S u;·,J\ w R. IT E< UJ , .+ 7 } AV r::
FAS1993:) FASl 99ttJ FAS199'..l0 FAS19960 FAS1997) FAS199S;J FAS19':J90 FAS2JOOJ FAS20Ql0 fAS2002J FAS2003J FAS200+0 FAS2.J05J FAS2uCbJ FAS20070 FAS2JCJJ FAS20090 FAS 201 :t J fAS2011J
Fu K .,l.H l/ I , 11 :; ( 1 rl '-' J ,/I ) FAS 2 0 l 2 J FJ~'lAT {!/T1J,35HU~Il=0~·>1 N:JR!'t•L STR!\IN DET~;<.;··111' .. EtJ = ,El4.7) FAS2Jl3J f\.~"'•-'.i:.T {/f 10,2LiT~i= ~v~.n . .:;c S!GX IS ,fl4.7) FAS2014J f1J1\,'-1AT (/,i'..:X,'t<.':S'JLTS AT LCA'.J I 1·JCPEtir:rJT 1 , I5,/lOX, 1 STPAit\1 I!~CRE.4tl;F.i\S2Jl3J
lT =1 ,El5.7,1JX, 1 TGT.\l ST?"~Ii'! =1 ,[15.7) FAS20160 fu,<.·:1\T (/,lJX,'et,Z:~JLTS ;\T U\~D I:1CPt::1c>1T',IS,/,lJ,(,•Tf'l?tR~TUQ.E INFAS.20170
lCRE:'·lt:•H =',[15.7,1 1JX, 1 Trl1.: TEHt'ERATCRE IS = 1 ,il5.7' Fi~S201~J FCR.~Ln (/,lJX,'r<.~SULlS .'\T Ll)~U It'!CF01EiH' ,15,/,l•JX, 'FCf.Cf H~Cf,E,;\c~JFt\S2Jl'J()
1 T =' ,c:1s. 7,1 J;<, 'TJT.~L FO.<CE =1 , ::is. 7) FAS2021JO F C ,-, " ;\ T { I , 1 J " , ' i·. E S ·J ;_ I S ~ T l L• I\ lJ I : ~ C ;:: E . ; c i ff 1 , I ') r I , l J X , ' '. C I) T U ::\ f I' J C R E F A S 2 J 2 1 J
l '<! E •'i T = ' , c l 5 • 7 , l J X , ' ~ iJ I S TU~~ l CG:\ ft i'JT I S ' , E 1 5 • 7 } FAS Z 0 2 2 "J JJ=0 FAS20230 IF (LI~C.~0.1.Gl.LI\C.l.C:.~INC~) JJ=l FASZJ240
I F ( L I ~'-l C • G [ • l L'J C P ~ ( l , L 01\ D ) • 1'.l. N CJ • l I NC • LE • L HJ CPR ( 2 , L 0 I\ 0 ) ) J J = 1 I F (i'~ PL u S • v T • 9 ) J J = l . Lii'K=LirlC+l IF (JJ.EQ.0) R~TUK~ IF {KEY(Z) .r·~E.0) G:J TO 1010 i"I k I T :: ( L ~1 , 1 ) ~~ R I T ;:: ( L 1,~ , 2 ) ( I , f.. X C I ) , E Y ( I ) , E L ( I } , E Y Z ( I ) , E X Z { I ) , E X Y ( I ) , I = l , i\ F 7'1 A X )
1010 IF (~EY(3).~E.O) GG TO 1J2J
FAS2025') FAS2u260 FAS2J27•J FAS202l3J FAS2J2'JJ Ft\S2.'J300 FAS20310 FAS2J32J
WRITE(LW,3) FASln33~ WRITE(LW,4) (l,SIGX(Il,S!GY{l),SIGZ(l),SIGYZ(!),SIGXZ(I),SIGXY(I) F1\SZ03-'tJ l,I=l,NE~AXl FAS20350
l FUR~~T (35H ,//,TlO,l3H** STRAINFAS20360 l S >'<>:: ,/ I f 2 , ~~ H '.:: L:.:: • '• l • , T L> , 2 f·Jr: X , T 3 J , 2 H E Y , T 4 5 , 2 ;-1 i: l , T r~ .J , 3' l f.: Y l , T 7 5 , 3 H E X f A S 2.. J 3 7 J 2Z,TJ0,3HEXY,ll) FAS20330
2 FOR~AT {T2,I~,TlS,~l4.7,T30,El4.7,T45,El4.7,T~O,El4.7,T75,El4.7, FAS20390 1T9J,~l~.7) FAS20400
.3 Fu,\>>~ T ( I 11 ' Tl •J, l 4-: I;: ~' s T '<. :~ .s s E s '!<"" ,/ I T 2 J J H c L [ • :'n • ' T 1 5 ' 4 H s I G x , T 3 ') ' 4 r ,\ s 2 J 4 l J 1HSISY,t~j,~HSI~L,16J,5HTAUYZ,T7~,SHTAUXZ,T90,5HTAUXY,/I) FAS2J420
4 FJ~~AT (T2,I5,Tl5,~L4.7,T30,El4.7,T45,El4.7,T&O,El4.7,T75,~14.7 1 T0FAS20430 10,:::14.7) FAS204tt'J
1020 IF (~EY(4).~E.G) GJ TO l03J FAS2J45J ~~ITE(L~,5) FAS2046J DO 7 I=l,N~1·L~X FAS20470
7 WR I TC { L 'r: , ,-_: > I , .Sr} 1 l I l , S ~12 { I ) , S :1.:J ( I ) , S ·)2 3 ( I ) , S 0 13 { I l , S \Jl 2 { I ) , f A IL ( I FAS 2 J <'t .10 1) FAS20490
j FU~~~T (l/l//,T20,~~H E0JIVALE~T SfR:SSES ,J,T2,4HELE.,Tl0,3HS~FAS205JJ 11 , T 2 :) , 311 SQ l , T 4 ,J , 3 rl S . ) .3 , l 5 '.J , 1; HS·} 2 3, T 7 ~J, '1 HS r~ l 3, T ;_: '.) , ti-HS ( 12 , fl G IJ , l1 HS • F • F ii S 2 J 5 l 0 2) FAS2052J
6 F J :-~ :': ;-\ T ( T 3 , I 4 , T l ,) , E l 2 • 5 , T 2 5 , :: l 2 • 5 , T I;- J , ~ l 2 • 5 , T 5 5 , E 12 • 5 , T 7 J , E 12 • 5 , T q F 1\ S 2 J 5] J 15,Cl2.~,TlJJ,~12.5) FASZ0540
l•J30 \..GrH I"'. J F. FAS205.JJ IF lK::Y(5).:,E.J) KcTUKN FAS2056tJ
WRITE(LW,S) FAS20570 ~RITE(LW,9) {1,~U{l,l),UU{l,2),UU(l,J),I=l,NDS). FAS205SO
8 FOK~AT [/////5X,31H TUTAL NOUAL OISPLACE~fNTS/lOX, 65HNLlDEFAS205J0
q l '.'JC. U-OISPLACE1·1t:;iH V-DISPLACE'."iENT ~-DISPLACEMEiH) FAS206)J
F d ,\ •·: 1\T ( 1.3 X , I 5 , 7 ~ , .:: L.: • ';) , 7 X 1 t 12 • ::> , 7 X , f 12 • 5 ) k. E T U ;-\j'.J ENO
FAS20610 FAS20620 FAS20630
The vita has been removed from the scanned document
NONLINEAR ANALYSIS OF BONDED JOINTS
WITH THERMAL EFFECTS
by
· Edward A. Humphreys
(ABSTRACT)
A numerical analysis of the nonlinear response of bonded joints is
presented. Mechanical and thermal loadings are considered. Material
stress-strain response is represented by Ramberg-Osgood approximations.
Temperature dependent properties including modulus percent retentions
and coefficients of expansion are modeled with linearly segmented
curves. Bonded joints with graphite-polyimide, boron-epoxy, titanium,
or aluminum adherends are analyzed using a quasi 3-dimensional finite
element analysis. In adhesive bonded joints, the adhesives considered
are Metlbond 1113 and AF-126-2.
Elastic results are presented for single and double lap joints, with
and without adhesives. It is shown that mechanically induced stresses
are greatly affected by longitudinal adherend stiffness. The effects
of adherend transverse stiffness are shown to be significant in some
cases. Residual curing stresses are shown to be significant in all
joints except those with similar adherends and no adhesive.
Norilinear results are presented for adhesive bonded joints. It is
shown that adhesive nonlinearities are only significant in the predicted
adhesive shear stresses. Adherend nonlinearities and temperature de-
pendent properties are shown to have little effect upon the adhesive
stress predictions under mechanical and thermal loadings.