Prepared by James G. Crose and Robert M. Jones
324
AIR FORCE REPORT NO. SAMSO-TR-71-103 SAAS III AEROSPACE REPORT NO. TR-0059(S6816-53)-1 FINITE ELEMENT STRESS ANALYSIS OF AXISYMMETRIC AND PLANE SOLIDS WITH DIFFERENT ORTHOTROPIC, TEMPERATURE-DEPENDENT MATERIAL PROPERTIES IN TENSION AND COMPRESSION Prepared by James G. Crose and Robert M. Jones 71 JUN 22 San Bernardino Operations THE AEROSPACE CORPORATION Prepared for SPACE AND MISSILE SYSTEMS ORGANIZATION AIR FORCE SYSTEMS COMMAND Ai r Force Uni t Post Offi ce Los Angeles, California 90045 Approved for publ ic release; distribution unlimited.
Transcript of Prepared by James G. Crose and Robert M. Jones
AIR FORCE REPORT NO.SAMSO-TR-71-103
SAAS III
AEROSPACE REPORT NO.TR-0059(S6816-53)-1
FINITE ELEMENT STRESS ANALYSIS OFAXISYMMETRIC AND PLANE SOLIDS WITH DIFFERENTORTHOTROPIC, TEMPERATURE-DEPENDENT MATERIAL
PROPERTIES IN TENSION AND COMPRESSION
Prepared by
James G. Crose and Robert M. Jones
71 JUN 22
San Bernardino OperationsTHE AEROSPACE CORPORATION
Prepared for SPACE AND MISSILE SYSTEMS ORGANIZATIONAIR FORCE SYSTEMS COMMAND
Ai r Force Uni t Post Offi ceLos Angeles, California 90045
Approved for publ ic release; distribution unlimited.
UNCLASSIFIEDSec:ulity Cl•••iflc.tion
DOCUMIMT CONTROL OATA· ...D(Secu""" cI•••Wee"" ., ,,,,., ... 0' ".'"0''''' ..............."... __.t be _t.NlII _.,. ... 0_.." "POrt .• cl...med)
I O"IOINATINO ACTtV,·V (Co....,. MI''''.') Ie. tlllE_O"T IIlCU'"'''' C t., ..... IPI'IC.TION
The Aerospace Corporation UnclassifiedSan Bernardino, California .. ellllou"
--J "I~OIltT TITLI SAAS III
Finite Element Stress Analysis of Axisymmetric and Plane Solids with DifferentOrthotropic, Temperature-Dependent Material Properties in Tension and Compression
.. OC'CIltI~TJV.NO"., (T..". ##1 ,.~" Mttl 1"./va/_ .,..)
Technical ReportS AUTHOIlli(') (I...., "MIl, H,., "...e. '''111101)
Crose, James G. and Jones, Robert M.
• till. "0 lit" OATI. ,.. TOTA ... NO. O~ ...... 1 'b, NOza; tIIlEpr•71 JUN 22 335
I. CONTlitACT 0" Illflt .... ,. NO. ... OfllleiNATOllll'. "1l_0tllT NUM•• tII(S)
F04701-70-C-0059 TR-0059(S6816-53)-1b PIlIO,JIlC T NO
, II. I"W." _'_0111" NO(') (A,,)' odl., """"...,. ..., "'0" be •••,.."'10 ,.,."
d SAMSO-TR-71-103
"I>. V" IL ••".ITY/LIMIT. TION NOTICII
Approved for public release; distribution unlimited.
11 SU""t. .MINTA." NOTII 11· '~ON'O.IHCI MILIT"'"., ACTIVfTV
Space and Missile Systems OrganizationAir Force Systems CommandNorton Air Force Base, California 92409
" AIIS"".CT
The finite element method is used to determine the displacements, stresses,and strains in axisymmetric and plane solids with different orthotropic,temperature-dependent material properties in tension and compressionincluding the effects of internal pore fluid pressures and thermal stresses.The mechanical loads can be surface pressures, surface shears, and nodalpoint forces as well as acceleration or angular velocity. The continuoussolid is replaced by a system of elements with triangular or quadrilateralcross sections. Accordingly, the method is valid for solids which are com-posed of many different materials and which have complex geometry. A
! listing of the resulting FORTRAN IV computer program and instructions forits use are given in appendices. Two-dimensional mesh generation andtemperature interpolation features allow the computer program to be readilyused. The convergence of the method to exact answers with diminishingelement size is demonstrated and discussed.
DD FORM 1413'l'A(:S' .... ILEJ
UNCLASSIFIEDSecurity Classification
••
UNCLASSIFIEDSecurity Cla..ificatiOll
KEY WO"OI
Finite Element MethodStress AnalysisThermal Stress AnalysisAxisymmetric SolidsPlane StrainPlane StressTemperature-Dependent PropertiesOrthotropicPorous MediaUnequal Properties in Tension and CompressionPlastic Analysis
Ab.tract (eMtinued)
UNCLASSIFIEDSecutity Classification
Air Force Report No.SAMSO-TR-71-103
Aerospace Report No.TR -0059(S6816 -53)-1
SAAS III
FINITE ELEMENT STRESS ANALYSIS OF
AXISYMMETRIC AND PLANE SOLIDS WITH DIFFERENT
ORTHOTROPIC, TEMPERATURE-DEPENDENT MATERIAL PROPERTIES
IN TENSION AND COMPRESSION
Prepared by
James G. Crose and Robert M. Jones
71 JUN 22
San Bernardino OperationsTHE AEROSPACE CORPORA TION
Prepared for
SPACE AND MISSILE SYSTEMS ORGANIZATIONAIR FORCE SYSTEMS COMMAND
Air Force Unit Post OfficeLos Angeles, California 90045
Approved for public release; distribution unlimited.
1.
FOREWORD
The computer program presented in this report was developed over aperiod of years. The initial development was sponsored by the NationalScience Foundation (NSF Research Grant G-18986) and performed byEdward L. Wilson while a student at the University of California, Berkeley,California. Additional developments we re made unde r NASA Contract NAS9 -1986 while Dr. Wilson waS employed by the Aerojet-General Corporation,Sacramento, California. The program was extended to orthotropic materialproperties by Dr. Wilson under purchase order to The Aerospace Corporation,San Bernardino, California. The technical monitor of this activity wasDr. Robert M. Jones.
The second version of the program, SAAS II, was prepared and publishedby Dr. Jones and Dr. James G. Crose as Aerospace Report No. TR-0200(S4980)-1. It included extensive revisions and was augmented with many newfeatures including modified input/ output, clarification of program logic, extension of the nonlinear material properties feature, automatic mesh generation,temperature interpolation, contour plotting, restart capability, and overlaystructure.
The program was further augmented by the addition of finite elementmethods for the stress analysis of porous media by Dr. Crose, reported inAerospace Report No. TR-0200(S4816-76)-1.
The present report, designated SAAS III, includes reV1SlOns to previouslydocumented program versions. A plane-strain/plane-stress option and anunequal properties in tension and compression capability are incorporated.Reorganization of the program logic and additional internal documentation makethe program easier for use by others. Improvements were also made in meshgeneration, plotting, and input/output procedures. A new method for calculating element strains reduces total running time by almost 20 percent. Othercontributions to the program by Mr. Brian Stocks of Lockheed Palo AltoResearch Laboratory, Dr. David Rodriguez and Dr. Frank Weiler of AerothermCorporation, and Mr. Leonard Bass of The Aerospace Corporation areacknowledged at suitable places in the text.
This report by The Aerospace Corporation, San Bernardino Operationshas been prepared under Contract No. F0470 1-70 -C -0059 as TR -0059(S6816-53)-1.The report was submitted by the authors in April 1971 to the Air Force programmonitor, Captain T. Swartz (RNSE), for review and approval.
This technical report has been reviewed and is approved.-"
~~~~~~~~ ~1r:f:caPt.,~RNSE
ii
UNCLASSIFIED ABSTRACT
SAAS III,FINITE ELEMENT STRESS ANALYSISOF AXISYMMETRIC AND PLANESOLIDS WITH DIFFERENT ORTHOTROPIC,TEMPERATURE -DEPENDENT MATERIALPROPERTIES IN TENSION AND COMPRESSIONby James G, Crose and Robert M. Jones
TR -0059(S6816-5 3)-171 JUN 22
The finite element method is used to determine the displacements, stresses,and strains in axisymmetric and plane solids with different orthotropic,temperature -dependent material propertie s in tension and compre ssionincluding the effects of internal pore fluid pressures and thermal stresses.The mechanical loads can be surface pressures, surface shears, and nodalpoint forces as well as acceleration or angular velocity. The continuoussolid is replaced by a system of elements with triangular or quadrilateralcross sections. Accordingly, the method is valid for solids which arecomposed of many different materials and which have complex geometry.A listing of the resulting FORTRAN IV computer program and instructionsfor its use are given in appendices. Two-dimensional mesh generationand temperature interpolation features allow the computer program to bereadily used. The convergence of the method to exact answers withdiminishing element size is demonstrated and discussed. (Unclassified Report)
iii
CONTENTS
I
II
III.
INTRODUCTION
METHOD OF ANALYSIS
A. Scope of Analysis
1. Axial Symmetry
2. Plane Strain
3. Plane Stress
4. Material Models - General Discussion
a. Orthotropic Elasticity
b. Orthotropic Bilinear Plasticity
c. Unequal Properties in Tension and Compression
d. Temperature Dependence and Thermal Stress
e. Porous Media
B. Equilibrium Equa tions and Finite Element Discretization
C. Linear Displacement Triangular Element Approximation
1. Displacement Model
2. Material Description
3. Thermal, Mechanical, and Pore Pressure Loads
D. Quadrilateral Element
E. Boundary Conditions
SUMMARY
iv
I- I
II-I
II - I
II-I
II- 2
II- 2
II-2
II- 2
II-3
II- 3
11-3
II-4
II-4
II-8
II-8
II-14
II-IS
II-20
II-21
III-I
CONTENTS (Continued)
APPENDIX A:
A.l
A.2
A.3
A.4
A.5
APPENDIX B:
B.l
B.2
B.3
B.4
APPENDIX C:
C.l
C.2
APPENDIX D:
D.l
D.2
D.3
D.4
D.5
APPENDIX E·
E.l
E.2
APPENDIX F:
SPECIAL COMPUTER PROGRAM FEA TURES
Finite Element Mesh Generation
Skew Boundaries
Nodal Point Temperature and Pore PressureInte rpola tion
Stres s -Strain Calculations
Restart and Multiple Case Capability
MA TERIAL MODELS
Orthotropic Linear Elastic Behavior
Orthotropic Plastic Behavior
Orthotropic Linear Behavior with Different ElasticModuli in Tension and Compression
Effect of Pore Pressures
SOLUTION OF LINEAR EQUA TIONS
Gaussian Elimination
Simplification for Band Ma trices
CONVERGENCE OF FINITE ELEMENT RESULTS
Introduction
Error Analysis
Methods of Improving Accuracy
Convergence of Stresses in SAAS III
Conclusions and Recommendations
COMPUTER PROGRAM OUTPUT
Printed Output
Plotted Output
COMPUTER PROGRAM INPUT INSTRUCTIONS
v
A-I
A-I
A -10
A-II
A-12
A -14
B-1
B-1
B-7
B-ll
B-18
C-l
C-2
C-3
D-l
D-l
D-3
D-5
D-9
D-9
E-l
E-l
E-2
F-l
CONTENTS (Continued)
APPENDIX G: FORTRAN IV COMPUTER PROGRAM G-1
REFERENCES
G. I
G.2
G.3
G.4
G.5
G.6
G.7
APPENDIX H:
H. I
H.2
H.3
H.4
H.5
H.6
H.7
H.8
Description of FORTRAN Auxiliary Units
Functions of Subroutines
IBM 360 FORTRAN IV COIT1puter PrograIT1 Listing
UNIVA C II 08 FORTRAN IV COIT1puter PrograIT1Listing
CDC 6600 FORTRAN IV COIT1puter PrograIT1 Listing
PLT360, IBM 1627 Plotting Routine
Modification of PrograIT1 Capacity
EXAMPLE PROBLEMS
Hollow Cylinder with UniforIT1 Internal and ExternalPressure (Lame Cylinder)
Hollow Cylinder with Noncylindrical Orthotropy Uniform Pressure
Hollow Cylinder with Noncylindrical Orthotropy Axial Load
Thick Spherical Shell of a Bilinear IsotropicMaterial under Uniform Internal Pressure
Hollow Cylinder Composed of Two Materials
Solid Porous Cylinder
Thick Spherical Shell of a Multimodulus IsotropicMaterial under Internal and External Pressure
Plane Stre s s Solution to the Bending of a CantileverBeam
vi
G -1
G-3
G-7
G- 65
G-7Z
G-75
G-II0
H-I
H-l
H-9
H-19
H-24
H-34
H-46
H-58
H-67
R-I
FIGURES
I.
2.
3.
4.
5.
A-I.A-2.
A-3.
A-4.
A -5.
A-6.
A-7.
A -8.
A -9.
A -10.
B-I.B-2.
D-I.D-2.
D-3.
D-4.
F-I.
F-2.
F-3.
The Finite Element Idealization
The Finite Element Idealization of Plane Solids
Triangular Element
Definition of Principal Ma terial Coordinates
Quadrilateral Element
Laplacian Grid with Equally Spaced Boundary Points
I-J Grid Transformed from Laplacian Grid in Figure A-I
Laplacian Grid with Unequally Spaced Boundary Points onTwo Sides
Laplacian Grid with Internal Line Specification
Laplacian Grid with Triangular Elements
I-J Grid Transformed from Laplacian Grid in Figure A-5(Note diagonal line segment. )
Mesh Plot Utilizing Eq. (A -1)
Mesh Plot Utilizing New Circular Region Option [Eq. (A - 2)J
Angle to Skew Boundaries
Triangular Area Determined by Input Temperature Points
Effective Stres s -Strain Relationship
Relation of Material Orthotropy to Principal Stress andBody Coordinates
SAAS I Convergence of Solutions - Lam'; Cylinder
Comparison of Old and New Integration Schemes withRespect to Discretization Errors
Illustration of the Use of Double Precision Arithmetic toImprove Accuracy
Convergence of New Stress-Strain Calculations
Orientation of Principal Material (MN) Axes Relative toBody (RZ) Axes
Boundary Pressure Sign Convention
Boundary Shear Sign Convention
vii
1-2
1-3
II-9
II-15
II- 21
A-3
A-3
A-5
A-5
A-6
A-6
A-7
A-9
A-IO
A-12
B-9
B-12
D-4
D-6
D-8
D-IO
F-16
F-18
F-20
FIGU RES (Continued)
H-l.
H-Z.
H -3.
H-4.
H -5.
H-6.
H-7.
H-8.
H-9.
H-IO.
H-ll.
H-IZ.
H-13.
H-14.
H-15.
H -16.
H-17.
H-18.
H-19.
H-ZO.
H-Zl.
H-ZZ.
H-Z3.
Four -Element Idealization of Hollow Cylinder
Computer Program Output for Example 1
Hollow Cylinder - Uniform Pressure
Computer Program Output for Example Z
Hollow Cylinder - Axial Load
Computer Program Output for Example 3
Schematic Diagram of Wedge-Shaped Ring
Computer Program Output for Example 4
Ten-Element Idealization of Hollow Cylinder Composedof Two Materials
Computer Program Output for Example 5
Radial Stress in Hollow Cylinder
Axial Stres s in Hollow Cylinder
Circumferential Stress in Hollow Cylinder
Ten-Element Idealization of Solid Cylinder
Computer Program Output for Example 6
Radial and Circumferential Stress in Solid Cylinder
Axial Stress in Solid Cylinder
Schematic Diagram of Wedge -Shaped Ring
Computer Program Output for Example 7
Element Plot
Computer Program Output for Example 8
Deformed Grid
Contours of Longitudinal Stress
viii
H-Z
H-3
H-ll
H-13
H-l9
H-ZO
H-Z5
H-Z6
H-34
H-36
H-43
H-44
H-45
H-46
H-48
H-56
H-57
H-59
H-60
H-68
H-69
H-83
H-84
TABLES
H-l
H-2
H-3
Exact and Computer Stresses for Hollow Cylinder ofFigure H-l
Exact and Computer Results for Stresses in a ThickSpherical Shell of a Bilinear Isotropic Material
Exact and Computer Results for Stresses in a ThickSpherical Shell of a Material with Different Tensile andCompressive Moduli
ix
H-2
H-33
H-66
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SECTION I
INTRODUCTION
The finite element terminology as applied to the analysis of continua
was first used in 1960 (Ref. 1). Prior to 1960, two-dimensional elements
were used in conventional methods of structural analysis as a means of
improving the stiffness idealization of complex aircraft structures (Ref. 2).
Since the introduction of the finite element method for the stress analysis
of plane stress bodies, the technique has been successfully applied to plates
(Ref. 3), axisymmetric solids (Refs. 4, 5, 6), axisymmetric shells (Refs.
7, 8), three-dimensional solids (Refs. 9, 10), torsion of shafts (Ref. 11),
and other boundary value problems (e. g., Refs. 12, 13).
Stress analysis of complex solids subjected to arbitrary loads is a
fairly common problem. At the present time, the solution of arbitrary
three-dimensional stress problems is impractical because of the large amount
of computer time required (Ref. 10). However, for a reasonable amount of
computer time, a large class of practical two-dimensional problems can be
readily solved (Ref. 6).
In the finite element approximation of solids, the continuum is replaced
by Cl system of elements which are interconnected at their corners (nodes).
In the case of an axisymmetric solid, the nodes are actually circles and are
called nodal circles. In the case of a plane body, the nodes are points. A
finite element idealization of a simple axisymmetric solid is shown in Figure
1, and the finite element idealization of a plane body is presented in Figure 2.
Two equilibrium equations, which are expressed in terms of unknown nodal
circle or point displacements, are derived for each node of the finite element
system. A solution of the resulting set of linear algebraic equations consti
tutes an equilibrium solution to the finite element approximation.
I- 1
z
z
a. Actual Solid
• R
b. Finite Element Approximation
Figure 1. The Finite Element Idealization
1-2
....------------+R
z
....------------.. Ra. Plane Solid
z
No
~l:l;:: b. Finite Element Idealization
Figure 2. The Finite Element Idealization of Plane Solids
1-3
The advantages of the finite element method over other methods are
numerous. The finite element method affords nearly complete generality in
the specification of geometrical and material properties, i. e., geometrically
complex bodies of many different materials are easily represented. Dis
placement or stress boundary conditions can be specified at any node of the
finite element system. Thermal and mechanical loads can be specified at
nodes and, if the number of nodes is sufficient, nearly arbitrary distributions
of the loads can be represented. In addition, the body as a whole can be
subjected to accelerations and/or angular velocities.
In the present report, the finite element method is applied to the
determination of stresses, strains, and displacements in arbitrary axisym
metric and plane bodies with orthotropic, temperature -dependent material
properties that can be different in tension and compression. The bodies can
be subjected to arbitrary axisymmetric mechanical, thermal, and pore pres
sure loading. The mechanical loads can be surface pressures, surface
shears, and nodal point forces as well as acceleration or angular velocity.
The equilibrium equations for a finite element system are derived, and the
corresponding computer program is described and displayed. This computer
program is named SAAS III for the third version of ~tress :6:nalysis of :6:xi
symmetric ~olids. The present report embodies extensive revisions of
Refs. 14 and 15 in which the SAAS I and SAAS II programs are described.
SAAS III is based on fairly extensive use of SAAS I and SAAS II, and the
revisions were undertaken to increase the capability of SAAS and to make it
easier to use.
The program's capability was improved by the introduction of plane
stress/strain options, unequal properties in tension and compression, and
a porous :media option. Program efficiency was improved by incorporating
a new procedure for calculating element strains (Ref. 16).
1-4
The SAAS III program is mOre user oriented than is SAAS Ior SAAS II.
Improvements consist of extensive internal documentation through the liberal
use of comments cards as well as reorganization and restructuring of the
program so that each subroutine represents a specific computing task. The
two-dimensional mesh generation scheme first implemented in SAAS II has
been improved by correcting some minor deficiencies and adding an option
which aids automatic mesh generation in circularly shaped regions. In
addition, a new pressure interpolation subroutine is employed to make the
inputting of surface pressures a much easier task. Material property input
was simplified so that one needs to input only those properties that are unique
for the material type. Contour plotting has been improved and an option was
added to permit plotting of the deformed grid. More extensive instructions
are included for use in modifying the program capacity, and detailed descrip
tions are given of modifications required in the program for its implementa
tion on CDC 6600 and UNIVAC 1108 computers.
All of the SAAS II features that have been retained for SAAS III are
also described in this report. The features include temperature interpolation,
restart and multiple case capability, skew boundaries, and elastic -plastic
analysis. In addition, numerical examples and convergence studies were
performed and are presented again to verify the computational accuracy of
the program.
1-5
(This page intentionally left blank)
1-6n
SECTION II
METHOD OF ANALYSIS
The finite element method and the general equations which govern the
equilibrium of the system are given in the literature. However, for com
pleteness and in order to define the various terms involved, the equations
are rederived here.
A. SCOPE OF ANALYSIS
The SAAS III computer program performs a static stress analysis of
three general two-dimensional structures: solids of revolution, solids in a
state of plane strain, and solids in a state of plane stre s s. In each of these
problems, three clas ses of material behavior can be modeled: orthotropic
linear elasticity, orthotropic linear behavior with different elastic moduli
in tension and compression, and orthotropic bilinear plasticity. In addition,
the materials may be porous with internal pore fluid pressures and tempera
ture dependent.
1. Axial Symmetry
Symmetrically loaded bodies of revolution are solved by
applying a triangular ring element idealization of the solid. The orthotropy
of material properties is as general as possible within the assumption of
axial symmetry. All mechanical loadings in the meridional plane can be
handled in addition to body forces due to acceleration and rotation. Arbitrary
axisymmetric temperature and pore pressure distributions are internally
converted to thermal and pore fluid stresses which eventually become equi
valent nodal point force s.
II-I
2. Plane Strain
The plane strain feature of the computer program can be
invoked by the input of a single quantity. It is accomplished internally by
applying a triangular plane element idealization of the solid. The total
transverse strain is set to zero, and all equations are modified accordingly.
All of the material and loading options are available as in the case of axial
symmetry. Distributed loads in the form of pressures are converted in
ternally to equivalent nodal point forces. Body forces due to an acceleration
in the plane are admissible.
3. Plane Stress
The plane stress feature of the computer program can be
invoked by inputting a single quantity just as in the case for plane strain.
The only difference is that the transverse stress is set to zero to obtain
the appropriate equations. All the program features are available in a
way similar to plane strain.
4. Material Models - General Discussion
The following is a general description of the material models
available in the computer program. Detailed discussions appear in Appen
dix B.
a.9rthotropic Elasticity
There are only two restrictions to linear elastic
material modeling; rotational symmetry in the axial symmetric mode of
operation and orthogonality of material axes. The input quantities are in
the form of Young's moduli and Poisson's ratios. These quantities are con
verted internally to stress-strain properties and put in matrix form. A
detailed description of the model and definitions of input quantities are pre
sented in Appendix B, Section B. 1.
II-2
b. Orthotropic Bilinear Plasticity
The computer program has provision for input of a
bilinear form of Young's moduli. By application of an orthotropic form of
the von Mises' yield criterion and through a recursive iteration procedure,
a final solution is obtained wherein the stress and strain results are con
sistent with the appropriate secant modulus description of an effective
stress -effective strain relationship. This is known as the deformational
plasticity approach to this class of problems. As such, the user should be
reminded that a specific history of loading cannot be accounted for. How
ever, the procedure is well-founded and accurate for proportional loading
problems of isotropic plasticity. A detailed description of the process is
given in Appendix B, Section B.2. This feature of the computer program
cannot be used simultaneously with the unequal propertie s option.
c. Unequal Properties in Tension and Compression
The computer program has provision for input of
differe nt orthotropic temperature -depe ndent mater ial pr ope rtie s in tens ion
and compression. By suitable definition of cross-compliance terms in the
resulting stress-strain relation and through a recursive iteration procedure,
a final solution is obtained wherein tension and compression properties are
consistent with stress magnitudes and signs. A detailed description of the
process is given in Appendix B, Section B. 3. This feature of the computer
program cannot be used simultaneously with the bilinear plasticity option.
d. Temperature Dependence and Thermal Stress
Material properties can be input as a multilinear
function of temperature. During solution of a problem, each element tem
perature is used to obtain element material properties from the tabular
input by linear interpolation. The coefficients of thermal expansion used to
compute thermal stresses are also input as a multilinear function of tem
perature and can be input as either "coefficients of thermal expansion" or
" free thermal strains. "
II-3
e. Porous Media
The effect of internal pore fluid pressures in porous
materials can be handled in the program by inputting a pore pressure field
similar to the temperature field. All program options relating to the handling
of temperature data are also available for pore pressure data. The theory of
deformation of porous elastic solids by M. A. Biot is specialized for appli
cation herein. A detailed description of the approach taken is presented in
Appendix B, Section B.4.
B. EQUILIBRIUM EQUATIONS AND FINITE ELEMENT DISCRETIZATION
Derivation of the matrix equations utilized in the finite element method
of analysis is given in the following discussion. At each step it is shown how
the pore pressures augment the relationships normally used for solid media
analyses.
The potential energy of a porous elastic solid is given by
v = U - fvol
w.F. dV 1 1 f
area
w.P. dA1 1
(1)
where U is the total strain energy of the solid, or
U = f [fCi
a. dC i ] dV1
vol 0
(2)
F. is a body force, P. is a surface traction, w. is a displacement, and1 1 1
'if. is the total stress due to both the solid and the pore fluid. The meaning1
of a. is more thoroughly explained in Section B.4 of Appendix B.1
For a body composed of M elements, the potential energy can be
written as
M
V =L:m=l
[ u= - fvol
mm fw. F. dV -1 1
area
II-4
m m ]w. P. dA1 1
(3)
For each element, assume a displacement field
where the vector of nodal displacements and Cd] m
(4)
is an undete r-
mined matrix of coefficients. In transposed form,
(5 )
In addition, let the strains of an element I € 1m
be given in terms of
nodal point displacements, or
(6)
and
(7)
where [a]m depends on the geometry of the problem and is undetermined
at this point.
The thermoelastic stress-strain equation for a porous material is
given in Appendix B, Section B.4, and is written for an element, m, as
where IT I are thermal stresses. They correspond to the state of stress
due to the complete restraint of thermal expansion. The I(J Im represents
the pore stress in the element. Note that it is implicitly assumed that the
pore pressure and thermal stress are constant throughout the element,
II-5
The strain energy of an element can now be formulated in terms of
the element displacements as
um
= tf IEj';: [C]m /El m dV - f lEI';: l-rjm dVvol
+ f lEI';: Hm dVvol
(9)
The total potential energy of the system can be found by a summation
of the element strain energies, body forces, and surface tractions as
M
V = EJt llEI';: [CJ m IElmdV
- LIEI;;HmdV
+ f H;; jujdV - f H;; IFlm dV - f /wi ;; Iplm dA]vol vol area
(IO)
By substitution of Eqs. (7) and (5) into Eq. (10), the potential energy
of the system can be expressed as a function of nodal point displacements.
Then, the potential energy is made stationary by requiring that
av = 0 i = 1, NaUi
where N is the total number of nodal point displacements.
Il-6
( 11)
The result is a set of N simultaneous equations which can be written
in matrix forIn as
(12 )
It is customary to introduce the following notation. The individual
element stiffness is
[a]T [C] [a] dVm m m
(13)
The body force vector for an element is
II-7
(14 )
Note that the effect of pore pressures is to augment the body force vector.
The surface force vector is
(15)
area
Only those elements having a portion exposed to the surface are involved in
the surface force vector. The system stiffness is obtained by a summation
of the element stiffnesses. That is,
M
[K] = Lm=l
(16)
The total load on the system is a summation of the element loads, or
(17)
Therefore,
(18)
This equation is recognized as the general equilibrium relationship for a
finite element system. The unknown displacements Iu I can be obtained by
solving the N simultaneous equations.
C. LINEAR DISPLACEMENT TRIANGULAR ELEMENT APPROXIMATION
1. Displacement Model
Let the body be idealized by a
plane elements as shown in Figures 1 and 2.
triangular element is illustrated in Figure 3.
II-8
system of triangular ring or
The cross section of a typical
The displacement of the
element is assumed to be a linear function of the coordinates. To simplify
documentation, rand z are chosen to be coordinate name s for the plane
problems instead of the usual x and y. The r -z displacements are
(19a)
(l9b)
or, expanded in matrix form,
(20)
This linear displacement field assures continuity between elements since
line s which are initially straight remain straight in their displaced position.
r.J
OJr.
0 ILtl
..M i
1Zi
MM
ZkN
~
Figure 3. Triangular Element
In the plane stress problem, there is a third displacement normal to the two
dimensional body. Since the third displacement is not required in the solution
process, it is ignored in the analysis.
II-9
When Eq. (20) is evaluated at the three nodal points of the
triangular element, the following matrix is obtained:
i i 1fb l b41u u r. z.
r Z 1 1
U j U j 1 (21 )= r. z.
lb2
bS jr Z J J
k k 1 b 3b6u u r k zkr Z
Note that the nodal point displacements are not in vector form. A conversion
to the vector form is necessary prior to their use in the theoretical equation,
Eq. (18). By inverting Eq. (21) and writing in vector form,
where
(22)
rjzk -rkzj0 rkz i -rizk
0 r.z.-r.z. 01 J J 1
Zj -Zk 0 Zk -Zi 0 Z.-z. 01 J
1 rk
-rj
0 ri
-rk 0 1'.-1'. 0[h] =- J 1 (23 )m )...
0 r/k -rkzj0 rkz i -r iZk 0 1".z.-r.z.
1 J J 1
0 Zj -zk 0 zk-zi 0 2.-Z.1 J
0 rk
-rj
0 ri-rk0 r. -r.
J 1
and
r. (z. - zk) + r k (z. - z.)1 J 1 J
II-10
(24 )
The element strains are obtained from Eqs. (19a) and (19b):
OwE
rb 2= fiT =rr
OW z b6Ezz = ()Z =
w=l...br + b 2 + z
b3E()(} = -r r I r
OW OWE = r + z b 3 + b
5=rz Oz or
(25a)
(25b)
(25c)
(25d)
For plane strain, c(}(} = O.
For plane stre s s, E () () is computed from the other strains
and the material properties with the condition, (J(}() = O.
These strains can be written in matrix form for axial
symITletryas
b l
c 0 I 0 0 0 0 b 2rr
Ezz 0 0 0 0 0 I b3
= (26 )I
Iz
0 0 0E(}(} r r b 4
c 0 0 I 0 I 0 b 5rz
b 6
II-ll
or, symbolically,
(27)
For plane problems, the third row of [gJ is set to zero.
Substitution of Eq. (22) into Eq. (27) yields
(28 )
Thus, the strain-displacement transformation matrix, as defined in Eq. (6),is
(29 )
With this definition of [aJm , the element stiffness matrix, Eq. (13), is
rewritten as
[k]m = f[h]';: [g]T [C]m [gJ [hJm dV
vol
Since [hJm
is not a function of rand z, Eq. (30) becomes
(30)
(31 )
Because of the need to perform the integration term by term,
the matrices under the radical are multiplied by hand. The result for axial
symmetry is:
II-12
The result for plane problems is
0 0 0 0 0 0
0 Cll C l4 0 Cl4 CIZ
[g]T [C][g] =0 C14 C44 0 C44 CZ4 (33)0 0 0 0 0 0
0 Cl4 C44 0 C44 C24
0 C IZ CZ4 0 CZ4 CZZ
II-13
Equations (32) and (33) are programmed directly, and the integrals in Eq. (32)
are evaluated numerically in the computer program. Note that for plane
problems, the result of integration is simply the element area. Thus, each
term is multiplied by that quantity.
2. Material Description
For bodies with orthotropic material properties, the principal
axes of which are not aligned with the body r-z coordinates, stress-strain
relations in the principal m-n coordinates are
, 1 ,O"mm Cll C l2 C 13
I , I
0" C l2 C22 C 23nn=
1 I 1
0"(j (j C l3C23 C 33
0" 0 0 0mn
o
o
o
fmm
fmn
Tmm
o
(34)
whereI 1 I
Tmm = T (C ll(t + C l2(tn + C 13(te)m
I 1 I
T = T (C l2(t + C22(tn + C
23(te) (35 )
nn m, I ,
Tee = T (C l3(t + C23(tn + C
33(te)
m
and the m-n coordinates are defined in terms of the r-z coordinates in
Figure 4.
II-l4
zn
m
'- .rFigure 4. Definition of Principal Material Coordinates
Equation (34) can be abbreviated as
(36)
where the subscripts refer to the coordinate system in which the quantities
are expressed. The stresses in the local m-n coordinates are transformed
into the body r-z coordinates by use of the transformation
(37)
where [t]T is the transpose of
2 . 20cos Cl sm Cl
. 2 20SIn CJ. cos ex
[t] =0 0 I
-2sinClcosCl 2 sinClcosCl 0
sinacosa
-sinacosa(38 )
o2 . 2
cos Cl-Sln Ct
The strains in the local m-n coordinates are transformed into the body r-z
coordinates by use of the transformation
(39 )
II-IS
The inverse of Eq. (39) is
(40)
Upon substitution of Eqs. (37) and (40) in Eq. (36), it is seen that
la Irz = [C]rz 1< Irz - IT Irz
where
[C]rz = [tf [C]ns[t]
ITlrz = [tf Hns
Equation (41) can be expanded to read
a C ll C IZ C 13 C l4 < Tlrr rr
a zz C IZ C ZZ C Z3 C Z4 < TZzz
=a()() C 13 C
Z3 C 33 C 34 fee T3
a C l4 CZ4 C 34
C44 f T4rz rz
where
Tl = T (CllCir + CIZCiz + C 13Ci()}
TZ = T (CIZCir + CZZCiz + C
Z3Ci e)
T3 = T (C
I3Ci
r + CZ3
Ciz + C
33Cie)
T4 = T (CI4
Cir + C
Z4Ci
z + C 34Ci()}
(41 )
(42 )
(43 )
(44)
(45)
The coefficients of thermal expansions,
r, z, and (j directions, respectively,
within the element.
Cir
and
, Ci z and Cie , are in the
T is the temperature change
II-16
For axial symmetry and plane strain, the stress-strain matrix
[C] and the thermal stress vector ITI are used in the form shown.rz rzFor plane stress, it is necessary to incorporate the condition a
OO= 0 in
[C] and IT I . The third equation of Eq. (44) becomesrz rz
(46 )
With this definition of too ' Eq. (44) can be rewritten as
a ell e lZ 0 e 14t T lrr rr
a e lZ e ZZ 0 e Z4t TZzz zz
= (47)0 C 13 CZ3 C 33 C 34 too T3
a e 14 e Z4 0 e44
t T4rz rz
where the barred quantities are
C 13Z
ell Cll (48a)= - C33
=
=
=
=
C 1Z
CZ3 C 13- C 33
C 14
C13
C34- C33
CZ3
Z
CZZ - C33
CZ4
CZ3 C 34- C 33
II-17
(48b)
(48c)
(48d)
(48e)
=
=
=
=
(48f)
(49a)
(49b)
(49c)
The barred quantities are used in Eq. (33) for plane stress analyses and in
setting up the thermal stress vector.
3. Thermal, Mechanical, and Pore Pressure Loads
The body force vector, Eq. (14), can be put in the following
form by combining it with Eqs. (4), (5), (6), (22), and (29):
/L\m = [hJ;:' f l[g]T !TI + [e]TIFI - [gJT/alldv (50)
vol
where the vector, being integrated, can be written explicitly for axisym
metric problems as
Fr
rFr
zFr
F z
rFz
1+ r (T3
-a)
+ Tl + T
3 - 2a
+ ~(T _ a) + T4r 3(51)
+
zFz +
II-18
The pore fluid stre ss is given by
a = - fp
and f is the porosity and p is the pore pressure.
(52 )
In the case of rotation of the body with angular frequency, w ,
the body force in the r -direction is
and, for acceleration of the body in the z -direction, a z ' the body force
in the z -direction is given by
F = -maz z
where m is the mas s density of the material.
(53 )
(54)
For plane strain and plane stress problems, Eq. (51) becomes
Fr
rF + 71 - a
r
zF + 7 4r(55 )
F z
rF + 74z
zF + 72 - a
z
where F is now interpreted as a body force in the r-direction due to anr
acceleration in the r-direction.
F = - mar r
( 56)
Integrations of Eqs. (51) and (55) are performed numerically in
the computer program. The vector 1L I is formed by standard matrix opera
tions and is added to the load vector ·1 Q I as indicated in Eq. (17).
II-19
D. QUADRILA TERAL ELEMENT
A typical quadrilateral element is composed of four triangular elements
as illustrated in Figure 5. The ten equilibrium equations for the quadri
lateral are developed by the application of Eqs. (31) and (50) and can be
written in the following matrix form,
[~;:---+---~~-] = (57)
where ua
and qa are associated with points 1 to 4 and ~ and qb are
associated with point 5. Equation (57) can be written as two matrix equations.
[kaall ua I[kba] Iu a I
+ [kab ] lubl = Iqal
+ hb] !ub ! = !qb!
(58 )
(59 )
Equation (59) can be solved for the displacements ub '
If Eq. (60) is substituted in Eq. (58), an expression is found which relates
the force s at points 1 to 4 to the unknown displacements at points 1 to 4
and the known thermal loads.
where the quadrilateral stiffness matrix is
(60)
(61 )
II-20
(62)
1...,... 4
2
......... ,/......... ./
.......... /'
...................... 5 ,//
--->(-- \\\
\
Figure 5. Quadrilateral Element
and the modified load matrix is
(63)
The use of the quadrilateral as a separate element is desirable since
the resulting set of equilibrium equations has fewer unknowns for a given
number of triangular elements. In the computer program, the above procedure
is applied to only 1 degree of freedom at a time for the center point. There
fore, the procedure first reduces [K] to a 9 x 9 and then to an 8 x 8
matrix. In this way, the inversion of [KbbJ is triviaL
E. BOUNDARY CONDITIONS
Equation (18) repre sents the relationship between all nodal point force s
and all nodal point displacements. Mixed boundary conditions are considered
by rewriting Eq. (18) in the following partitioned form:
(64)
II-2l
where
IOal = the specified nodal point forces,
lObi = the unknown nodal point force s,
hi= the unknown nodal point displacements, and
jUb I = the spec ified nodal point displacements.
The first part of Eq. (64) can be written as a separate matrix equation,
(65)
and then expre s sed in the following reduced form,
(66 )
where the modified load vector is given by
(67)
In the computer program, the above procedure is performed for 1
degree of freedom at a time by row and column manipulations. For displace
ment boundary conditions, the load vector is modified as in Eq. (67), and then
the corresponding rows and columns are set to zero except for the diagonal
terms which are given the value 1. Then, the corresponding terms in the
load vector are given the value of the specified displacements. Force
boundary conditions are implemented by simply modifying the load vector.
Note that this procedure preserves the order of the original system; that is,
specifying a displacement does not reduce the number of equations being
solved.
II~22
Q
SECTION III
SUMMARY
The governing equations are developed for the finite element stress
analys is of com plex axis ymmetric and plane solids. The as sodated com
puter program is very general, but spedal options make it particularly
suitable for the thermal stress analysis of solids with orthotropic, tempera
ture-dependent material properties. In addition, unequal properties in
tension and compression, internal pore pressures, and elastic plastic
behavior can be accounted for in plane and axisymmetric solids. Due to
the requirement for iteration in plastic and unequal properties problems,
the two features cannot be implemented simultaneously.
Several special computer program features are described in
Appendix A. These s pedal features include: automatic mesh generation,
skew boundaries, temperature and pore pressure interpolation, orthotropic
material properties, and restart and multiple case capability.
A complete description of the various material models that can
be implemented in SAAS III is given in Appendix B. These models are very
general, and a great deal of effort was expended to make them eas y to use.
A very important and time consuming part of the computer program
is the solution of the equilibrium equations. The technique used is the
well- known Gauss elimination method. A complete desc ription of this
me thod as a pplied to band matrices is contained in Appendix C.
An important consideration in the use of an approximate method
such as the one presented here is whether or not convergence can be demon
strated. Convergence of the approximate solution to the exact solution for
a Lam;:;' cylinder problem is discussed in Appendix D. Relative magnitudes
of discretization and round-off errors are identified as a function of element
size and computational accuracy.
III-l
Printouts of the computer program and the plotted output are given
in Appendix E. The computer program input sequence described in Appendix
F has been made simple for the user. Only that information necessary
for solving a particular problem need be input. In Appendix G, the complete
IBM 360 computer program is listed with instructions for modification of
capacity, implementation on CDC 6600 and UNIVAC 1108 computers, and
a des c ription of the plotting package. Eight simple numerical example s
with known solutions are presented in Appendix H. These examples illustrate
the use of various program capabilities and provide test cases for the
SAAS III program.
1II-2Q
APPENDIX A
SPECIAL COMPUTER PROGRAM FEATURES
Certain special computer program features not implied by the report
title are discussed in this appendix. These special features include finite
element mes h generation, skew boundaries, tern pe rature interpolation,
stress-strain calculations, and restart and multiple case capability. In
addition, it may be noted from Appendix F, Computer Program Input Instruc
tions, that a constant temperature can be specified for the body (a feature
which can be utilized in cool-down problems), and free thermal strains
can be input as an alternative to coefficients of thermal expansion.
A. 1 FINITE ELEMENT MESH GENERA nON
A. 1. 1 Introduction
The finite element mesh generation scheme was obtained
from Mr. Brian Stocks of the Lockheed Palo Alto Research Laboratory and
adapted by the authors for use first with SAAS II and now with SAAS III.
Mesh generation is accomplished in three steps.
The first step in mesh generation is to define the perimeter
of the two-dimensional region (in right-handed R-Z coordinates) in terms of
a finite number of line segments. The line segments are defined by the locations
of the end points. Circular line segments are defined by one intermediate
point, or the center, in addition to the end points. Intermediate points on the
perimeter are generated by linear interpolation. A two-dimensional indexing
scheme determines the number of finite elements into which the area is divided.
The second step in mesh generation is to determine the
coordinates of the nodal points which are interior to the perimeter. This step
is accomplished by satisfaction of Laplace's equation over a corresponding
A -1
(transformed) grid in the 1- J plane (I and J are right-handed coordinates).
The use of Laplace's equation results in finite elements which are similar in
size and shape to adjacent elements.
The final step in mesh generation is to renumber (index)
the two dimensionally generated nodal points and elements in accordance with
the one-dimensional numbering scheme used in the analysis.
Any reasonable combination of internal and external
line segments which represent circles, straight lines, or points in the R-Z
plane and horizontal, vertical, or 45-degree diagonal straight lines in the
I-J plane can be used to generate a finite element mesh.
A. 1. 2 Point Generation
A region in the R - Z plane which is to be divided into
finite elements is shown in Figure A -1. The perimeter of the area is defined
by a series of line segments which, in turn, are defined by the R-Z coordinates
of a set of points such as are designated by dots in Figure A -1. The remainder
of the perimeter points are determined by linear interpolation along the
prescribed line segments. The region in the R- Z plane can be transformed
into a region in the I-J plane as shown in Figure A-2. Note the 1:1 corre
spondence between points in the R-Z plane and points in the I-J plane.
The interior points of the finite element grid are found
by satisfying Laplace's equation in finite difference form over the transformed
grid for each of the coordinates, Rand Z. That is, for each point of the
transformed grid, the following equations must be satisfied:
Z + Z + ZI+l,J I-l,J I,J+l
+ RI , J - 1
+ ZI J-l,
o
o
(A -1 a)
(A-lb)
The above linear simultaneous equations are solved by specifying the R- Z
coordinates on the boundary and working into the interior by a relaxation
technique. It is seen that the R-Z coordinates at each point (I, J) take on
values which are the average of the R-Z coordinates of the four surrounding
points.
A-2
z
a
Figure A-L Laplacian Grid with Equally SpacedBoundary Points
J
12d
11e
109
876 f c5432
'" 1 a b0
'"l:l 0t:l 0 1 2 3 4 5;::
Figure A-2. 1- J Grid Transformed from LaplacianGrid in Figure A-I
A-3
R
A. 1. 3 Special Features
A considerable
the present type of mesh generation.
following parag raphs.
variety of effects can be obtained with
Two special effects are discussed in the
Internal line segments can be utilized in addition to
boundary line segments in the generation of a finite element mesh. Consider
the effect of moving points c and f in Figure A-I in an attempt to generate
a finer mesh in the region cdef as shown in Figure A-3. The mesh in
Figure A-3 can be considerably improved by specifying the internal line
segment cf to be a straight line as in Figure A -4.
Diagonal line segments in the I-J plane are an additional
feature of mesh generation. For example, the finite element mesh in Figure A-4
is refined in the region abcf to yield the mesh shown in Figure A-5 by specifying
a 45-degree diagonal line segment ga (Figure A-6). Note that line segment
gha is a circular segment in the R-Z plane. Diagonal line segments can have
only a 45-degree inclination in the I-J plane so that all intersections of lines
in the I-J plane occur at integer values of I and J.
A. 1. 4 Mesh Generation in Circular Regions
It was found through use of SAAS II that circular regions
(in the R-Z plane) were not always modeled adequately by Eq. (A-I). The
deficiency is illustrated in Figure A-7 where it can be seen that the elements
are too small near the inner boundary and too large near the outer boundary
of the circular region. At the suggestion of Dr. Frank Weiler, an option was
incorporated in SAAS III whereby the I-J coordinate system could be
interpreted as a polar system through inputting variable parameters to the
program to adjust the relative curvatures in the I-J plane. The equations
which are equivalent to Eq. (A-I) and are used in SAAS III are:
RItl , J + R I _I , J + R I , HI + R I , J-l - 4RI , J + Ci (RItI , J
- R I _ l , J)/2 (I + Is) + Cj (RI , HI - R I , J_l)/2 (J + J s ) = 0
A -4
(A - 2a)
z
a
Figure A-3. Laplacian Grid with Unequally SpacedBoundary Points on Two Sides
...
z
a
Figure A-4. Laplacian Grid with InternalLine Specification
A-5
d
e
R
...
z f
Figure A-5. Laplacian Grid with Triangular Elements
h
e
f
)/
/
d
c
...a
1234567b I8 9 10
Figure A-6. 1- J Grid Transformed from LaplacianGrid in Figure A-5 (Note diagonalline segment.)
A-6
e.
s.
0.0
-2.
-4.
-6.
-e.
c
b
a
d
ELEMENT PLOT
C/J....x~ -to.N
0.0 2.R-AXIS
s. e.
Figure A-7. Mesh Plot Utilizing Eq. (A-l)
A-7
+
When C. = C. = 0,1 J
This occurs as a default in the program.
appropriate quantities are input. When
(A-2b)
ZI, J_l)/2 (J + J s ) = °Eq. (A - 2) reduces to Eq. (A -1).
That is, Eq. (A-l) is used unless
C. = 1, C. = 0, and J = 0, J isJ 1 s
the radial component. With C. = C. = 1 and I = J = 0, both I and J1 J S S
are radial components. The variable parameters Is and J s determine the
location of the origin of the I-J coordinate system. When C. and/or C.1 J
are set to positive values other than 1, the effect is emphasized or
de-emphasized according to these values in comparison with 1. The
parameters C., C., I ,J are input to the program and, with some1 J S s
experimentation, curved regions can be modeled very satisfactorily.
The results of applying Eq. (A-2) to mesh generation in
a circular region are shown in Figure A-S. Line segments ab, be, cd, and
da were input along with C. = 1, C. = 0, I = 10, J = 0. It can be seen that1 J s s
the resulting mesh is superior to that of Figure A-7.
A. 1. 5 Nodal Point Numbering
The final step in mesh generation is to number the
nodal points and elements in accordance with the one-dimensional numbering
scheme. This procedure is initiated on the line represented by the least
value of J. The point on that line with the least value of I is then designated
as the first point (say, a in Figure A-2). A simple counting is initiated by
proceeding in the direction of increasing 1. When the maximum value of I
corresponding to the current value of J is reached, the next point is found
by incrementing J. This sequence continues until the maximum value of J
is reached. Element numbers are assigned by a similar counting procedure.
A-8
8.
6.
0.0
-4.
-8.
ELEMENT PLOT
({J.....xCf -10.N
0.0R-RXIS
6. 8.
Figure A-S. Mesh Plot Utilizing New Circular Region Option[Eq. (A-Z)J
A-9
A.2 SKEW BOUNDARIES
If the number in Columns 21-30 of the BOUNDARY CONDITION
CARD or in Columns 6-15 of the NODAL POINT CARD is negative, it is
interpreted as the magnitude of an angle (in degrees) illustrated in Figure A -9.
2
No
~
z
... .. r
1
Figure A -9. Angle to Skew Boundaries
The term in Columns 36-55 of the NODAL POINT CARD is
then interpreted as follows:
XR is the specified load in the I-direction.
XZ is the specified displacement in the 2-direction.
The angle must be input as a negative angle and can range
from -0.001 to -180 degrees. Hence, +1. 0 degree is the same as -179.0
degrees. The displacements of these nodal points which are printed by the
program are:
u = the displacement in the 1 -directionr
u = the displacement in the 2-directionz
The stresses and strains are calculated in the rand z directions.
A-IO
A.3 NODAL POINT TEMPERA TURE AND PORE PRESSUREINTERPOLA TION
If not specified on nodal point information cards, nodal point
temperatures and pore pressures are interpolated from a finite set of input
points. That is, the temperature or pressure and the corresponding rand
z coordinates are specified at a finite number of points in the region of
interest. The temperature or pressure "surface" (in r, z, T space) which
is defined by the input is approximated by a set of triangular-shaped planes.
The vertices of the planes are located at the intersection of the actual surface
with the input points.
The interpolation procedure consists of searching the list of
input points for the three closest to the nodal point for which the temperature
is to be determined. If the area in the r-z plane of the triangle defined by
the three closest input points is too small relative to an area determined by
one of the sides of the triangle and the dis tance from the nodal point to the
closest input point, then the fourth closest point replaces the third closest
point. The area test is repeated until a suitable trio of points has been found.
Four input points, a, b, c, d, are shown in the vicinity of
nodal point n in Figure A -1 O. Points a, b, and c are the closest to n so
they would be the initial choices for the area test. Distance ab is a measure
of the fineness of the input temperature field. Distance an: is a measure of
the proximity of the input temperature points to the nodal point. To satisfy
the area test, the area of triangle abc must be greater than k· ab . an where
k is a constant which can be adjusted to make the test more or less severe.
It was found for the cases studied that a value of k = 0.1 was satisfactory.
If the point c is located such that the area test cannot be satisfied with
triangle abc, point d (the fourth closest point) is used in place of c. By
inspection, it is apparent that the triangle determined by points a, b, and d
would satisfy the area test and hence be used for interpolation.
A-l1
d
a
n
c
Figure A-I O. Triangular Area Determined by InputTemperature Points
A.4 STRESS-STRAIN CALCULATIONS
After the unknown nodal point displacements have been found,
the total strains, stresses, and mechanical strains are computed and output.
. In SAAS II, strains were calculated from displacements by a method that
required reformulating the quadrilateral stiffness matrix. Since stiffness
generation is a significant part of the problem in terms of time usage, this
method proved to be very expensive. In SAAS III, a new method by Robert
D. Cook (Ref. 16) has been incorporated. It employs a linear displacement
function derived from a least squares fit of the nodal point displacements.
The result of Cook's derivation is
(' = (X3
Y l XzYZ)/D (A-3a)rr
(' = (Xl Y3 - XZY4 )!D (A-3b)zz
(' = XIYZ
+ X 3Y4 - XZ(Y I + Y3 )/D (A-3c)rz
A-IZ
f.()() = (t: U i ) /4r 0
where
Xl L z= r.1
i
Xz = L r.z.1 1
i
X 3 L z= Z.1
i
Y l = L r.u.1 1
i
YZ = L Z.ll.1 1
i
Y3 = L z.v.1 1
i
Y4 = L r.v.1 1
i
D = XiX 3 - X
ZZ
(A-4a)
(A-4b)
(A-4c)
(A-4d)
(A-4g)
All summations run from 1 to 4. The terms u. and v. are the radial and1 1
axial displacements at nodal point i, r. and z. are measured from the centroid1 1
of the element to nodal point i, and r 0 is the radius of the centroid. For
plane strain and plane stress, the operations are the same except that the
calculation of f.()(j is ignored.
The stresses are calculated in the usual way by formulating
the stress-strain matrix (taking into account any temperature dependence) and
applying Eq. (8). Mechanical strains are found by inverting the stress-strain
matrix and multiplying by the stress vector.
A -13
Although it was necessary to use double precision arithmetic
for evaluating Eqs. (A-3) and (A-4) when working on the IBM 360, the above
procedure saves roughly 20 percent of total computer running time for average
problems.
The old method of computing stresses is retained as an option
and is required when one needs to approximate the fundamental frequency of
the structure. This procedure requires the stiffness matrix to be reformulated
so, when it is elected, the old stress calculation procedure is used. Con
vergence of stresses by these two methods is discussed in Appendix D,
Section D.4.
A.5 RESTART AND MULTIPLE CASE CAPABILITY
For the purpose of operational efficiency, the SAAS III program
can be stopped and restarted at appropriate places by use of FORTRAN Unit
10. The first stopping location occurs after finite element mesh plotting and
is useful when it is desired to view only the mesh without expending the effort
to obtain the full solution. The second location for stopping is after the solution
is obtained, but before any contour plotting is performed. A tape can be saved
so that the problem can be restarted with contour plotting being the first
operation. This second starting location is especially useful when an initial
view of a few contour plots is desired. The tape can be used to generate more
contour plots without significant execution time.
Multiple cases can be run on the SAAS III program as described
in the input instructions. However, the restart capability can be used only in
conjunction with the last case as information for preceding cases is destroyed
in the process of solving multiple cases.
A-14Q
APPENDIX B
MATERIAL MODELS
There are several material models (stress -strain relationships)
available in SAAS III: (1) orthotropic linear elastic behavior, (2) ortho
tropic plastic (nonlinear elastic) behavior, (3) orthotropic linear behavior
with different elastic moduli in tension and compression, and (4) porous
media. Orthotropic linear elastic behavior is described in Section II,
Method of Analysis, and in Section B. 1 of this appendix. Orthotropic
plastic behavior is described in Section B. 2 and multimodulus behavior in
Section B. 3. The effect of internal pore pressures in porous media is
discus sed in Section B. 4.
B.l ORTHOTROPIC LINEAR ELASTIC BEHAVIOR
For an axisymmetric elastic body under axisymmetric loading, the
most general orthotropic material model (stress-strain relationship) is
given by the following strain-stress relations:
1 V VmOmn 0 a(0 E"" -~ -~mm mmm m m
vmn 1 VnO
(0-~ E - E"" 0 a
nn nnm n n
= Vm 0 vnO 1 ( B-1)
(000 -~ - E"" EO0 a
OOm n
1(0 0 0 0 c;-- amn mn
mn
B-1
where m and n are the principal material directions in the r-z plane
and () is the circumferential direction (and is, of course, a principal
material direction by virtue of the axial symmetry). For many materials,
the principal material directions, m and n, are in the rand z (radial
and axial) directions. However, for materials such as tape -wrapped rein
forced phenolics, the principal material directions are not aligned with the
rand z directions. Note that in Eq. (B-1) there are seven independent
elastic constants, Em' En' E(), vmn' Vm ()' Vn ()' and Gmn ; there are
no relationships between any of these moduli for a truly orthotropic material.
An example of an orthotropic material is a three dimensionally reinforced
composite material with unequal numbers and/or sizes of fibers in the three
directions.
a (all other stresses zero)
directionmodulus in the ()
for the loading am = a (all other stresses zero)
for the loading
Young'sEnn
Young I S modulus in the n direction
_ E ()()
Emm
---Emm
E =m
E =n
E() =
V =mn
Vm () =
The elastic constants in Eq. (B-1) are further defined as
Young's modulus in the m direction
= _ E()()Enn
for the loading a (all other stre s se s zero)
= shear modulus in the mn plane
The quantities V ,V ()' and V () are often called Poisson's ratios ortnn rn n
strain ratios. Note that the considered body is axisymmetric and is loaded
axisymmetrically; hence, there is no shearing in the m() and n() planes.
B-2
An alternative way of writing Eq. (B-1 ) is
I vmn vme0<
~ - --r- -~ ammmm m m mvnm I Vne
0< -~ E -~ annnnn n n
= vern ven (B-2)1
fee -~ -Ee Ee0 a
ee
0 0 0I
< c-- a mnmnmn
wherein there are apparently ten constants. However, by virtue of the sym
metry inherent in the definition of stre s s -strain re lations for an orthotropic
material, certain of the constants are related (i. e., not independent):
=
= (B-3)
=
Equation (B-3) is an alternative expression of the reciprocal relations of
orthotropic elasticity. The Poisson's ratios that have not yet been defined are:
= a (all other stresses zero)
Vnm =
=
for the loading a =n
for the loading <7e
a (all other stresses zero)
a (all other stresses zero)==<nn
- -,- for the loading aeee eNote from the above definitions and Eq. (B-3) that V is obviously not
mnequal to V •nm
B-3
B. 1. 1 Transverse Isotropy
The strain- stress relations for a transversely isotropic
material (a typical example is A TJ -S graphite) are
f1 v' v 0mm E - ET -E (jmm
V' 1 V' 0f nn - ET E"' -r (jnn
= (B-4)
f8f)V v' 1
0 (j()()-E" ET E
0 0 01
f GT (jmn mn
wherein there are five independent elastic constants: E, E', v, v', and G'.
The plane of isotropy for the material in Eq. (B-4) is the m-() plane. For
ATJ-S graphite, the plane of isotropy is "with the grain." Thus, perpendicular
to the plane of isotropy is "across the grain. "
The elastic constants in Eq. (B-4) are further defined as:
E =E' =
Young's modulus in the plane of isotropy
Young's modulus perpendicular to the plane of
isotropy
Vf mm
for a (all other stresses zero)= --- a() =f()()
V =f()()
for a = a (all other stresses zero)-f-- mmmf
v' mmfor a (j (all other stresses zero)= --- =
t nnn
v' = - f()() for a = a (all other stresses zero)f nn
n
G' = shear modulus for planes normal to the plane of
isotropy
B-4
Note that the definition of v' requires that the stress be applied perpendicularly
to the plane of isotropy and the lateral strains be measured in the plane of
isotropy. If the stress were applied in the plane of isotropy and the lateral
strains measured perpendicular to the plane of isotropy, a different Poisson's
ratio, vIr, would govern the behavior where
v" = v' ~E' (B- 5)
Thus, the terminology "with and across grain Poisson's ratios" is inadequate
since the direction of loading is not specified. Note that v" is not an
independent constant but is defined in the terms of the independent constants
by use of the reciprocal relations
( B-6)
Finally, for a transversely isotropic material, the terms in Eq. (B-l)
are identified by comparison with Eqs. (B-4) and (B-5) as
E = E, E = E' EO = Em n,
V = ,E v = v, v'lJ EI, vnO =mn mO
G = G'mn
B.l.2 Isotropy
In the context of Eq. (B-l), the strain-stress relations for
an isotropic material are
1 v v 0E -E -E
v 1 v-E E -y 0
Emm
Enn
=
E 00
Emn o o
B-5
1E
o
o
2(1 + V)E
(Jmm
(Jnn
(B-7)
(Jmn
whe rein there are only two independent elas tic cons tants. E and V.
The terms in Eq. (B-1) are identified by comparison with
Eq. (B-7) as
E = E = EO = Em n
vmn = Vme = lJ ne = V
G = E/[2(1 + lJ )]mn
B. 1. 3 Engineering Use of Material Models
Despite the fact that a certain number of elastic constants are
required to properly describe a particular material model, often the available
material data do not include all the constants. In such cases, it is necessary
to make some engineering approximations in order to accomplish an analysis.
However, the validity of such approximations is always open to question, and
analyses with such approximations must be clearly labeled as approximate.
The validity of the approximations Can be established in two ways: (I) experi
mental determination of the unknown constants, and (2) parametric numerical
examination of the importance of the unknown constants. Experimental
determination is preferable, but difficult and expensive. Parametric studies,
on the other hand, are of considerable aid to the analyst and to the experi
mentalist as well.
An example of a commonly missing property is G' in
Eq. (B-4). Often it is approximated by
G' = E + E'4(1+V')
( B-3)
or similar relations. For a woven material such as three-dimensional (3-D)
quartz phenolic, it should be anticipated that the shear stiffness is considerably
lower than that given by Eq. (B-3). Lenoe, Oplinger, and Serpico (Ref. 17)
compared the use of a relation like Eq. (B-3) with experimentally determined
values (lower than Eq. B-3) in a buckling analysis and found discrepancies of
about 30 percent in the buckling loads calculated with the two shear moduli.
Thus, considerable reservation about the validity of an analysis should be
expressed when approximations such as Eq. (B-3) are used.
B-6
B. 2 OR THOTROPIC PLASTIC BEHA VIOR
In this section, an attempt is made to define the behavior of an ortho
tropic material which has a bilinear effective stress-effective strain curve.
This work should not be construed as being anything more than a rough
approximation since practical orthotropic materials mayor may not be
representable by the effective stress function used here. Accordingly, this
theory should be applied with considerable reservation when orthotropic
materials are treated. However, for isotropic materials, the effective
stress function is well-founded (it is related to the von Mises yield criterion)
and can be used with confidence as is demonstrated in Appendix H.
A method of successive approximations is used to solve for displace
ments, stresses, and strains in bodies with nonlinear material properties.
Basically, this method involves the repeated solution of the following equation
for the displacements of the system:
(B-9)
where
[K]. 1 = an estimate of the effective stiffness of the system based1-
on the previous solution (i-l)
= an estimate of the loads acting on the system (since the
thermal loads are a function of the stiffne s s)
= the displacements of the system for the ith approximation.
The load and stiffne s s used in the first approximation (i= 1) are based on the
initial linear material properties. Since deformations are assumed to be
small, the development of the effective stiffness depends only on the estima
tion of an effective stress-strain relationship for each element in the system.
It is apparent that this approach has certain disadvantages. First, the
procedure is not guaranteed to converge. However, experience has indicated
that, for systems where the nonlinear effects are small compared to the initial
linear analysis, the procedure does converge. Second, to obtain a unique
solution, the method is restricted to elastic materials -- in other words,
materials with single-valued stress-strain relationships.
B-7
Apparently, a general stress -strain relationship for an orthotropic
nonlinear tnaterial has not been fortnulated. The specific relationship used
in this cotnputer progratn has little experitnental justification, however, it
does degenerate to the von Mises yield condition in the case of isotropic
tnaterial.
The von Mises yield condition for isotropic tnaterial is given by
(- - )2(J _ (J
2 3(B-IO)
where
yield
(Jl' (J2' and (13 are the principal stresses, and (Jy is the uniaxial
stress. Equation (B-IO) can be rewritten in nortnalized fortn
1 J(- -)2 (- -)2 (- -)2= _1_ ~ _ (J2 + ~ _ (J3 + (J2 _ (J3
~2 (J (J (J (J (J (J'Ie. y y y y y y
(B-ll)
For orthotropic tnaterials, Eq. (B-ll) is tnodified without justification to
11=
f2(B-12)
where and (J 3y are the yield stresses in the principal directions.
Now, for all values of stress, the "nortnalized effective stress" for
orthotropic tnaterials is defined as
1(J =
IZ
B-3
(B-13 )
The normalized effective stress -strain relationship is shown in Figure B-1.
An approximate consideration of strain hardening can be included in the
formulation as shown in the figure.
(J
enen o .... Icc>-en...>>c.:>......... 1.0...Cl...N
-'C2;...oz
______'- '!!- -+e1.0 e j
NORMALIZED EFFECTIVE STRAIN
Figure B -1. Effective Stres s -Strain Relationship
The solution procedure for the nonlinear problem is as follows:
1. After each approximate solution, the normalized effective stres s is
calculated for each element (linear properties are used in the first
approximation).
E-9
(B-14)R.
1
2. The ratio of nonlinear properties to linear properties for each element
for a given approximation i is defined by
a.1
=e.
1
Therefore, the corresponding normalized effective strain is
(R. is set equal to 1 for the first approximation. )1
(B-16)e.
1
=
3. If the strains are as sumed not to change, an estimation of the pIa sticity
ratio for the next approximation is
+ n (e. - 1)1
4. In the next approximation, the linear propertie s for E , E , E e .rn n
and G are multiplied by the ratio in Eq. (B-16) before they aremnused. The Poisson's ratios are changed according to the following
relation:
= 1/2 (1/2 (B-17)
This procedure must be repeated until convergence is obtained. The
displacements, stresses, and strains are printed by the program after each
approximation. The nurnber of approximations required will depend on the
specific problem and must be specified as computer input. Hence, a certain
amount of experience is required in using the nonlinear option. An arbitrary
number of approximations can be requested, but the program will stop when
R itl for 'all elements changes by 0.5 percent or less.
B -10
B.3 OR THOTROPIC LINEAR BEHA VIOR WITH DIFFERENT ELASTICMODULI IN TENSION AND COMPRESSION
Although many materials exhibit orthotropic behavior, a significant
subclass of those materials exhibits a further complexity of behavior, namely,
different orthotropic moduli under tensile and compressive loading.
Ambartsumyan (Ref. 18) has developed a material model (stress-strain
relationship) for such materials, but does not correctly treat the shear
moduli. A new material model (derived by R. M. Jones) is presented in
this section and is incorporated in the SAAS III program.
Because the material properties depend on the stress state and vice
versa, the basic problem is statically indeterminate. However, the indeter
minancy can be resolved by an apparently convergent iterative procedure
consisting of four steps. First, displacement and stress calculations are
performed based on an initial assumption of stress signs which, in turn,
implie s an initial choice of material properties. Second, the appropriate
material properties are selected based on the principal stresses calculated
in the previous step. Third, displacements and stres se s including the new
principal stresses are recalculated. Fourth, steps two and three are
repeated until convergence to the desired accuracy is achieved.
In this procedure, three different coordinate systems are required:
(1) body (r -z) coordinate s, (2) principal material (m-n) coordinates, and
(3) principal stress (p-q) coordinates. All three coordinate systems are
shown in Figure B-2 along with the definition of the angles between the systems.
As will be shown in the following discussion, transformations of stresses and
material properties must be made between the various coordinate systems.
For an axisymmetric body under an axisymmetric loading field, the
following strain-stress relations in principal stress (p-q) coordinates are
B-ll
proposed by the second author for orthotropic materials that exhibit different
moduli in tension and cOlTIpression:
£ spq spq spq spq GP 11 12 13 14 P
£ spq spq spq spqG qq 12 22 23 24
= (B-18)£0 spq spq spq spq
GO13 23 33 34
Y spq spq spq spq 0pq 14 24 34 44
z
q
Figure B-2.
m
....:::...- I-. .l.- r
Relation of Material Orthotropy to PrincipalStress and Body Coordinates
B -12
Note that the principal stress directions do not coincide with the principal
strain directions. The compliances. s~q, are as signed according to the1)
signs and magnitudes of the principal stresses:
if a > 0 spq = S·pqP 11 11 t
if a < 0 spq = spqP 11 llc
if a > 0 spq = spqq 22 22t
if a < 0 spq = spqq 22 22c
if a() > 0 spq = spq33 33t
if a() < 0 spq = spq33 33c
if a > 0 and a > 0 spq = spqp q 12 l2t
(B"19)
if a < 0 and a < 0 spq = spqp q 12 l2c
if a > o and a < 0 spq = k spq + k spqp q 12 ppq l2t qpq l2c
if a < o and a > 0 spq = k spq + k spqP q 12 ppq l2c qpq 12t
if a > o and > 0 spq pqa() = S13tp 13
if a < o and a() < 0 spq = spqP 13 13c
if a > 0 and a() < 0 spq = k spq + k spqp 13 ppt 13t tpt 13c
if a < o and a() > 0 spq = k spq + k tpt sljtp 13 ppt 13c
B-13
if <7 > 0 and <7() > 0 spq = spqq 23 23t
if <7 < 0 and <7() < 0 spq = spqq 23 23c
if <7 > 0 and <7() < 0 spq = k spq + ktqt s~jcq 23 qqt 23t
if <7 < 0 and <7 > 0 spq = k spq + k spqq () 23 qqt 23c tqt 23t
if <7 > 0 spq = spqP 14 14t
if <7 < 0 spq = spqP 14 14c
if <7 > 0 spq = spqq 24 24t
if <7 < 0 spq = spqq 24 24c
(B- 19)
if <7() > 0 spq = spq (Cant. )34 34t
if <7() < 0 spq = spq34 34c
if <7 > 0 and <7q > 0 spq = spqP 44 44t
if <7 < 0 and <7 < 0 spq = spqp q 44 44c
if a > 0 and a < 0 spq = k spq + k spqP q 44 ppq 44t qpq 44c
if a < 0 and a > 0 spq = k spq + k spqP q 44 ppq 44c qpq 44t
B-14
etc. could be chosen as some other
where
k =lapl
k =Iaql
ppq! apl + laql qpq I apl + laq I
k = rpl ktpt =
lao I(B - 20)ppt Iapl + ! aol lap! + laol
klaq !
k tqt =laol
=qqtlaql + Iaol laql + laol
The weighting factors, k ,ppq
function of the principal stresses (full qualification of the form of the weighting
factors awaits definitive experimental work). Note that only two of the three
principal stresses are used to determine each of the cross compliances Sli,
S15' and S~5' Furthermore, a single principal stress is used to determine
each of the cross compliances sti, S~4' and S~4- Although S~4 is not
required in principal stress (p-q) coordinates, it is required to be defined
there so that transformations to any other coordinate system reduce properly
to the results of anisotropic elasticity when the tensile and comprIOssive
moduli are the same. This last point was neglected by Ambartsumyan
(Ref. 18).
The compliances S?qt and S?q in the principal stress (pq) coordi-1J 1JC
nates are related to the compliances S'::tn and S,::n in the principal material1J 1JC
(mn) coordinates by the usual transformations of anisotropic elasticity:
spq mn 4 (2Smn + Smn ) . 2fJ 2fJ + Smn . 4fJ= Slltc cos j3 + 44tc Sln cos 22tc SlnlItc 12tc(B-21)
pq mn + (Smn + mn 2Smn mn . 2 2S12tc = SlZtc S22tc - S44tc) sm/3 cos fJlItc 12tc
B-15
pq Smn 2 + Smn . 2f3Sl3tc = cos f3 23tc SIn13tc
spq mn mn2S~~c) cos
3f3 sinfJ14tc = (S44tc 25 lltc +
mn mn + 2S~~c) sin3iJ cos fJ- (S44tc - 2S22tc
spq mnsin4f3 + (2S~~c mn . 73 2 mn 4
= Slltc + S44tc) sm cos f3 + S22tc cos fJ22tc
spq Smn . 2 mn 2= sm fJ + S23tc cos iJ23tc 13tc
spq mn mn + 2S~~c) sin3fJ cosfJ= (S44tc 2S lltc24tc
(B-21)(Cont. )
Spq =33tc
pq . =S34tc
mnS33tc
pq Inn mn mn Inn Inn . 2 2S44tc = S44tc + 4(Slltc + S22tc - 2S 12tc - S44tc) sm fJ cos iJ
where the subscript t or c is taken as appropriate, and iJ is the angle
between the principal material and the principal stress coordinates.
The compliances in principal material coordinates,
related to the technical constants (direct moduli, Pois son I s
shear moduli) by:
mnSijtc' areratios, and
mnSlltc = liE
mmtc
Sml
n = _ IJ IE = _ IJ IE2tc mntc mmtc nmtc nntc
B-Ib
(B-22)
smn = -v IE = -Vemtc/Eeetcl3tc mete mmtc
smn = liE22tc nntc
smn = -v IE = - v IEee (B-22)23tc netc nntc entc tc(Cont. )
Smn = l/Eeetc33tc
Smn = l/G44tc ITlntc
where V t = -E lEfor a = at and all other stresses are zero.ron n m 1TI
There are apparently seven independent material propertie s in tension in
Eq. (B-22) and the same number in compression. However, the compliances
s:~ and S:~ (l/Gmnt and l/Gmnc respectively) cannot be measured in a
shear test on an orthotropic material with different moduli in tension and com
pression since one principal stress is tension and the other is compression.
Rather, in accordance with a suggestion by Tsai (Ref. 19), the tension
modulus at 45 degrees to the principal material axes, E't' is measured
d Smn. b' d fan 33t 1S 0 ta1ne rom
1 4
(Em~t1 2v )Smn = = E' - + r- - E rnnt (B-23)
33t Gmnt t nnt mmt
A similar relation is us e d to define Smn when E' is known.33c c
In the body (r-z) coordinates, the compliances,
from the sPq according to1J
rzS .. ,
1Jare obtained
(B-24)
where the D matrix is given in Section II, Method of Analysis.
B-17
At this stage, all required compliances have been defined for an
orthotropic material with different moduli in tension and compression.
Thermal stress terms are incorporated in the principal material directions
in an obvious manner in Subroutine MPROP, and further description of the
procedure is given there and in Subroutine STRESS at the appropriate locations.
An example problem is shown in Section H. 4 of Appendix H.
B.4 EFFECT OF PORE PRESSURES
The material in this section has been published separately (Ref. 20)
and is repeated here, for convenience, with slight modification.
B. 4. 1 Introduction
The theory of deformation of porous materials containing a
viscous fluid has been developed by Biot (Ref. 21). This theory is applicable
to a wide variety of practical problems (Refs. 22, 23, 24). The essential
elements of the theory are:
(l) Definition of a stress tensor for porous material
which includes the stress in the pore fluids.
(2) Generalization of Hooke's law to include the
deformational characteristics of the pore fluid.
(3) Addition of Darcy's law to obtain complete solutions
where a coupling occurs between the displacements
of the pore fluid and the solid.
Although Biot's theoretical developments are extensive, there
is a notable deficiency in the availability of methods of analysis that can be
used to obtain solutions for practical problems involving porous media. The
application of the finite element method of analysis to a particular class of
problems involving porous media is discussed in Ref. 20 where it was also
shown that, by making certain reasonable assumptions, SAAS II could be
adapted to this class of problems. The necessary assumptions are described
in the following paragraphs, and derivations presented of the theoretical
B-18
relations. The SAAS III computer program has been given this capability,
it being only necessary to input a point by point description of the pore
pressure field and the effective porosity of the material.
B. 4. 2 Basic Theoretical Eguations
The state of stress in a porous material has been given by
Biot (Ref. 21). The stress tensor is:
a + a axx xy
ayx ayy + a
a azx zy
with the symmetry property
a .. = a ..1) Jl
axz
ayz
a + azz
(B-25 )
(B-26)
where a.. are the forces and tractions transmitted by the solid material1)
across the face of a unit cube of the material.
a = - fp (B-2?)
is the total normal tension force applied to the fluid part of the faces of the
cube, f is the porosity of the material, and p is the pore pressure.
The total stress field of the bulk material satisfies the
equilibrium equations (Ref. 1):
o~ (axx + a) +oaxy
+oaxz + px 0=
oY Oz
oayx+ o (a + a) +
oayz + pY = 0 (B-28)oX Oy yy oz
oazx +oazy + ..£
( °zz + a) + pZ 0ax =OY Oz
B·19
where p is the mass density of the bulk material, and X. Y. Z are the body
forces per unit mass.
It is convenient to introduce a quantity a .. where1J
a.. = a .. + a11 11
and a.. = a..1J 1J
(B-Z9)
If a stress -strain relationship can be found for a.. , then11
stress analyses can be performed for porous media using the same methods
as those employed for solid media. Now, in general, the state of stress in
an elastic porous material is related to strains by a matrix of material
constants (Ref. Zl) as follows:
axx Cll C IZ - - - - - C17 e xx
ayyCZl CZZ
eyy
\
a \e
zz zz\ (B-30)
a = \ eyz
\yz
a \ ezx zx
\
axy\ e
\xy
a C71 - - - - - - - Cn E
where the e .. are elastic strains in the conventional sense and E is the1J
volumetric strain of the pore fluid. This relationship is seen to be a
generalization of Hooke's law.
An important class of problems can be solved by making the
assumption that pore fluid pressure is independent of the deformations of the
solid. With this assumption and the symmetry property,
Ci7 = C 7j = 0 (i, j = 1, 6)
B-ZO
(B-31)
Therefore, the pore stress can be uncoupled frOITI the stress-strain relation
of the solid ITIaterial. It should be noted that conventional testing procedures
will yield the reITIaining C .. for the porous solid as long as area ITIeasureITIents1J
are not ITIodified by deducting pore areas.
With the assUlTIption that the pore pressure is known as a
function of the space coordinates, C77
and I' can be ignored in the stress
analysis probleITI. PresuITIably, the given pore pressures could be obtained
frOITI consideration of Darcy's law and therITIodynaITIic relationships.
The stress-strain relationship can then be written as
where
= [CJ + 1(7\ (B-32)
(7 + (7xx
(7 + (7yy
(7 + (7
la-Izz
(7yz
(7zx
(7xy
and
I'xx
I'yy
I'
Ie Izz
=Eyz
Ezx
Exy
B-21
(B-33)
(B-34)
[c] is a 6 by 6 matrix of constants to be determined by experiment.
(1
(1=ooo
where
(J = - fp
and
f = !lx, y, z)
and
p = pIx, y, z)
which are assumed to be known.
(B-35)
(B-36 )
(B-37)
(B-38)
In Biot's theory, f is the ratio of the area of pore spaces
accessible by fluids to the total area over a plane section through the porous
material. Biot indicates that this ratio can be shown to be equal to the
volumetric porosity. In many cases, and especially in the case of particulate
material, this has not been correlated through experiment. Although it appears
to be true for ductile solids, brittle and particulate materials may require a
special interpretation of f. Therefore, the selection of volumetric porosity
for the factor f should be made with caution.
This approach to the stress analysis of porous media is
consistent with the assumption of small deflections, and is valid when the
porosity and permeability change insignificantly during deformational processes
and when Hooke's law holds for the solid material. It also requires that the
fluid flow characteristics be relatively unaffected by the stress state and
deformation of the solid. Moreover, it is implicitly assumed that deformations
B-22
will not open new pore areas, e. g., opening a closed crack. It is anticipated
that the approach will be valid for the analysis of transpiring pressure vessels
of porous metal materials where the reservoir pressure is held in a steady
state. In addition, problems of diffusing gases can be approached utilizing
these assumptions if the change in pore volume (and, therefore, in pressure,
volume, and temperature of the gas) is assumed to be small.
B-23
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B-24n
APPENDIX C
SOLUTION OF LINEAR EQUATIONS
The equilibrium equations can be written in the following form:
AllX l + AlZXZ + A 13 X3 + .... + AlNXN = B l
(C-la)
AZlX l + AZZXZ + AZ3
X3 + + AZNXN = B
Z(C-lb)
A 3lX l + A3Z
X Z + A33
X3 + .... + A
3NX
N = B3
(C-lc)
or, symbolic ally,
where
[A] = the stiffness matrix
IX I = the unknown displacements
IB) = the applied loads
C-l
(C-lN)
(C-l)
(C-Z)
C.l GA USSLAN ELIMINATION
The first step in the solution of the above set of equations is to solve
Eq. (C-la) for Xl:
Xl = Bl/A ll - (AlZ/All)XZ-(A13/All) X3----(AIN/All)XN
If Eq. (C-Z) is substituted into Eqs. (C-Ib, c, •••• , N), a modified set of
N-I equations is obtained:
I I IB
IAZZXZ + A Z3 X
3 ---------- + AZNXN = Z (C -3«)
I I I I(C-3b)A 3Z XZ + A
33X
3 ---------- + A 3NXN
= B 3
where
(- -)
(C-4a)
Bl = B.i 1
(C-4b)
A similar procedure is used to eliminate Xz from Eq. (C-3), etc. A
general algorithm for the elimination of X can be written asn
X = (Bn -l IAn-I) - L(An~l IAn-l)X. J = n + 1, .... , N (C-5 )
n n nn nJ nn J
A:'.n-l n-l (An~l/An-l) i, . n + 1, .... , N (C-6)= A .. A. =
IJ IJ In nJ nn J
B:' n-l n-l (Bn-l/An-l)i n + 1, .... , N (C-7)= B. A. =1 1 In n nn
C-Z
Equations (C-5), (C-6), and (C-7) can be rewritten in compact form:
x = D LH .X. J = n + 1, .... , Nn n nJ J
n n-l n-lH i, j 1,A .. = A .. A. = n + .... , N
1J 1J In nj
B:'n-l n-l
D i 1,= B. A. = n + .... , N1 1 In n
(C-8)
(C-9)
(C-IO)
where
D = Bn-l/An - ln n nn
Hn-l/ n-l
= A. Anj nJ nn
After the above procedure is applied N -1 times, the original set of equations
is reduced to the single equation
which is solved directly for XN
:
X = BN-I/A N - IN N NN
In terms of the previous notation, this is
(C-il)
The remaining unknowns are determined in reverse order by the repeated
application of Eq. (C-8).
C. 2 SIMPLIFICATION FOR BAND MA TRICES
For the present class of problems, the stiffness matrix occurs in a
"band" form which results in the concentration of the elements of the stiff-
ne s s matrix along the main diagonal. Therefore, the following simplifications
C-3
in the general algoritlun [Eqs. (C -8), (C-9), and (C-IO) ] are possible:
X = D LH .X. j = n + I, .... , n + M - 1 (C-12)n n nJ J
A?n-l n-l
H j I, M I (C-13)= A .. A. 1, = n + .... , n + -IJ IJ m nj
B?n-l n-l
D i I, M I (C_14)= B. A. = n + .... , n + -1 1 m n
where M is the bandwidth of the matrix.
The number of numerical operations can further be reduced by
recognizing that the reduced matrix at any stage of the procedure is sym
metric. Accordingly, since
n nA .. = A ..
Jl 1J
Eq. (C-13) can be replaced by
n-l n-l i = n + l, .... , n + M - 1A? = A .. A. H
1J IJ m nji, + M IJ = .... , n
(C-IS)
The number of numerical operations required for the solution of a
band matrix is proportional to NM2
as compared to N 3 which is required
for the solution of a full matrix. Also, the computer storage required by the
'cand matrix procedure is NM as compared to N2
required by a set of N
·trb·i: 'ary equations.
Furthermore, the complete matrix is not required to be in high speed
storage while the matrix is being reduced. The equations which are not
being operated on can be placed on tape or disk storage and then moved into
the high speed storage area when required.
C-4Q
APPENDIX D
CONVERGENCE OF FINITE ELEMENT RESULTS
This appendix was a part of the SAAS I! documentation. It is repeated
here for convenience with slight modification. The results of this study apply
equally to the SAAS II! computer program.
D.l INTRODUCTION
Tong and Pian (Ref. 25) show that finite element solutions converge to
exact solutions for linear elastic problems as element size decreases. The
only apparent proviso is that the geometrical and loading parameters must be
smooth functions (1. e., well-behaved, differentiable, etc.) of the independent
variables. The error, termed a discretization error, is shown to be of order
A1+ where)... is an element dimension. For comparative purposes, finite
difference methods can be shown to have a discretization error of order >--2due to truncation of the approximating function. Evidently, the finite element
procedure has not been proved to be as rapidly convergent as finite difference
procedures.
The very important practical consideration of round-off error
resulting from the inherent limitations of digital computers is excluded from
the foregoing discussion. It must be recognized that only with large digital
computers is the finite element procedure practical. Thus, the problem of
round-off error cannot be ignored.
The magnitude of round-off error increases with an increasing
number of elements because a larger and larger system of equations must be
formulated and solved. This situation must be contrasted with the
discretization error which diminishes with an increasing number of elements.
D-l
Thus, there is not necessarily any convergence in an absolute sense. That
is, round-off error can dominate the solution and cause it to diverge from the
exact solution. Round-off error, per se, is a phenomenon which is difficult
to isolate and study. Although not generally conclusive, well-defined
computational experiments lead to a practical assessment of the accuracy of
resulting answers.
During the development of SAAS II, certain discrepancies were noted
in the results from the various computers on which the program was used.
(SAAS I had a capacity of 600 elements and was exercised on the IBM 7094
computer. SA AS II and SAAS III have a capacity of 1000 elements and were
exercised on the IBM 360 and the CDC 6600 computers for this study.)
Specifically, oscillatory behavior was noted in the stresses while using the
IBM 360 (single precision results). This had never been observed in
IBM 7094 results. The problem was traced to round-off error, which is
related to the number of significant figures carried in the arithmetic
calculations of each computer. Five to six significant figures are carried
in the IBM 360 (13-14 in double precision), seven to eight in the IBM 7094
(l5-16 in double precision), and fourteen to fifteen in the CDC 6600 (27-28
in double precision). Thus, round-off error is more likely to occur on the
IBM 7094 than on the CDC 6600, and is much more likely to occur on the
IBM 360 than on the CDC 6600 if single precision arithmetic is used in each
computer.
The effect of round-off errors was investigated by performing
meaningful computational experiments on all three of the digital computers
on which SAAS is used. It was found that a divergence of results occurs
when very large systems of equations are solved. The divergence is
evidently related to the number of elements in each direction as well as to
the total number of elements. That is, when divergence occurs with nd
elements in one direction for a one-dimensional problem, divergence will
not occur until the num.ber of elements in one of two directions in the
equivalent two-dimensional problem exceeds nd
.
D-2
Two predominant sources of round-off errors were isolated by
performing computational experiments wherein certain program operations
were carried out in double precision. For the problem studied, the
occurrence of round-off error is strongly associated with the situation where
a single element dimension is very small compared to its nodal point
coordinate values. In this case, the computed element area is in considerable
error and this error is propagated through the solution. Also, round-off
errors accumulate in the Gauss elimination procedure. The use of double
precision arithmetic in these operations results in considerable improvement
in accuracy in the IBM 360 version of the program. The CDC 6600 version
does not require double precision arithmetic.
D.2 ERROR ANALYSIS
In order to study the convergence of solutions with an increasing
number of elements, a Lame' cylinder problem for which the exact solution
is available was idealized by use of SAAS I and SAAS II with a decreasing
element size for a fixed-dimension problem. First, the problem was
simulated by a one-dimensional plane strain model. Second, a two
dimensional model was used where axial displacements on the boundary were
suppressed. The radial displacement of the inner cylinder wall as a function
of finite element size is shown in Figure D-l. Curve A is from the CDC 6600
program, Curve B is from the IBM 7094, and Curve C is from the IBM 360.
Curve D was obtained by using double precision arithmetic to solve the
governing simultaneous equations in the IBM 360 program. Curve E was
obtained for the two-dimensional grid. It may be seen that the CDC 6600
and IBM 7094 give superior results due to the inherently higher accuracy
of the machine s. This occurred in spite of the fact that double precision
arithmetic was used for the Gauss elimination procedure in the IBM 360
version of the program.. In fact, this use of double precision arithmetic
actually made the solution worse because the inaccurate stiffness matrix
was used in an accurate equation-solving routine.
D-3
LEGEND:• DOUBLE PRECISION "SOLVE"
•• SOLUTION NOT OBTAINED
ELEMENT SIZEMESH MACHINE
0.167 in. 0.002 in.
1-D CDC 6600 2.435101 2.455551-D IBM 7094 2.435098 2.44104l-D IBM 360 2.435064 1.995851-D IBM 360' 2.435049 1. 808812-D IBM 360' 2.434969 ••
2.456
2.455
2.454....C> 2 453-x.!E 2.452
.... 2.451~
ex:....ac 2.450::l(I)
ac 2.449....zi!:.... 2.448Q
.... 2.447z....:IE.... 2.446~ex:.....a.. 2.445(I)
Cl
2.444
A (CDC 6600) "EXACT" SOLUTION
2. 443~~"'!!!"!!"!!~!!'!""~!"'!"~!!"!!!"!"~~~o 0.01 0.020.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10FINITE ELEMENT SIZE - in.
Figure D-l. SAAS I Convergence of Solutions - Lame Cylinder
D-4
The divergence of results is apparently more a function of the element
size than the number of elements. An element size of 0.0715 inch corresponds
to 7 elements in the I-D mesh and 49 elements in the 2 -D mesh. Although
solution of the 2-D problem requires far more arithmetic operations, the
solution error is about the same (0.0046 inch compared to 0.0043 inch). This
may be due to the absence of variations in the unknown parameters
(displacements) in one direction, it may be attributed to the particular way
in which the equations are set up, it may be due to the way in which errors
accumulate in the Gauss elimination procedure, or it may be indicative of
the fact that the discretization error is independent of the number of elements.
The closeness of solutions for a relatively coarse mesh (0. 167 inch) suggests
that the observed error is due entirely to discretization.
Since Curve A is apparently converging for all element sizes attempted,
it is reasonable to conclude that Curve A represents solutions in which round
off error is unimportant compared to discretization error. It is, of course,
expected that Curve A would eventually diverge in the same manner as
Curves B to E. Round -off errors become relatively important for element
sizes of about 2 percent of the cylinder thickness for problems on the
IBM 7094 and of about 8 percent for the IBM 360. Element sizes smaller
than 0.2 percent are evidently feasible on the CDC 6600 for this particular
problem.
Observation of the errors at a mesh size of 0.002 inch leads to the
conclusion that round-off errors can cause serious errors in the solution.
It would be better to use a coarser mesh where the primary error is due to
discretization.
D.3 METHODS OF IMPROVING ACCURACY
The integration procedure in SAAS I was found to contribute signifi
cantly to discretization errors. The program subroutine used a three-point
numerical integration scheme. Dr. Frank Weiler developed a subroutine in
D-5
which a five -point numerical integration scheme is used and also a subroutine
in which an exact integration scheme is used. Results of the three -point and
five -point schemes are illustrated in Figure D-Z. Curve A, which is redrawn
from Figure D-l, represents a Lame cylinder solution for varying element
sizes and was obtained by use of the old (three -point) method. Curve B was
obtained with the new (five -point) method. The relative benefit in using the
new procedure is readily apparent, as the exact integration procedure offers
very little improvement over the five-point procedure and utilizes over twice
the computer storage. Therefore, the five-point integration method is used
in SAAS II and SAAS III.
2.456
...C) 2.455....x
C 2.454
....~ 2.453CI:....c::::::lc;) 2.452c::....z 2.451!:....C)
.... 2.450z....:2:~ 2.449CI:.....D-c;) 2.448is
"EXACT" SOLUTION
A (old-SAAS I)
B (new-SA AS II)
No
i 2.447 0....- ...·0·...0·2- ....0....04-...-0•.•06----0.....08-..........10. 10
FINITE ELEMENT SIZE - in.Figure D-Z. Comparison of Old and New Integration Schemes
with Respect to Discretization Errors
D-6
A study of round-off errors was accomplished by solving the Lame"
cylinder problem with double precision arithmetic used in locations where it
was expected that round-off errors might be serious. Mr. Leonard Bass of
The Aerospace Corporation made an extensive number of computer runs in
which the use of double precision arithmetic was varied in an attempt to
isolate the locations for which double precision is required. Two distinct
locations in the program appear to be especially prone to the accumulation
of round-off errors. The first is in the Gauss elimination process. This
can be expected and requires little explanation. The other location is the
computation of element area (or volume) in the integration subroutine. When
an element dimension is small compared to the coordinate values of the
element nodal points, the area computation is relatively inaccurate. This
inaccuracy is cumulative throughout the body. These effects are demonstrated
in Figure D-3. Curve A represents a solution of the Lame cylinder problem
for varying element sizes and was obtained without any double precision On
the IBM 360. Curve B represents a solution where the aforementioned
operations were accomplished in double precision. Curve C is redrawn from
Figure D-2 and represents solutions obtained from the CDC 6600. The
differences between Curve C and the others can be attributed to round-off
errors. The difference between Curves A and B is attributed to the use of
double precision arithmetic in the Gauss elimination scheme and in the
elemental area computation.
Due to storage limitations of The Aerospace Corporation IBM 360,
not all computations could be performed with double precision arithmetic.
Thus, the best IBM 360 results (Curve B in Figure D-3) are affected by
round-off error in a different manner from the single precision IBM 360
results (Curve A in Figure D-3). In fact, the round-off error causes the
finite element solution to exceed the exact solution. This situation would
appear to be contradictory to the established theorem that finite element
D-7
2.455
2.453~
CI-x.5. 2.451....
uC101-ICI::::::IenICI: 2.449....~101CI
~
ffi 2.447:::E....uc-'D-~CI 2.445
B
C
A
(SAAS II on IBM 360 with selecti ve O. P. )
"EXACT" SOLUTION
(SAAS II on CDC 6600)
(SAAS II on IBM 360 without O. P.)
2. 443!-__~~__~~_~~__~~_.-.o.!~
o O. 02 O. 04 O. 06 O. 08 O. 10
FINITE ELEMENT SIZE in.
Figure D-3. Illustration of the Use of Double PrecisionArithmetic to Improve Accuracy
D-8
displacements approach the exact displacements from below as the number of
elements increases. The theorem proof is based on the presence of
discretization error only and ignores the effect of round-off error. Thus,
the theorem is not contradicted since the presence of round-off error
invalidates the necessity for the finite element displacements to be less than
the exact displacements.
SAAS III includes the new integration subroutine and utilizes double
precision operations in the aforementioned locations in the IBM 360 version.
It is apparent that these modifications are essential in obtaining reasonable
accuracy from the IBM 360 computer. Double precision arithmetic is not
required in the CDC 6600 version of SAAS III, but is used in the UNIVAC 1108
version.
D.4 CONVERGENCE OF STRESSES IN SAAS III
The SAAS III computer program embodies a faster method of
calculating stresses and strains from the resulting displacement field than
SAAS II offered. This new procedure is discussed in Appendix A, Section A-4.
It will be shown in the ensuing discussion that the new method converges as
well as the old. The same Lame" cylinder problem illustrated in the previous
sections was solved using SAAS Ill, and displacement results were virtually
unchanged, as expected. However, comparison of the stresses revealed
some differences as illustrated in Figure D-4. Although, in this case,
SAAS III results appear to be better, this should not be expected to be true
for all problems. However, since discretization errors should be of the
same order with both methods, it is expected that the new method will prove
to be superior because it is so much faster.
D.5 CONCLUSIONS AND RECOMMENDATIONS
The convergence of solutions with decreasing element size has been
demonstrated for the SAAS III finite element computer program. Two kinds
of errors were identified (discretization and round-off) and their relative
D-9
r = 0.75 (center of cylinder)
U r = 8704 psi }U z = 7000 psi "EXACT"
U t = 14,630 psi
200
100
0U zen ...........~".... '~ .......
~ .........., .......IX " ....Cl " ' ....IX " .... U ZIX -100 , , ........ , , ....en , , , ,en , ,ur ,.... , , ,IX.... , , ....en , , ....
-200 ut- ', ,, , ...., , ,, , ....,, , ,, ,, ,
-300"NEW" STRAIN CALCULATION ' ,, ,1 ELEMENT
- - - "OLD" STRAIN CALCULATION, ,, ,
MAX ERROR = 5% ' ,\ \ •
31 ELEMENTS , ,-400
ELEMENTS ' ,
r\ ,
7 ELEMENTS \
3 ELEMENTS\
N0 -5000
"' 0 . 1 .2 .3 .4 .5'"'"NFINITE ELEMENT SIZE~ in.~
Figure D-4. Convergence of New Stress-Strain Calculations
D-IO
values as a function of element size presented in the context of a particular
problem (Lame' cylinder). A substantial portion of the discretization error
experienced with SAAS I was eliminated by the use of the more accurate
numerical integration procedures in SAAS II and III. Two primary sources
of round-off error were isolated and demonstrated with the use of double
precision arithmetic.
It has been found also that the relative accuracy of one computer
versus another (e. g., CDC 6600 versus IBM 360) is a matter of considerable
importance when solving large complex problems with SAAS III.
It is recommended that:
1. More accurate numerical integration schemes such as
those presented in SAAS III be used with finite element
computer programs.
2. Double precision arithmetic be used in the computation
of elemental areas and in the Gauss elimination process
when SAAS III is exercised on computers with limited
accuracy (such as the IBM 360, which carries only five
to six significant figures) to solve large problems.
3. A coarse mesh be used wherever practical to avoid
the possibility of solution divergence due to round
off error.
D-ll
(\
(This
\page intentionally lei \ blank)
D-l2n
APPENDIX E
COMPUTER PROGRAM OUTPUT
In a computer program as comprehensive as the SAAS Ill, it is easy
to generate enough information to overwhelm the interpretive capabilities of
the user for a considerable period of time. Consequently, much thought was
given to the problem of making the job of interpretation by the user as easy
as possible. There are two types of output: printed and plotted. The
plotted output serves as a quick guide to the more extensive information
printed by the program.
E.l PRINTED OUTPUT
The following information is printed by the program:
1. Input data. (An image of each data card is printed prior to
execution of any data case. Input data are also printed out
and described for each data case as it is executed.)
2. Mesh generation information.
3. Nodal point locations and interpolated temperatures.
4. Element makeup (with respect to nodal points) and
interpolated temperature s.
5. Material properbes.
6. Pressure boundary conditions.
7. Shear boundary conditions.
8. Nodal point displacements.
Preceding page blank
E -1
9. Stresses and mechanical strains (total strain minus free
thermal strain) at the center of each element.
10. An approximate fundamental frequency if desired at the
cost of extra time. (If the mass densities of all the
materials are given, the displacements for the given
load condition are used as an approximate mode shape
in the calculation of a frequency by Rayleigh's procedure.
A considerable amount of engineering judgment must be
used in the interpretation of this frequency. )
E.2 PLOTTED OUTPUT
The following information is optionally plotted by the program:
1. Finite element grid. (Two options are available. In one,
the grid is plotted to a large scale suitable for use as a
working drawing. In the other, the grid is plotted with a
maximum dimension of 10 inches.
2. Contour plots of stresses, strains, and temperatures.
(Contour locations are determined by linear interpolation
between function values at the center of each element.
The contour plots are the same scale as the finite
element grid plots, and are obtained only if the latter
are requested.)
3. Deformed grid. (With appropriate input parameters, a
plot of the finite element mesh deformed by exaggerated
nodal point displacements can be obtained. When this
option is elected, contour plots are drawn on the
resultant deformed shape.)
E-2{J
APPENDIX F
COMPUTER PROGRAM INPUT INSTRUCTIONS
The first step in the computer stress analysis of an axisymmetric
solid is to select a finite element representation of the two-dimensional
cross section of the solid. The definition of positive directions of the R-Z
coordinate system for axisymmetric and plane problems is shown in
Figures 1 and 2 in the main text. The R-Z coordinate system IS right
handed as must be the I-J coordinate system in mesh generation. The
following punched cards define the problem to be solved. Recall that all
integer (I) fields must be right-justified.
Only the last data case can utilize the complete restart feature.
That is, results are written on FORTRAN Unit 10 starting at the same
location so only the last data case results are preserved. However, any
data case can be stopped at the allowed locations. Succeeding data cases
are executed if no data for the stopped case appears in the input data deck
after the stop request. That is, after a stop, the next card to be read must
be the first card of the next data case.
After each data case, a card with END rbF CASE beginning in Column 1
must appear. After the last data case, a card with END rbF DATA must
appear (after END rbF CASE).
F-l
NUMBER OF CASES INPUT
This card is not to be repeated for each successive case.
Format (IS)
Columns 1-5
TITLE CARD
Total number of cases in data deck
This card must be present for each case.
Format (20A4)
Columns 1-80 Title
JOB CONTROL CARD
This card must be present for each case.
Format (12, 13,
Columns 1-2
3-5
6-10
11 -12
13 -15
IS, 12, 13, SIS, 13, 12, 215, F5.0, 215)
Plane s'train/ stress option. (If 1, plane
strain; if 2, plane stress
Start parameter (If 1, start at beginning.
If 2, start with contour plotting. If 2,
a card with READ in Columns 1-4 must be
inserted in the data deck in front of the
next data to be read.)
Stop parameter (If 1, stop after mesh
plotting. If 2, stop before contour
plotting. )
Deformed grid parameter (If 1, plot deformed
grid. )
Plot parameter (If 1, small plot with 10 -inch
maximum dim.ension. If 2, large plot with
10-inch minimum dimension. If 101, small
plot with specified dimensions. If 102, large
plot with specified dimensions. If 101 or 102,
the PLOT SCALE CARD must be included
F-2
16-20
21-25
26-30
31 -35
36-40
41-43
44-45
46-50
51-55
56-60
61-65
66-70
before the MATERIAL PROPERTY INFORMATION.
If the field is blank, no mesh is plotted nor can
there be any contour plots generated. )
Number of nonlinear approximations
Mesh generation parameter (If 1, the mesh is
generated. )
Number of temperature cards (If zero,
temperatures are given on nodal point cards.
If -1, temperatures are given on input tape 14.
If -2, a constant temperature is specified. )
Number of nodal points (1000 maximum)':'
Number of elements (1000 maximum)':'
Number of pore pressure cards (If zero, the
pore pressures are given on nodal point cards.
If -1, pore pressures are given on tape 14.
If -2, a constant pore pressure is specified.)
Number of different materials (6 maximum)
Number of boundary pressure cards
Number of boundary shear cards
Reference temperature
Number of tension-compression approximations
Natural frequency parameter (If 1, compute
natural frequency and input next card with
accelerations and angular velocities. )
':' It is not necessary to specify this value if the mesh is generated.
F-3
ACCELERATION CARD
This card is req\lired only if Columns 66-70 of thq previous ~ard (JOB
CONTROL CARD) is nonzero.
Format (2F10.0)
Columns 1 -10
11 -20
If plane problem, R acceleration. Ifaxisymmetric
problem, angular velocity.
Z acceleration
F-4
MESH GENERATION CONTROL CARD
This card is included only if a mesh is generated and Column 25 of the
JOB CONTROL CARD contains the integer I.
Format (615, 2FlO.0, 215)
Columns 1-5 MAXI, maximum value of I in mesh':' (not to
exceed 25)
6-10
11 -15
16-20
21-25
26-30
31-40
41-50
51-55
56-60
MAXJ, maximum value of J in mesh':' (not to
exceed 100)
Number of line segment cards
Number of BOUNDARY CONDITION CARDS
Number of material block cards
Number of iterations in relaxation technique
(If 0, program automatically sets appropriate
value. )
Polar coordinate parameter I (See Appendix A,
Section A. 1. )
Polar coordinate parameter J (See Appendix A,
Section A. 1. )
I curvature modification (See Appendix A,
Section A. 1. )
J curvature modification (See Appendix A,
Section A. 1. )
Note that Columns 31-60 are left blank for mesh
generation in noncircular regions.
>:< Note that because of the limitation to 1000 nodal points the product ofMAXI and MAXJ cannot exceed 1000 unless the I-J grid is nonrectangular.
F-5
LINE SEGMENT CARDS
These cards are included only if a mesh is generated.
The order of LINE SEGMENT CARDS is immaterial.
The number of cards must agree with the numbers m Columns 11-15 of
the MESH GENERATION CONTROL CARD.
Format (3(213 • 2F8. 3), 15)
Columns 1-3 I coordinate of 1st point
4-6 J coordinate of 1st point
7 -14 R coordinate of 1st point
15 -22 Z coordinate of 1st point
23-25 I coordinate of 2nd point
26-28 J coordinate of 2nd point
29-36 R coordinate of 2nd point
37 -44 Z coordinate of 2nd point
45-47 I coordinate of 3 rd point
48-50 J coordinate of 3rd point
51-58 R coordinate of 3rd point
59-66 Z coordinate of 3rd point
67 -71 Line segment type parameter
If the number in Column 71 is
o point (input only 1st point)
1 straight line (input only 1st and 2nd points)
2 straight line as an internal diagonal (input
only 1stand 2nd points)
3 circular arc specified by 3 points with the
1st and 3rd points at the ends of the arc. The
2nd point on the arc can be anywhere between
the 1st and 3rd points.
4 circular arc specified by 1st and 2nd points
at the ends of the arc with the coordinates of
the center of the arc given as the 3rd point
(delete I and J)
F-6
BOUNDARY CONDITION CARDS
These cards are included only if a mesh is generated.
Each card assigns a particular boundary condition to a block of elements
bounded by 11, 12, Jl, J2. For a line, 11 =12 or Jl =J2. For a point,
11 = 12 and J 1 = J2.
The number of cards must agree with the numbers in Columns 16-20 of the
MESH GENERATION CONTROL CARD.
Maximum
Minimum11-15
16-20
Format (415, 3F10.0)
Columns 1 -5 Minimum 1
6-10 Maximum 1
J
J
21-30
31-40
41-50
Boundary condition code ~:~
Radial boundary condition,
Axial boundary condition,
XR
XZ
If the number in Columns 21-30 is
O.
1.
2.
3.
!XRis the specified R -load and
XZ is the specified Z-load
!XRis the specified R -displacement and
XZ is the specified Z -load
!XRis the specified R-load and
XZ is the specified Z -displacement
!XRis the specified R-displacement and
XZ is the specified Z -displacement
All loads are total forces acting on a one-radian segment.
':' See Appendix A, Section A. 2 for instructions to input skew boundaryconditions.
F-7
MATERIAL BLOCK ASSIGNMENT
These cards are included only if a mesh is generated.
Each card assigns a material definition number to a block of element,
defined by the I, J coordinates.
The number of cards must agree with the number in Columns 21 -25 of the
MESH GENERATION CONTROL CARD.
Format (SIS, FlO.O)
Columns 1-5 Material definition number (1 through 6)
6 -10 Minimum I
11-15 Maximum I
16-20 Minimum J
21-25 Maximum J
26-3.5 0, material principaL property inclination :1.nglc
F-8
NCASE
TEMPERATURE FIELD INFORMATION
These cards are not required if Columns 26-30 of the JOB CONTROL CARD
are blank.
If the temperature field is given on the cards (i. e .. a number greater than 1
appears in Columns 26-30 of the JOB CONTROL CARD), one card must be
supplied for each point for which a temperature is specified.
Format (3Fl0.0)
Columns 1-10 R
11 -20 Z
21-30 Temperature
If the temperature field is given on tape (i. e •• -1 in Columns 26-30 of the
JOB CONTROL CARD), a single card is used to indicate the file record
desired.
Format (IS)
Columns 1-5
If a constant temperature field is specified (i. e •• -2 in Columns 26-30 of
the JOB CONTROL CARD), the value is given on a single card.
Format (FlO. 0)
Columns 1-10 Temperature
F-9
.~
PORE PRESSURE CARDS
These cards are not required if Columns 41-43 of the JOB CONTROL CARD
are blank.
If the pore pressure field is given on the cards (i. e., a number greater than 1
appears in Columns 41-43 of the JOB CONTROL CARD), one card must be
supplied for each point for which a pore pressure is specified.
Format (3FI0.0)
Columns 1-lOR
11-20
21 -30
zPres sure
If the pressure field is given on tape (i. e., -1 in Columns 41-43 of the JOB
CONTROL CARD), a single card is used to indicate the file record desired.
Format (15)
Columns 1-5 NCASE
If a constant pore pressure field is specified (i. e., -2 in Columns 41-43 of
the JOB CONTROL CARD), the value is given on a single card.
Format (FlO. 0)
Columns 1-10 Pressure
F-IO
NODAL POINT CARDS
One card is required for each nodal point in numerical sequence. These
cards are not included if a mesh is generated.
Format (15, 7 FlO. 0)
Columns 1-5 Nodal point number
6-15 Boundary condition code ,;,
16-25 R -coordinate
26-35 Z-coordinate
36-45 XR
46-55 XZ
56-65 Temperature (if not given on temperature cards
or on tape)
66-75 Pore Pressure (if not given on cards or on tape)
If the number in Columns 6-15 is
O.
I.
2.
3.
tR is the specified R -load and
XZ is the specified Z -load
!XRIS the specified R -displacement and
XZ is the specified Z-load
!XRis the specified R-Ioad and
XZ is the specified Z -displacement
\XRis the specified R -displacement and
XZ IS the specified Z -displacement
All loads are total forces acting on a one-radian segment.
If NODAL POINT CARDS are omitted, nodal points will be generated at equal
intervals along a straight line between the defined nodal points. The
boundary condition code is then O. along with XR and XZ . The
temperature s are linearly interpolated from the defined temperature s.
,;, See Appendix A, Section A. 3 for instructions to input skew boundaryconditions.
F-ll
ELEMENT CARDS
These cards are not included if a mesh is generated.
One card is required for each element in numerical sequence.
Material identification
Ci, material principal property
angle
Format (615, FIO.O)
Columns 1-5 Element number
6 -IONodal point 1
J
111-15 Nodal point
16-20 Nodal point KLJ
21 -2 5 Nodal point
26-30
31-40
1.
2.
Order nodal points counterclockwise (see Figure 1 ofmain text) around element.
Maximum difference betweennodal point numbers mustbe less than 25.
inclination
If ELEMENT CARDS are omitted, the program generates the omitted elements
by incrementing by 1 the preceding I, J, K, and L. The material
identification code for the generated elements is the value specified on the
first nongenerated card. The last ELEMENT CARD must be supplied.
Triangular elements are specified by repeating the last nodal point number,
i. e., I, J, K, K.
F-12
PLOT SCALE CARD
This card is not included if plots with a 10 -inch maximum or minimum
dimension are elected and/or if a deformed grid plot is not desired. If
101 or 102 appears in Columns 11-15 of the JOB CONTROL CARD, the
following information must be supplied to describe the scale of the plots:
Format (4FlO.0)
Columns 1-10 Radial location of plot orIgIn
11-20 Axial location of plot origin
21-30 Plot scale (number of units per inch of
plot paper)
31 -40 Factor by which displacements will be
multiplied for purposes of plotting nodal
point deflections to an exaggerated scale
on the element plot
F-13
MATERIAL PROPER TY INFORMATION
The following group of cards must be specified for each of a maximum of
six materials.
a. MATERIAL IDENTIFICATION CARD
Format (2.I5, FIO.O, 15)
11 -20
21 -30
31-40
41-45
Columns 1-5
6-10
Mate rial identific ation num be l'
Number of temperatures for which properties
are given (12 maximum)
Mass density of material (if required)
Thermal expansion parameter (If 1, free
thermal expansions are on the material
property cards. Otherwise, coefficients of
thermal expansion are on the material
property cards.)
Material effective porosity
Isotropy parameter, ISO (If 0, all moduli must
be input. If 1, material i$ transversely isotropic
and E Q, VMN
need not be input. Also, GMN
is
input in place of E~. If 2, material is isotropic
and only EM' VMN , I;tM' and aM need be input. )
b. MATERIAL PROPERTY CARDS
Three cards are required for each temperature.
TENSILE PROPERTIES CARD Format (8FlO.O)
Columns 1-10
11 -20
21-30
31-40
41-50
Temperature
Modulus of elasticity, E MtModulus of elasticity, E
NtModulus of elasticity, E QtPoisson's ratio, V
MNt
F-14
51-60
61-70
71-80
(Note that vMNt =
are zero.)
Poisson's ratio, VMGt
Pois son's ratio, VNGt
Modulus of elasticity at 45 degree s to MN
coordinate system':' E~ (If ISO = I, input
shear modulus GMNt
• )
EN- EM for aM = at and all other stresses
(Note that E~ can be calculated from
+ + IE
Nt
where GMNt
is the shear modulus in an all-tension stress field. )
COMPRESSIVE PROPERTIES CARD
Format (lOX, 7FIO.0)
Columns II -20 Modulus of elasticity, EMc
21-30 Modulus of elasticity, ENc
31-40 Modulus of elasticity, EGc
41-50 Poisson's ratio, vMNc
51-60 Poisson's ratio, vMGc61-70 Poisson's ratio, v
NGc71-80 Modulus of elasticity at 45 degrees to MN, ,
coordinate system E where VMN
and Ec c c
defined in a manner similar to VMNt and
(If ISO = 1, input shear modulus GMN c' )
,;, The relative location of the M-N coordinate system to the R -2 bodycoordinate system for the angle ct input on the MATERIAL BLOCKASSIGNMENT CARD or the ELEMENT CARDS is described III
Figure F-l.
F-15
THERMAL AND YIELD PROPERTIES CARD
Format (lOX, 7FIO. 0)
Columns 11-20
21-30
31 -40
41-50
51-60
61-70
71-80
N
No:;;~
Fre e the rmal s tra in, aM T, or
Coefficient of thermal expansion, aM
Free thermal strain, aNT, or
Coefficient of thermal expansion, aN
Free thermal strain, a g T, or
Coefficient of thermal expans ion, a g
Yield stress, JM
Yield stress, aN
Yield stress, ag
Ratio of effective plastic to elastic
modulus, PEMR
z
M
... ...... R
Figure F-l. Orientation of Principal Material (MN) AxesRelative to Body (RZ) Axes
F-16
BOUNDARY PRESSURE CARDS
The numbe r of cards must corre spond to the value of NUMPC input in
Columns 46-50 of the JOB CONTROL CARD. If the number is positive, the
pressure input corresponds to the element bounded by nodal points I and J.
The positive sense of pressure is shown in Figure F-2a. Surface normal
tensile force is input as a negative pressure.
Format (215, FIO.O)
Columns 1-5 Nodal point I
6-10 Nodal point J
1 I -20 Normal pressure
If NUMPC is negative, pressures are linearly interpolated between end points
identified by the two-dimensional (mesh generation is required) nodal point
numbering system. One card is required for each part of the boundary so
designated. The positive sense of pressure is shown in Figure F-2b which
also illustrates the meaning of the input quantities.
Format (415, 2FIO.0)
Columns 1 -5 II
6-10 Jl
11 -15 12
16-20 J2
21 -30 PI
31-40 P2
F-17
z
12. J2---4--,
P2
NoN
'"Ml=.l
(a)
NORMALPRESSU R.-=E:..-,....o...
11, J 1
P1 (b) R
Figure F-2. Boundary Pressure Sign Convention
F-18
BOUNDAR Y SHEAR CARDS
The number of cards must correspond to the value of NUMPC input in
Columns 51-55 of the JOB CONTROL CARD. If the number is positive, the
shear input corresponds to the element bounded by nodal points I and J
The positive sense of shear is shown in Figure F-3a.
Format (215, F10.0)
Columns 1 -5 Nodal point I
6-10 Nodal point J
11-20 Surface shear
If NUMPC is negative, shears are linearly interpolated between end points
identified by the two-dimensional numbering system. One card is required
for each part of the boundary so designated. The positive sense of shear is
shown in Figure F-3b which also illustrates the meaning of the input
quantities.
Format
Columns
(415, 2FlO. OJ
1-5 Il
6-10 Jl
11-15 12
16-20 J2
21-30 Sl
31-40 S2
F-19
z
12. J2 S2---
'- t----- J '-
----- ---------- ----\ \
\ \\ ~ACE \
\ SHEAR 11 , J1 \
Sl
(a) (b)N0 R'"'"t:lN
:::
Figure F-3. Boundary Shear Sign Convention
F-20
CONTOUR PLOTTING CARDS
a. PLOT CONTROL CARD
This card must be included if stop parameter (Columns 3 -5 on JOB
CONTROL CARD) equals 0 or 1. It can be blank if no contour
plots are desired.
Format (16)
Columns 1-6 Number of plots
b. PLOT CARD
One card is required for each contour plot.
Format (216, 1OF6. 0)
Columns 1-6 Plot code (see below)
7-12 Number of contours
13 -18
19 -24
25 -30
31 -36
37 -42
43 -48
49-54
55-60
61-66
67-72
Value of each contour (10 maximum)
The following plot code numbers (Columns 1-6) specify the functions to
be plotted:
1 R stress 1 a R2 Z stress, a Z3 9 stress, a g
4 R-Z stress, T RZ
F-21
5 Maximum principal stress in R-Z plane
6 Minimum principal stress ifl R-Z plane
7 Angle to maximunl principal stress
8 M stress, aM
9 N stress, aN
10 M-N stress, TMN
I I R strain, ER
12 Z strain, EZ
13 Q strain, EQ
14 R -Z strain, YRZIS Maximum principal strain in R-Z plane
16 Minimum principal strain in R-Z plane
17 Angle to maximum principal strain
18 M strain, EM
19 N strain, EN
20 M-N strain, YMN21 Temperature, T
22 Pore pres sure, P
If the plot code is a negative integer, the first plotted contour for the
lowest and highest contour value is labeled with the contour value.
Additional contours of the same value are not labeled. The contour
values need not be input in any special order as they are arranged in
ascending order in Subroutine CONTR.
F-ZZ
END OF CASE
This card is the last card for each case and must always be included for
each case.
Format
Columns
END OF DATA
(20 A4)
1 -1 1 END OF CASE
This card is the last card in the data deck and must always be included
as such.
Format
Columns
(20 A4)
1-11 END OF DATA
F-23
(This page intentionally left blank)
F-24Q
APPENDIX G
FOR TRAN IV COMPUTER PROGRAM
The SAAS III program has been made operational on three "third
generation" computers, the IBM 360/65, the UNIVAC 1108, and the CDC
6600. Most of the development work on the program was performed on the
IBM 360/65 at The Aerospace Corporation, San Bernardino Operations.
An alternative version was simultaneously made operational on The
Aerospace Corporation, El Segundo Operations' CDC 6600. More recently,
a third version was made operational on the Southern Methodist University /
Alpha Systems, Inc. UNIVAC 1108. The IBM 360/65 program listing is given
in this appendix along with those changes required to make the program
operational On the UNIVAC 1108 and the CDC 6600. First, however, the
FORTRAN auxiliary units are described as are the functions of each of the
subroutines. Finally, The Aerospace Corporation plotting routine PLT 360
is described to aid in conversion of the program to other computer systems.
G.l DESCRIPTION OF FORTRAN AUXILIARY UNITS
Six FORTRAN auxiliary units, 1, 2, 3, 9, 10, and 14, are used in
addition to the standard input and output files (Units 5 and 6 respectively)
as well as about 216 10 K bytes of IBM 360 core storage, 1428
K words of
CDC 6600 core storage, and about 50 10 K words of UNIVAC 1108 core
storage. The auxilary units can be tapes, disks, or drums.
Unit 1 is a scratch unit used for storage of the reduced simultaneous
equations in SOLVE and, after that storage is no longer needed, for storage
of arrays SIG and EPS as well as element center coordinates in CONTR. Unit
2 is a scratch unit used for storage of the stiffness matrix as generated in
G-l
STIF. Unit 3 is a scratch unit used to transfer the principal stresses from
STRESS to MPROP. Unit 9 is a scratch unit used as an input file after all
data cards have been written on Unit 9 and on the output file. Unit lOis a
"restart" or saved unit provided for the optional feature of restarting the
program as described in Appendix A, Section A. 5. Unit lOis also used as
a scratch unit to transfer stresses and strains from STRESS to CONTR. Unit
14 is an input unit for temperature information in TEMI (which may also be
the output unit for a thermodynamics program used to predict temperatures
in the body for which the thermal stresses are desired). On the UNIVAC
1108, auxiliary Units .. and 31 are used for plotting and encode/decode of
numbers respectively. Unit 4 is assigned in MAIN, and Unit 31 is called in
BDF.
G.2 FUNCTIONS OF SUBROUTINES
The functions of the subroutines which comprise SAAS III are as follows:
1. MAIN Program
The input data are read and written, certain parameters are initialized,
and Subroutines DATA, MESH, PNTIN, FLDIN, PLTM, MATLP,
PRESIN, STIFF, SOLV, STRESS, and CONTR are called.
2. Subroutine DATA
All data case card images are read, written on the output file, and
written on FORTRAN Unit 9 for access by the calling routine.
3. Subroutine REST
Certain blocks of COMMON are written on, or read from, FORTRAN
auxiliary Unit 10.
4. Subroutine MESH
The input data for mesh generation are read and used to define the
mesh in two dimensions. Subroutines MNIMX, ANGLE, CIRCLE,
and POINTS are called.
5. Subroutines MNIMX, ANGLE, CIRCLE
These subroutines perform minor functions as noted in the comment
cards.
6. Subroutine POINTS
The two-dimensional mesh is transformed into a one-dimensional
mesh by use of function NODE.
7. Subroutine PNTIN
This subroutine enables the user to input nodal point and element data
without two-dimensional mesh generation. It is consistent with the
original SAAS I program.
G-3
8. Subroutine FLDIN
This subroutine inputs temperature and pressure field data in the form
of an arbitrary set of point values. A two-dimensional linear inter
polation subroutine (Subroutine TEMP) transfers field values to the nodal
point set.
9. Subroutine TEMI
FORTRAN auxiliary Unit 14 is read to obtain temperatures at arbitrary
Rand Z coordinates.
10. Subroutine TEM2
A constant temperature is read and assigned to all nodal points.
11. Subroutine TEMP
Temperatures at arbitrary Rand Z coordinates are interpolated to
nodal point values.
12. Subroutine MA T LP
The material properties are read from FORTRAN Unit 9 and written on
the output file.
13. Subroutine PRESIN
This subroutine inputs pressure or shear boundary conditions. It
calls Subroutine PBNDR Y.
14. Subroutine PBNDRY
This subroutine converts boundary pressures and shears to equivalent
nodal point forces and stores the results in the boundary condition
vectors.
15. Subroutine STIFF
Subroutine QUAD is called. The quadrilateral stiffness matrices are
combined to form the complete stiffness matrix in blocks. Subroutine
MODIFY is called prior to writing the blocks on FORT RAN auxiliary
Unit 2.
G-4
16. Subroutine QUAD
The 10 by 10 quadrilateral stiffness matrix is formed for the nth
element by calling Subroutine TRISTF four times. Subroutine MPROP
is called.
17. Subroutine MPROP
This subroutine finds the stress-strain relationship in body coordinates.
Subroutines SYMINV and ROTATE are called.
18. Subroutine ROTATE
The transformation matrix is calculated for a rotation.
19. Subroutine TRISTF
The 6 by 6 stiffness matrix for a triangular cross-section ring element
is formed. Subroutine INTER is called.
20. Subroutine INTER
Numerical integration is performed over the triangular ring elements.
The contribution of this subroutine by Dr. Frank Weiler while working
for The Aerospace Corporation is gratefully acknowledged.
21. Subroutine MODIFY (NEQ, N, U)
The stiffness matrix A and load matrix B are modified for a specified
displacement U at Eq. No. N.
22. Subroutine SYMINV (A, NMAX)
The symmetric matrix A (NMAXxNMAX) is inverted.
23. Subroutine SOLV
The stiffness and load matrices are read from FORTRAN auxiliary Unit
2 and solved for the nodal point displacements which are stored in the B
arra y. Note that, for problems with more than 50 nodal points, the B
array will overflow into the A array, but the storage is available at this
time.
G-5
24. Subroutine STRESS
The stresses and mechanical strains are calculated output and written
on FORTRAN auxiliary Unit 10. Subroutines QUAD, MPROP, SYMINV,
and REST are called.
25. Subroutine PLTM
The mesh is plotted with a maximum dimension of 10 inches if IPLOT = I,
or with a minimum size of 10 inches if IPLOT = 2.
26. Subroutine CONTR
The stresses and strains are transferred from FORTRAN auxiliary
Unit 10 to Unit 1. Subroutine REST is called. The plotting of stress,
strain, and temperature contours is controlled from this subroutine.
Subroutines DRAW, BDF, and PLT are called in addition to function
NODE.
27. Subroutine DRAW
DRAW is a standard Aerospace Corporation subroutine for contour
drawing.
28. Subroutine BDF
A number is converted to a hollerith array that can be used as contour
plot annotation. It has IBM 360, CDC 6600, and UNIVAC 1108 versions.
29. Subroutine PLT
PLT is a standard Aerospace Corporation plotting subroutine. Its use
is described in Section G. 4.
G-6
G.3 IBM 360 FORTRAN IV COMPUTER PROGRAM LISTING
G-7
c••••**•••* MAIN 1C SAAS III. FINITE ELEMENT STRESS ANALYSIS CFAXISYMMETRIC AND PLANEMAIN 2C SOLIDS WITH DIFFERENT ORTHOTROPIC. TEMPERATURE-DEPENDENT MATERIAL MAIN 3C PROPERTIES IN TENSION AND COMPRESSION INCLUDING THE EFFECTS OF MAIN 4C INTERNAL PORE FLUID PRESSURES AND THERMAL STRESSES. MAIN 5C 8Y JAMES G. CROSE AND ROBERT M. JONES MAIN 6C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAIN 1
COMMON/BASIC/NUMNP.NUMEL.NUMPC.NUMSC.ACELl.ANGYEL.TREF.YOL.IFREQ MAIN BCOMMGN/MATP/ROI61.AOFTSI61.EI12.22.61.EEI211.POROTYI61 MAIN 9COMMCN/NPDATA/R1I0001.CODEII0001.XR1I0001.l1I00DI.XlIIDDOI.TIIOOOIMAIN 10COMMON/ELDATA/IXII000.51.EPRII0001.ALPHAllDOOI.PST1I0001 MAIN 11DOUBLE PRECISION CRl.Xl.RR.ll.S.RRR.lll MAIN 12COMMCN/ARG/RRRI51.lll151.RRI41.ll141.S110.101.CRlI4.41.XIIIOI. MAIN 13
I P1I01.TTI41.HI6.101.HHI6.101.ANGLEC41.SIGII01.EPSII01.N MAIN 14DOUBLE PRECISION X.Y.TEM MAIN 15COMMON/SOLYE/XC17001.YI17001.TEMI17DOI.~UMTC.MBAND MAIN 16COMMON/PTT/IPLOT.TITLEI201.RMIN.lMIN.DELP.TILT.FACT.IDEF MAIN 17COMMON/TD/IMIN1I001.IMAXIIOOI.JMINI251.JMAXI251.MAXI.MAXJ.NMTL.NBCMAIN IBCOMMON/CONYRG/IPDONE,ITCDON.NNLA.HTCA.NTITER.OLOSIGI41 MAIN 19COMMCH/PLANE/NPP MAIN 20
C••••••••••••••••••••**•••••••••••••••••••••••••••••••••••••••••••••••••MAIN 21C REAC AND WRITE INPUT DATA CARD IMAGES fOR ALL CASES MAIN 22C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••HAIN 23
CALL OATA1NCASESl MAIN 24NCASE=O MA I N 25
100 HCASE=NCASE+l MAIN 26REWIND 10 MAIN 27REiolND 14 MAIN 28
C•••••••••••** ••••••••••••••••••••••••••••••••••••••••••••••~••••••••••*MAIN 29C READ AND WRITE CCNTROL INFORMATION MAIN 30C•••••••••••••••* MAIN 31
READI9.10DOI TITLE.NPP.ISTART.ISTOP.IDEF.IPLOT.~NLA.IMESH.NUMTC.MAIN 32INUMNP.NUMEL.NPORPR.NU~MAT.NUMPC.NUMSC.TREf.NTCA.lfREQ MAIN 33
WRITE16.20001 TITLE.ISTART.ISTOP.IDEF.IPLCT.NNLA.IMESH.NUMTC. MAIN 34INUMNP.NUMEL.NPORPR.NUMMAT.NUMPC.NUMSC.TREf MAIN 35IFINPP.E,.11 WRITE16.20041 MAIN 36IFC~PP.E'.21 WRITEC6,200S1 MAIN 31.RITEI6.20301 NTCA MAIN 3BANGYEL=O. MAIN 39ACELl=O. MAIN 40IFllfREQ.EQ.OI GO TO 150 MAIN 41REA019.10011 ANGYEL,ACELl MAIN 42WRITE16.20011 MAIN 43IfINPP.EQ.OI WPITEC6.20021 ANGYEL.ACELl MAIN ~4
IFINPP.EQ.l.OR.NPP.EQ.21 WRITE16.20031 ANGYEL.ACELl MAIN 45150 GO TO 120e, 0;001.1 START MAIN ~6
c••**••**.***•••••***••***.****.**•••****•••••••••**••**.******•••*••••*HAIN 41C GENERATE FINITE ELEMENT MESH MAIN ~B
c••••••••• **.*****.*.***•••••••••••*••******.**•••***.*••••••••***•••••*MAIN 49200 IfIIMESH.NE.OI CALL MESH MAIN 50
C MAIN 51C INIT IALllE MAIN 52C MAIN 53
DC 250 II=I.NUMNP MAIN 54PSTlIlI = O. MAIN 55
250 TIl Il=O. MAIN 56C•••••**$.*****.****.***•••*.**••*********•••****••*******••••••**.*•••*MAIN 57C READ AND WRITE NODAL POINT AND ELEMENT tATA MAIN 58c*.****.*.***********••*••****•••***•••***.*•••••***•••*••••••••••••*•••MAIN 59
IFIIMESH.EQ.OI CALL PNTIN MAIN 60c••••••••••••••**•••••••••••••••••***•••••••••**.*.** •••••••••***•••*•••MAIN 61C READ AND WRITE TEMPERATURE OATA MAIN 62c•••••••••******•••••• *••••••••**•••**.****••****••••••• ***••* ••••••*•••MAIN 63
IFINUMTC.NE.OI CALL fLDINIT.R.l.NUMTC.NUM~P.IMESH.NU~ELI MAIN 64c••*•••*••••*•••••*•••••**•••••••••••*••••*•••••••••••*****•••*•••••**••HAIN 65C READ AND WRITE PORE PRESSURE DATA MAIN 66c••*.*••••**•••••••••**••*••**••********•••••**.*.*****.***••**••••••*••MAIN 61
IFINPORPR.NE.OI CALL FLOINIPST.R,I.NPORPR.NUMNP.IMESH.NUHELI MAIN 68C••••••••••*•••••••••••••**•••**••••*•••••••***•••••*****••••••***••••*.HAIN 69C OUTPUT ELEMENT DATA MAIN 70
G-8
c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAIN 71MPRI~T=O MAIN 12DO 350 N=I.NUMEL MAIN 13IFIMPRINT.NE.OlGO TO 300 MAIN 14wRITE16.20081 MAIN 15MPR I NT=40 MA IN 16
300 HPRI~T=MPRINT-I MAIN 11350 wRITE(6.200~1 N.IIX(N.II.I=I.5l.ALPHAI~I.TINI.PSTINI MAIN 78
DELP=O. MAIN 7~
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAIN 80C PLOT MESH MAIN 81C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••HAIN 82
IFIIPlOT.LE.~~1 GO TO 400 MAIN 83IPLCT=IPLOT-IOO MAIN 84READI~.IOOII RMIN.lMIN.DELP.FACT MAIN 85
400 IFIIPLOT.EC.I.OR.IPLOT.EC.ll CALL PlTMll1 MAIN 86IFIISTOP.EQ.II GO TO ~IO MAIN 87
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAIN 88C READ AND WRITE MATERIAL PROPERTIES MAIN 8~C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••HAIN 90
500 IFI~~MMAT.EQ.OI GO TO 600 MAIN ~I
CALL MATLPINUMMATI MAIN ~2
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MAIN 93C READ AND WRITE PRESSURE AND SHEAR 8GUNCARY CONDITICNS MAIN ~4
C MAIN 95600 IF(N~MPC.NE.OI WRITE16.20131 MAIN ~6
IFINUMPC.NE.Ol CALL PRESININUMPC.I.1 MAIN ~1
IFlhUMSC.NE.OI WRITEI6.20ISl MAIN ~8
IFINUMSC.NE.OI CALL PRESININUMSC.D.l MAIN ~~
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••~AIN 100C CCNVERT PORE PRESSURES TO EQUIVALENT STRESSES MAIN 101C••••••••••••••••••••••••••••••••*••••••••••••••••••••••••••••••••••••••MAIN 102
IFINPORPR.EQ.OIGO TO 100 MAIN 103IFIISTART.EQ.2lGO TO 700 MAIN 104DO 650 N=I.hUMEL MAIN lOSMTYPE=IXIN.51 MAIN 106
650 PSTINI=-PSTINI*PO~OTYIMTYPEI MAIN 107C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••~AIN 108C DEIER~lhE BANDWIDTH. 1~ITIAlIIE ELASTI(-Pl~STIC RATIO. MAIN 10~
( AND CONVERT 8ETA FROM DEGREES TO RADIANS MAIN liD, ••••••••••••••••••••*••**••••••••*•••••••••••••••••••••••••••••••••••••MAIN 111
100 J=O MAIN 11200 710 N=I.NUMEL MAIN 113DO 110 1=1.4 MAIN 114DO 110 L=I.4 MAIN 115KK=IA8SIIXIN.II-IXIN.lll MAIN 116IFCKK.GE.JI J=KK MAIN III
110 CONTINUE MAIN 118M8AND=2*J+2 MAIN 11~
DO 120 N=I.NUMEL MAIN 120EPRINI=I. MAIN 121
120 ALPHAINl=ALPHAINI/S7.29S7" MAIN 122C*•• 3 •• $ •••••••••••••••**•••••••**•• **•••• $ ••••••*.********•••••**••••••MAIN 123C SOLVE NONLINEAR EL~STIC PRUtiLEM 8Y SUCCESSivE APPROXIMATIONS MAIN 124C••••••••••••••••••••••_•••••**•••••••••••••••••••••_•••••••••••••••••••MAIN 125
DO 820 NPITER=I.NNlA MAIN 126( MAIN 121C SOLvE TENSICN A~J CU~PkESSICN PRUBLEM ey SUCCESSIVE ~PPROXIMATIONSMAIN 128( MAIN 12~
DO BCD NTITER=I.NTCA MAIN 13CC MAIN 131C FORM STIFfNESS MATRIX MAIN 132( MAIN 133
(All STIFF MAIN 134C MAIN 135C SOLVE FUR UISPlACEMENTS MAIN 136C MAIN 137
CALL SOLv MAIN 138C MAIN 13qC COMPUTE STRESSES MAIN 140
G-9
C MAIN 141CAll STRESS MAIN 142
C MAIN 143IFIITCCCN.E'.ll GO TC 810 MAIN 144
800 CONT 1NUe lolA I N 145C MAIN 146
810 IFlITCDCN.E'.I.ANC.NTCA.GT.11 WRITEl6.20201 NTITER MAIN 147IFlITCDCN.NE.I.ANU.NTCA.GT.ll WRITE16.20211 NTITER MAIN 148IF( IPDCNE.E'.I) GO TO 830 MAIN 149
C MAIN 150820 CONTINUE MAIN lSI
C MAIN 152830 IFIIPDONE.tQ.I.AND.NNLA.~T.llWRlTtl6,2022) NPlTER MAIN 153
IFlIPOONE.NE.I.ANC.NNLA.GT.Il .RITEl6.20231 NPITER MAIN 154IFIISTOP.EQ.21 ,,0 TO 910 MAIN 155
C••••••••••••••**•••••••••••••••••••••••••••••••••••••••••••••••••••••••~AIN 156C PLOT OUTPUT CONTOURS MAIN 157C••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••MAIN 158
900 CAll CONTRIISTARTI MAIN 159910 IFINCASE.lT.NCASESI GO TC 100 MAIN 160
CAll PLTl6l MAIN 161C MAIN 162
1000 FORI'lAT C20A4/12,13, I~, 12, 13,515,13, 12,215,F5 .. 0,,z15) HAIN 1631001 FORMAT 14flC.Ol MAIN 1642000 FORMAT 12HI ,20A41 MAIN 165
I 33~O START PARAMtTER---------------�4� MAIN 1662 33~C STOP PA~AMETER----------------141 MAIN 1673 33~C IF I, PLuT OEFLECTWNS 141 MAIN 1684 :l2~C IF I, SMALL PLCT. IF 2, LARGE! MAIN 16<;5 3jHG FLCT. uTHc~.ISE Ne PLUT.------I41 MAIN 1706 33~C NUM8ER U' APP~uXIMATIONS------I41 MAIN 1717 33~0 IF I, GtNERATE MESH-----------141 MAIN 1728 3J~C ~U~rltH Of Tc~PEHATUk~ CARDS---14/ MAIN 1734 33HC NUMdEk Uf NCLAL PCINTS--------141 MAIN 1741 J3HU NUMDtR UF tLEME~TS------------I41 MAI~ 1752 33HC NUMBER UF INTERNAL PRESSURES--141 MAIN 1763 33HO NUMBER OF MATERIALS-----------141 MAIN 1774 33HO NUMBER OF EXTERNAL PRESSURES--141 MAIN 1785 33~C NUMBER UF SHEAR CARDS---------141 MAIN 1796 33HC REFERENCE TEMPERATURE---------EI2.41Il MAIN IBC
2COI FUR~AT 177H A FUNDAME~TAL FREQUENCY hiLL 8E COMPUTED. A LONGER RUNMAIN 181I TIME WILL 8E UBSERVED/80H DUE TO THE NEED TO RECOMPLTE EACH ELEMEMAIN IB22NT STIFFNESS MATRIX IN SUBROUTINE STRESS) MAIN IB3
2002 FORMAT 124H THE ANGULAR VELOCITY IS,EI2.4./3IH AND THE AXIAL ACCELMAIN 184IERATION IS ,E12.41 MAIN 185
2003 FOR~AT 123H THE R ACCELERATION IS .EI2.4/27H AND THE Z ACCELERATIOMAIN IB6IN IS ,E12.4l MAIN 187
2004 FORMAT 11142H THE PLANE STRAIN CPT ION HAS BEEN SELECTEOl MAIN 1882005 FORMAT ll142H THE PLANE STRESS OPTION HAS BEEN SELECTEOl MAIN 1892008 FORMAT 167H1 EL I J K L MATERIAL ANGLE TEMPERATURE MAIN 190
1 PRESSUREl MAIN 1912009 FORMAT 115.414.IB.F11.1,2FI3.3l MAIN 1922013 FORMAT l30HI PRESSURE 80UNDARY CUNDITIONS) MAIN 1932015 FORMAT l27H1 SHEAR 80UNDARY CONOITIC~Sl MAIN 1942016 FORMAT 126H THE SYSTEM CONVERGED IN 12.IIH ITERATIONS) MAIN 1952017 FORMAT l33H THE SYSTEM 010 NOT CONVERGE IN 12.11H ITERATIONS) MAIN 1962020 FORMAT 1/2gH THE PROCEDURE CONVERGED IN .12,34H TENSION - COMPRESMAIN 197
ISION ITERAT1UNS I MAIN 1982021 FORHAT 1/36H THE PROCEDURE 010 NCT CC~VERGE IN .12.33H TENSION - MAIN 19S
lCOHPRESSION ITERATIONSl MAIN 2002022 FORMAT 1/29~ THE PROCEDURE CONVERGED IN .12.30H ND~LINEAR ELASTICMAIN 201
1 ITERATICNS ) MAIN 2022023 FORMAT 1/36H THE PROCEDURE 010 NCT CCNVERGE IN ,I2,30H NONLINEAR MAIN 203
lElASTIC ITERATIONS I MAIN 2042030 FORMAT l/51H NUM8eR OF TENSION-COMPRESSION APPROXIMATIONS----, MAIN 205
1 14//1 MAIN 206STOP MAIN 207END MAIN 208
G-IO
SUB~OUTINE CATAINCASESI DATA 1, •••••••••••••••••••••••••••**•••••••**.** ••••••••••**••••••••••••••••••OATA 2C ALL DATA CASE CARD IMAGES ARE READ, oRITTE~ C~ THE DUTPLT FILE, AND DATA 3C ~~ITTEN CN FORTRAN UNIT 9 FOR ACCESS BY THE CALLING RCUTINE. THE DATA 4C PARAMETER NCASES IS USEu IN THE CALLING PROGRAM FOR CCNTRULLING DATA 5C THE NUMBER OF CASES TC EXECUTE DATA 6C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••OATA 1C THE RE'UIRED DATA CARD CECK SETUP FOR N DATA CASES IS DATA BC CARD WiTH TUTAL hUMBER uF DATA CASES, NcASES, RIGHT-JUSTIFIED DATA 9C IN COLUMNS 1 THROUGH? DATA 10C CASE 1 DATA CARDS DATA 11C CARD WITH 'END OF CASE' IN COLUMNS 1 THROUGH II DATA 12C AND SO ON UNTIL DATA 13C CASE N DATA CARDS DATA 14C CA~D WITH 'END OF CASE' Ih C'-'LUMNS I THKGUGH II DATA 15C CARD WITH 'END Of DATA' IN COLUMNS I THRGUGH II DATA 16C•••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••OATA 11
DATA TEST/4hENO I DATA 18DIMENSICN CARO(20) DATA 19
C•••••••••••••••••••••••••• ••• •••••••• • •••••••••••••••••••••••••••••••••OATA 20REWINC 9 DATA 21NCASE=O DATA 22
C••••••••••••••••••• ••••••••••••••• •••••••••••••••••••••••••••••••••••••OATA 23C ~EAD AND oRITl THE NUMBER CF DATA CASES DATA 24c••••••••••••••••••••••• $ ••••*•• ~ ••*•••••••••• *•••***••••••***.*****••••OATA 25
REAOI5.1COOI CARO DATA 26oRITE(6,1001l CARC DATA 27WRITE(9.IDOOI CARU DATA 28
c••••••••••••••••••• ***.**••••••••••••••••• *••••••••••••••••••••••••••••OATA 2~
C READ AND WRITE INPUT DATA CAKU IMAGES FOR ALL CASES CN fCRTRAN UNIT 9DATA 30c••••••••••••••••••••••••*•••••••••••••• ** •••••• ** ••••••••••••••••••••••OATA 31
10 NCARD=O DATA 32NCASE=NCASE+1 DATA 33~PRlhl=O DATA 34
20 REACl5,IOOCI CARO DATA 35IFICAROlll.EQ.T[ST.ANC.NCARG.l~.OI GO TO 40 DATA 36NCARC=NCARC+I DATA 37IFICAROlll.NL.HSTI WRlTEI9,lOOOI CARD DATA 38IFIMPRINT.NE.Ol GU TO 30 DATA 39WRITEl6,lOOll NCASE DATA 40MPRINT=34 DATA 41
30 MPRIN1=MPRINT-l DATA 42_RITEI6,1003) CARD DATA 43IF1CARDI ll.NE. TEST) GO TD 20 DATA 441F(~CARC.NE.ll GC Te 10 DATA 45
l •••••••••••*.*****.*******************•••*.****.*****••• ***••••**••••••OATA 46C INITIALIZE AND TEST ACT CAL NUMBER UF DATA CASES AGAINST INPUT VALUE DATA 47c•••••••••••*•••••**••• *•••••••*_.*_ _ _..•.***••_•••••••••••• OATA 48
40 NCASE:NCASE-l OATA 49RE_INO 9 DATA 50REACI9,l004) NCASES DATA 51IFI~CASE.NE.NCASESI .RITE16,10051 DATA 52RETURN DATA 53
C.***••*****************.********~*.*****************************••*****OATA 541000 FORMAT (2DA41 DATA 551001 FORMAT 19H1 NCAStS=,20A4) DATA 56L002 FORMAT (118Hl NCASE=1211L3X,2HIO,8X,2H20,8X,2H30,8X,2H40,8X,2H50, DATA 57
18X,2h6C,eX,2H70,dX,2H80/5X,8CH12345678S012345678901234567890123456DATA 582789012345678901234567890123456789012345678,01 DATA 59
1003 FOR~AT 15X,20A4) DATA 601004 FORMAT (15) DATA 611005 FORMAT 172H THE NUM8ER GF INPUT CASES DeES NOT AGREE _ITH THE VALUDATA 62
IE CF NCASES INPUT) DATA 63END DATA 64
G -II
SUB~OUTIN~ RESTIll REST 1C REST 2
COMMON/BASIC/Alql REST 3COM~ON/MATP/BI16l31 REST 4CCM~CN/NPDATA/CI60001 REST 5COMMON/ElDATA/DIBOOOI REST 6COMMCN/ARG/EI4411 REST 1COMMON/SDlV~/SI40001 REST BCOM~CN/PTT/FI211 REST q
COMMCN/TD/Gl254l REST 10COMMCN/CONVRG/Tl91 REST 11COM~CN/PlANflNPP REST 12
C REST 13IFII.EQ.41 GO TO 300 REST 14IFII.EQ.2.0R.1.~Q.31 GC TO 100 REST 15REwIND 10 REST 16WRITEl101 A,B,C,O,E.F,G,NPP,S,T REST 17HTURN REST 18
C REST lq100 REwiND 10 REST 20
REAC (101 A,B,C,C,E,F,G,NPP,S,T REST 21IFII.EQ.3l RETURN REST 22DATA R/4HREAOI REST 23
200 REAC19,l0001 X REST 24IFIR.NE.XI GO TO 200 REST 25RETURN REST 26
C REST 27C UNIT 10 IS READ TO AN END OF FILE IAT THE END OF COM~ONI TO SAVE REST 28C TIME OVER USING RESTl31 REST 2qC RE ST 30
300 REWiND 10 REST 31READ 1101 REST 32RETURN REST 33
C REST 341000 FORMAT I A41 RE ST 35
END REST 36
G-12
SUBRCUTINE ~ESH MESH 1C ~~ 2
COM~CN/TO/IMINII001.I~AXIIOOI.J~INI251.J~AXI251.MAXI.~AXJ.NMTL.NBCMESH 3CO~~CN/NPDATA/RIIOOOI.CODEIIDDOI.XRIIOOOI.IIIOOOI.XZII0001.TII0001HESH 4COMMCN/ELUATA/IXII00C.51.EPRIIOOOI.ALPHAIICOOI.PSTIIOO01 HESH 5OI~E~SICN ARI25.1001.AI125.1001.NCCOEI25.IOCI MESH 6EQUIVALENCE IRIII.ARII.III.IIIII.AIII.III.IIXII.II.NCODEII.111 MESH 7
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MESH 8C MES. CONTROL INFORMATION MESH 9C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MESH 10
REACC9.10001 MAXI.MAXJ.NSEG.NBC.N~TL.NLIM.CONI.CONJ.ISET.JSET MESH II.RITEI6.200CIMAXI.MAXJ.NSEG.NBC.N~TL.~LIM.CONI.CONJ.ISET.JSET MESH 12
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MESH 13C INITIALIZE MESH 14C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MESH 15
ISEO:-l MESH 16PI:3.1415927 MESH 17DC 110 J:l,lOO MESH 1800 leo 1:1.25 MESH 19NCCCEII.JI:O MESH 20ARC I.JI:O. MESH 21Alll.JI:O. MESH 22JMAX 111:0 ME SH 23
100 JMINI II:MAXJ MESH 24IMINIJI=MAXI MESH 25
110 IMAXIJI:O MESH 26C••••••••••••••••••••• $ •••••••••• **•••••••••••••••••••••••••••••••••••••HESH 27C LIM SEGMENT CARCS ME SH 28C••••••••••••••••••••••*•••••** ••••••••••••••••••••••••*••••••••*•••••••M~SH 2q
200 ISEG:ISEG+1 MESH 30IfIISEG.EQ.~SEGI GO TO 500 MESH 31REAC(9,1001J Il,J1,Rl,11,12 t J2,R2,12 t 13,J3 t R3,13,IPTICN MESH 32~RITEI6.2001)ll.JI.Rl.11.12.J2.R2.Z2.13.J3.R3.13.IPTICN MESH 33IFIII.E'.-11 GO TO 500 MESH 34IPTICN=IPTION+1 MESH 35ARIIl.Jll:RI MESH 3tAllll.JlI:ll MESH 37NCOOEIII.JI):1 MESH 38CALL MNIMXIII.JII MESH 39GO TG 12CO.300.3CC.400.4001.IPTION MESH 40
C••••••••** •••••••** ••*••••••••••••••••••• ** ••••••••••••••••••••••••••••MESH 41C GENERATE STRAIG~T LINES ON BOUNDAR V MESH 42C••••••••••••* •••••••*.** •••••••••••••••*•••••••••••••••••••••••••••••••MESH ~3
300 OI:ABSIFLOATI12-1111 MESH 44OJ:ABSIFlOATIJ2-Jll) MESH 45ARI12.J21:R2 MESH 46AII12.J2):12 MESH 47NCDDEI12.J21:1 MESH 48CAll MNIMXI 12.J2) MESH 49ISHT=1l MESH 50ISTp:12 MESH 51JSTRT:JI MESH 52JSTp:J2 ME SH 53OIFF:AMAXIIDI.OJI MESh 54ITER:OIFF-I. MESH 55IlNC:O MESH 56JINC:O MESH 57
dfI12.NE.lll I1NC:112-11I/IABSI12-1l1 MESH 5BIFIJ2.NE.JII JINC:IJ2-JII/IABSIJ2-JII MESH 59KAPPA:I MESH 60IFI12.NE.ll.ANO.J2.NE.JI.ANO.IPTION.NE.3) KAPPA:2 MESH 61IFIKAPFA.EQ.21 0IFF:2.*OIFF MESH 62RINC:IR2-RII/0IFF MESH 63ZINC:112-111/0IFF MESH 64~RITEI6.20021 OI.OJ.OIFF.RINC.lINC.ITER.IINC.JINC.KAPPA MESH 65
C MESh 66C CHECK FOR INPUT ERROR MESH 67C MESH 68
IFIIPTION.E'.3 .ANO.CI.NE.OJI GG TO 310 MESH 69IFIKAPPA.NE.2.0R.OI.EQ.OJI GG TO 32C MESH 7U
G-13
310 ~RITElb,20031 MESH 71GO TO 200 MESH 72
C MESH 73C INTHPCLATE MESH 74C MESH 75
320 1= 11 MESH 7bJ=Jl MESH 77~RITElb,20041 MESH 7800 340 ~=1, IHR MESH 79IFIITER.EQ.0.ANO.IPTI0~.E~.21GO TO 340 ~ESH 80IFIKAPPA.H.21 GO TO 330 MESH 81IOLe=1 MESH 821=1+ I H,C ME SH 83JOLC=J ~ESH 84J=J+JINC ME SH 85ARII,JI=ARIIOLO,JOLOI'RINC MESH 8bAZII,JI=AlIIOLD,JOLDI.lINC MESH 87~RITElb,20051 I,J,ARII,JI,AZII,JI MESH 88CALL MNIMXII,JI MESH 89NCCCEll,JI=1 MESH 90GO TC 340 MESH 91
330 10LD=1 MESH 92I=I'IINC MESH 93ARII,JI=ARIIOLU,JI+RI~C MESH 94AllI,JI=AlIICLD,JI+llNC MESH 95~RITE16,20051 I,J,ARII,JI,AZII,JI MESH 96NCCCEII,JI=1 MESH 97CALL MNIMXII,JI MESh 98JDLO=J MESH 99J=J+JINC MESH 100ARII,JI=ARII,JOLDI+RINC MESH 101AlII,JI=Alll,JOLOI+lINC MESH 102NCODEII,JI=1 MESh 103WRITEl6,20D51 I,J,ARII,JI,AZII,JI MESH 104CALL MhlMXll,JI MESH IDS
340 CG~TlNUE MESH lObIFIKAPPA.EQ.11 GO TO 200 MESH 10710LD=1 MESH 108I=I+IINC MESH 109ARII.JI=ARIIOLD.JI+RINC MESH 110AlII,JI=AZIIOLD,JI+lINC MESH IIINCCOEII.JI=I MESH 112WRITEI6.20DSI I.J.ARII,JI,AZII.JI MESH 113CALL MNIMXII.JI MESH 114GO TC 200 MESH US
C••••••••••••••••••*•••******••***.******.***••••••***••*.*••••*********MESH 116C GENERATE CIRCULAR ARCS CN BOUNCARY MESH 117C••••••••••••••••••••••**•• *****••••*** ••***••••••••**••••••**••••*••••*HESH 118
400 ARI12,Jil=R2 MESH 119Al112,J21=Z2 MESH 120NCOCElli,J21 = I MESh 121CALL Mh1MX112.J21 MESH 122IFIIPTION.EQ.SI GO TO 420 MESH 123
C MESH 124C FINe CENTER OF CIRCLE MESH 125C MESH 126
AR113,J31=R3 MESH 127AZ113,J31=Z3 MESH 128NeOCEI13.J31=1 MESH 129CALL MNIHXI13.J31 MESH 130SLAC=122-l11/IR2-RII MESH 131SLBF=-I./SLAC MESH 132SLCE=ll3-Z21/IR3-R21 MESH 133SLCF=-I./SLCE MESH 134
C MESH 135C CHECK FeR INPUT ERROR MESH 136C MESH 137
IFIABSISLAC-SLCEI.GT ••OOll GO TO 410 MESH 138WRITE16.20061 Rl.ll,R2.li,R3,Z3.SLAC.SLCE MESH 139GO Te 200 MESh 140
G-14
410 R4=Rl+(Rl-Rl1/2. MESH 14114=11+(12-111/2. MESH 142R5=R2+IR3-R21/2. MESH 14315=12+113-121/2. MESH Ilt4BBF·14-SlBF*R4 MESH Ilt5BDF=15-SLDF*R5 MESH 146RC=leBF-BOF1/ISLDF-SLBFI MESH 1101lC=SIBF*RC+BBF MESH HBWRlTElb,20011 RC,lC MESH 1109KAPPA=I MESH 150GO TO 430 MESH 151
420 KAPPA=2 MESH 152RC=R3 MESH 1531C=l3 MESH 154
430 ISTRT=II MESH 155IS1P=12 MESH 156JSHT=Jl MESH 151JSTP=J2 MESH 158kS IA I=R I MESH 159kS TP=R2 MESH 160IST~T=11 MESH 161ISTP=12 MESH Ib2
440 CAll ANGlEIRSTRT,lSIRT.RC,lC,ANGIJ MESH Ib3CALL ANGLElRSTP,lSTP,RC,lC,ANG21 MESH lblt
C MESH 165C FINe ANGULAR lNCRE~ENT MESH lbbC MESH Ib7
OI=ABSIFLOATIISTP-ISTRTII MESH 16BDJ=AeSIFLuATIJSTP-JSTRTII MESH 169IINc=e MESH 110JI~C=O MESH 111IFIISTRT.NE.ISTPI IINL=IISTP-lSTRTl/IABSIISTP-ISTRTI MESH 112IFIJSTRT.NE.JSTPI JINC=IJSTP-JSTRTI/IA8SIJSTP-JSTRTI MESH 113LA~CA= I MESH 114IFIIINC.NE.C.ANO.JINC.NE.OI LAMDA=2 MESH 115OIFF=AMAXIIOI,DJI MESH 116ITER=DIFF-l. MESH 111IFILAMOA.EQ.21 DIFF=2.*0IFF MESH 118OEL=ANG2-ANGI MESH 119IFICEl.GE.PIIDEL=-12.*PI-DELI MESH lBOIFIOEL.LE.-PIIDEL=DEL+2.*PI MESH 181DELPHI =D ELI 0 I FF MESH 182WRITEI6,20081 ANGI,ANG2,DIFF,OELPHI MESH 183
C MESH 184C CHECK FOR INPUT ERROR MESH 185C MESH 18b
IFllAMDA.NE.2.0R.DI.E~.CJIGO TO 450 MESH 181WRITEl6,20031 MESH 188GG IC 200 MESH 189
C MESH 190450 10=ISTRT MESH 191
JO=JST RT MESH 192WRlTEl6,20041 MESH 193
C MESH 194C INTERPOLATE MESH 195C MESH 196
DO 480 k=I,ITER MESH 191IFILAMOA.EQ.21 GO TO 460 MESH 198I=IC+lINC MESH 199J=JC+JINC MESH 200CALL MNIMXll,JI MESH 201NCOOEII,JI=1 MESH 202CALL CIRCLElANG1,OELPHI,RSTRT,lSTRT,RC,lC,ARlI,Jl,AllI,JII MESH 203WRITEl6,20051 I,J,ARll,Jl,Alll,JI MESH 204GO TO 410 MESH 205
C MESH 20b460 I=IO+IINC MESH 201
J::lIlJ( MESH 208NCOCEIl,JI=1 MESH 209CALL kNIMXII,JI MESH 210
G-15
CALL CIRCLEIANGl,DELPHl,RSTRT,ZSTRT,RC,ZC,ARll,Jl,AZll,Jl1 MESH 211wRITEle,2005l I,J,ARll,Jl,AZll,JI MESH 212J=JC+J [NC MESH 213NCDUEII,JI=1 MESH 214CALL M~IMXII,Jl MESH 215CALL CIRCLEIANGI,DELPHI,RSTRT,ZSTRT,RC,ZC,ARII,JI.AZII,Jll MESH 216wRITEl6,20051 I,J,ARII,Jl,AZII,JI MESH 217
410 10=1 MESH 218480 JO=J MESH 219
IFILAMDA.NE.2l GO TO 490 MESW2201=IC+IINC MESH 221NCCCEII,Jl=1 MESH 222CALL MNIMXll,Jl MESH 223CALL CIRCLEIANGI,DELPHI,RSTRT,ISTkT.RC,ZC,ARII,JI,AZII,Jll MESH 224wRITEl6,20051 I,J.ARll,Jl,AZII,JI MESH 225
4~O IFIXAPPA.EO.2l GO TC 200 MESH 226ISTRT=12 MESH 227ISTP=J3 MESH 228JSTRT=J2 MESH 229JSTP=J3 MESH 230RSHT=R2 MESH 231RS TP=R3 MESH 232ISTRT=Z2 MESH 233ZS TP= l3 MESH 234XAPPA=2 MESH 235GO TC 44D MESH 236
L••••••••••••••••••••••••••••••••••••••••••••••••••••• ••••••***•••••••••MESH 231C CALCULATE CCORDINATES OF INTERICR PCINTS MESH 238C•••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••MESH 239
50C IFIMAXJ.LE.21 GO TO 530 MESH 240J2=MAXJ-I MESH 241IFINLIM.LT.II NLIM = lOa MESH 24200 5LO N=I,NLIM MESH 243RESID=O. MESH 244CO 510 J=2.J2 MESH 24511=IMINIJI+I MESH 24612=HAXIJI-I MESH 241DO 510 1=11,12 MESH 248IFINCODEII,JI.EO.II GU TO 510 MESH 249DR=IARII+I,JI+ARII-I,JI+ARII,J+II+ARII,J-III/4.-ARII,J1 MESH 250
I +CCNI * lAR(J+l,JI - ARII-I,JlllflCATlS*II+ISETlI MESH 2512 + CCNJ * IARII,J+II - ARII,J-Ill 1 FLOATIS*IJ+JSETll MESH 252DZ=IAIII+l,Jl+AIII-I,Jl+AZII,J+II+AZII,J-lll/4.-AZII,J1 MESH 253
I + CONJ * IAIII,J+Il - AZII,J-Ill/FLDATIS*IJ+JSETII MESH 2542 + CONI * IAZII+l,JI - AZII-I,Jll 1 FLCATI8*11+ISETII MESH 255
RESIC=RESID+ABSIDRl+ABSIDZl MESH 256ARII,JI=ARII,JI+I.8*DR MESH 251AIII,Jl=AIII,JI+1.8*DI MESH 258
510 CONTINUE MESH 259IFIN.EO.ll RESI=RESID MESH 260IFI~.E_.I.ANC.RESIC.EO.OI GO TO 530 MESH 261IFIRESI0/RESI.LT.I.E-41 GO TO 530 MESH 262
520 CO~T1NUE MESH 263530 wRITEl6,20091 N MESH 264
c••••••••••••••••••••••••••••••••••••••••••••••••*******••••••**••••••**MESH 265CALL PCINTS MESH 266
c•••••••••••••••••••**••••••••••••••••••••••••**••*******•••••••••••*•••MESH 267RETURN MESH 268
C MESH 2691000 FORM~T 1615,2FIO.O,2151 MESH 21C1001 FORMAT 131213,2F6.3I,151 MESH 2112000 FORMAT 130Hl MESH GENERATION INFORMATIONII MESH 272
I 41HO MAXIMUM VALUE OF I IN THE MESH--------131 MESH 2732 41HC MAXIMUM VALUE OF J IN THE MESH--------131 MESH 2143 41HC NUMBER OF LINE SEGMENT CARDS----------131 MESH 2754 41HC NUMBER OF BOUNDARY CONOITICN CARUS----131 MESH 2765 41HO NUMBER OF MATERIAL BLOCK CARDS--------131 MESH 2716 41HO NUMBER OF ITERATIONS------------------131 MESH 2787 41HC POLAR COORDINATE PARAMETER 1----------EI2.41 MESH 2198 41HO POLAR COORDINATE PARAMETER J----------EI2.41 MESH 2BC
G-16
9 ~1~0 I CURVATURE ~OOIFICATION--------------131 ~ESH 2811 ~IHO J CURVATURE ~GOIFICATI0N--------------1311Il ~ESH 282
2001 FORMAT 11/88H INPUT 11 Jl RI II 12 J2 R2 Z~ESH 28312 13 J3 R3 Z3 IPTION/8X,31214,2F8.3l,161 ~ESH 284
2002 FORMAT ISH GI=F4.0.5H OJ=F4.0,lH OIFF=F4.0.1H RI~C=F8.3.1H IIMESH 285INC-F8.3.1H ITER=13,1H IINC=13,l~ JINC=13.8H KAPPA=lll ~ESH 286
2003 FORMAT I1X,38H ••8AD INPUT--THIS LINE IS ~GT DIAGONAL) MESH 2812004 FORMAT 130H I J AR AI) MESH 2882005 FORMAT 1215.2FIO.31 MESH 2892006 FORMAT 151H •• 8AD INPUT - THESE POI~TS DC NCT DEFINE A CIKCLE,I. MESH 290
13X,6FI2.4,10X,2E20.81 MESH 2912001 FORMAT 121H CENTER COORDINATE IF8.3.1H.F8.3.IHII MESH 2922008 FORMAT ITH ANGl=F9.6.1H ANG2=F9.6,lH DIFF-F3.0.9H DELPHI=F9.6IMESH 2932009 FORNAT 11130H COORDINATES CALCULATED AFTER 13,IIH ITERATIONSI MESH 294
END ~ESH 295
G-17
SL8~CUTI~E MNIMXII,JI MNIM 1C M~IM 2
COMMON/TO/IMINII001.IMAXII001,JMINI251,JMAXI251,MAXI,MAXJ,NMTL,NaCMNIM )C M~IM "
IFIJ.LT.JMINIIII JMINIII=J MNIM 5IFIJ.GT.JMAXIIII JMAXIII=J MNIM 6IFII.LT.IMINIJIl IMINIJI=I MNIH 1IFII.GT.IMAXIJII IMAXIJI=I M~IM a
C M~IM 9RETURN MNIH 10END MNIM 11
G-18
SUBROUTINE ANGLE IR,Z,RC,ZC,ANGI ANGl IC•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••ANGl 2C FI~C ANGLE (F INCLINATION BEThEEN 0 AND Z.PI ANGl 3C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••AhGl 4
PI=3.1415~Z7 ANGl 5Ol-II-ICI ANGl bOZ= IR-RCI ANGl 7IFIABSIR-RCI.GT.I.E-BI GO TL 100 ANGl BANG.PI/Z. ANGl ~
IfIDI.GT.I.E-81 RETURN ANGl 10ANG=I. ;.Pl ANGl IIRETURN ANGl IZ
C••••••••••••••••••••••••••••••*••••••••••••••••••••••••••••••••••••••••ANGL 13C Alll~ CIRCLE Tl CRGSS AXiS ANGl 14l •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••A~Gl 1,
100 AhG=ATANZIOI,DZI ANGl 16IFIANG.LE.l.E-5IANG=2 •• Pl+ANG ANGl 17
C ANGl 18RETURN ANGl I~
END ANGl ZO
G-19
SU8PCUTINE CIRCLE(A~~I.DELPHI.RSTRT.lSTRT.RC.lC.AR.All CIRC IC CIRC 2, •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••ClRC 3C fiNe INTERSECTIG~ Of LI~E AND CIRCLE z NE. RAND l CIRC ~
c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••CIRC 5PI·3.1~159~7 CIRC 6ANGI=ANGI+DELPHI CIRC 7IFIANGI.GT.2.*PIIANGI=ANGI-2.*PI CIRC 8RRzSQRTIIRSTRT-RCI**2+(lSTRT-lCl**21 CIRC 9AR.PC+RR*COSIANGIl CIRC 10Al=lC+RR*SINIANGII CIRC II
C CIRC 12RETURN CIRC 13END CIRC 14
G-20
FU~CT(ON NOUEII.JI NODE 1C NODE 2
COMMCN/TD/IMINIIDOI.IMAXIIDDI.JMIN(251.J~AXI251.MAXI.MAXJ.NMTL.N~CNOOE 3C NODE 4
NOOE"'O NODE 5DC 100 JJ=I.J NODE b~START=IM1NIJJ) NODE 7NSTOP=IMAXIJJ) NODE 800 100 ll=NSTART.NSTCP NODE 9NOOE=NODE+ I NODE 101FIJJ.EQ.J.AND.1[.E~.11 RETURN NODE 11
100 CONT 1NUE NODE 12C NODE 13
RtTURN folCOE 14END NODE 15
G-21
SUBROUTINE POINTS POIN 1C POIN 2
CCHHON/BASIC/NUHNP.NUHEl.NUHPC.NUHSC,ACEll,ANGVEL,TREF.VOL,IFREQ PCIN 3COH~CN/NPDATA/RII0001.CODEII0001.XRII0001.ZIIOOOI.XZIIOOOI,TIIOOOIPOIN 4COHHCN/ELDATA/IXII000.5I,EPRIIOOOI,ALPHAIICOOI.PSTIIOO01 POIN 5DOUBLE PRECISION X.Y.TEH POIN 6CUM~ON/SOLVE/XI11001.YII100I,TEHI1100I,NUHTC.HBAND POIN 1CCHMON/TD/IHINIIOOI.IMAXIIOOI.JHINI25I,JHAXI25I,MAXI.HAXJ.NHTL.NBCPOIN BCOM~ON/PLANE/NPP POIN 9OIMENSICN ARI25,1001.AZI25.100I,HATRILI6.51,BLKANGIIOI POIN 10EQUIVALENCE IRIII,ARII.III,IZIII.AZII,111 POIN II
C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• POIN 12C ESTABLISH NODAL POINT INFORHATION POIN 13C POIN 14
NEt=C POIN 15NCDSlJM=O POIN I~
00 100 J=I.HAXJ POIN 11NSTART=IMINIJI PCIN 18NSTCP=IHAXIJI PCIN 19Db 100 I=NSTART,NSTOP POIN 2C
100 NODSUH=NDOSUH+I PCIN 21NELSUH=O POIN 22JJMAX=MAXJ-I POIN 23DO 110 JJ=I,JJHAX PLIN 24NSTCP=MINOIIHAXIJJI,IMAXIJJ+III-1 POIN 25NSTART=MAXOIIHINIJJI,IMINIJJ+III PCIN 26DO 110 II=NSTART,NSTOP POIN 27
110 NELSUH=NELSUM+l POI N 28NUMNP=NODSUH POIN 29NU~Et=NELSUH POIN 30DO 120 J=I,HAXJ PCIN 31NSTART=IMINIJI POIN 32NSTOP=IHAXIJI PCIN 33Rl=ARINSTART,JI POIN 34II=AIINSTART,JI POIN 35R3=ARINSTOP,JI POIN 36Z3=AIINSTOP,JI POIN 37DI=tR3-RII**2 + 123-211**2 POIN 38DO 120 I=NSTART,NSTDP POIN 39NP=NOOEII,Jl POIN 40RINPl=ARlI,JI POIN 41
120 Z1NPI=Alll,JI POIN 42C••••••••••••••••••••••••••••••••••••••••••••••••••••••••**•••••••••••••POIN 43C READ AND ASSIGN BOUNDARY CONDITIONS POIN 44C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• POIN 45C INIT !AllIE POI N 46C POIN 47
DO 200 I=l,NUHNP POIN 48CODEIII=O. POIN 49IFCRCI1.EQ.0•• ANO.NPP.EQ.OI CODEIII = I. POIN 50XRIII=O. POIN 51XLI 11-0. POIN 52IFINUMTC.NE.-4l TIII=O. POIN 53
200 CONTINUE POIN 54C POIN 55
IFINBC.EQ.OI GO TO 220 POIN 56DO 210 IBCON=I.N~C POIN 57READ19.10021 1I,12,JI.J2,CON,RCON,ICON POIN 58DO 210 1=11,12 POIN 59DO 210 J=JI,J2 POIN 60NP=NOOEII,JI POIN 61CODE I NP I =CON PO IN 62XRCNPI=RCON POIN 63
210 XlINPI=lCON POIN 64220 MPRINT=O POIN 65
DO 240 J=I,MAXJ POIN 66NSTART=IMINIJI POIN 67NSTCP=IHAXIJI POIN 68DO 240 I=NSTART,NSTOP POIN 69NP=NODEII,JI POIN 70
G-22
IFIHPRINT.NE.OI GO TO 230 POIN 11WRITElb,20001 POIN 72HPR INT=40 POI N 73
230 HPRINT=HPRINT-I POIN 74240 WRITElb,20011 I,J,NP,COOElNPI,RINPI,IINPl,XRINPI,XIINPI POIN 75
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••POIN 16C ASSIGN HATERIALS IN BLOCKS POIN 77C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• POIN 78
DO 300 Hl=I,NUHEL paiN 79300 IXIHI,51=0 PUIN BO
DO 310 IHTL=I,NHrL POIN 81READ 19,10001 HTL,IHATRILlIHTL,IHI,IH=2.SI.BLKANGIHTL) POIN B2
310 HATRILIIHTL,11=HTL paiN 83c.··.· ..···· POIN 84C ESTABLISH ELEHENT INFORHATION POIN B5c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••POIN 86
JJHH=HAXJ-I paiN B7N=O POI N B8HlL= I POI N B9DO 440 JJ=I,JJHAX POIN 90NSTOP=HINOIIHAXIJJI,IHAXIJJ'III-1 paiN 91NSTART=HAXOIIMINIJJI,IHINIJJ'III POIN 92DO 440 II=NSTART,NSTOP POIN 93NEL=NEL+l PLJIN 94DO 400 I HTL=I.NHTL POIN 95IFlll.LT.HATRILIIHTL,211 GO TO 400 POIN 9bIFlll.GE.HATRILIIHTL,311 GO TO 400 POIN 97IFIJJ.LT.HATRILIIHTL,411 GO TO 400 paiN 9BIFIJJ.GE.HATRILIIHTL,511 GO TO 400 POIN 99HAT=HATRILIIHTL,11 paiN 100
400 CONTINUE paiN 101IFlHAT.EQ.HTLl GO TO 410 paiN 102Hll=HAT POI N 103GO TO 420 paIN 104
C PLJIN 105410 IFIII.EQ.NSTARTI GO TO 420 POIN lOb
IFIJJ.NE.JJHAX.UR.II.NE.NSTOPI GU TO 440 paiN 107420 I=NODEIII,JJI POIN lOB
J=I+1 paiN 109K=NODEIII'I,JJ'II POIN 110L=K-I paiN IIIH=NEL POI N 112IXIH.II=1 POIN 113IXIH,21=J paiN 114IXIH,31=K PCIN 115IXIH,41=L POIN libIXlH,51=HTL POIN 117ALPHAIHI=BLKANGIHTLI POIN 118
430 N=N'l paiN 119IFIH.LE.NI GO TO 440 POIN 120IXIN,II=IXIN-I,II'1 paiN 121IXIN,21=IXIN-I,21'1 paiN 122IXIN,31=IXIN-I,31'1 paiN 123IXIN,41=IXlN-l,41'1 paiN 124IXIN,51=IXIN-I,51 paiN 125ALPHAINI=ALPHAIN-11 paiN 12bIFIH.GT.NI GO TO 430 POIN 127
440 CONTINUE POI N 128IFINUHNP.GT.I000) WRITElb,20021 POIN 129
c••••••••••••••*••••••••**••****•••••••••*••*•••••• ** •••••••••••••••••••POIN 130C SET NOOAL PCINT TEHPERATURE TO REFERENCE TE~PERATURE POIN 131c•••*.·****••••••••****.*******.******.********•••****.*******•••••*••• *POIN 132
IFI~U~TC.NE.OI RETURN POIN 133DO 500 N=I,NUHNP paiN 134
500 T(NI=TREF paiN 135RETURN paiN 13b
C paiN 1371000 FORHAT 1515,FI0.0) POIN 13B1002 FORHAT 1415,3FI0.0) POIN 1392000 FORHAT 1104HI I J NP TVPE R-ORDINATE l-OROINAPOIN 140
G-23
ITE R LOAD OR DISPLACEMENT Z LOAD Ck DISPLACEMENTI2001 FORMAT 1215,16,FI2.I,FI2.3,FI4.3,E26.7,E24.712002 FORMAT· 135H BAD INPUT - TOO MANY NODAL pelNTSI
END
G-24
POIN 141POIN 142POIN 143PIJIN 144
SUB~OUTINE PNTIN PNTN Ic•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PNTN 2C THIS SUB~UUTINE ENABLES THE USER TC INPlT NCOAL PCINT AND ELEMENT PNTN 3C DATA ~ITHOUT TMC-OIMENSIONAL MESH GENE~ATION. IT IS CONSISTENT PNTN 4C ~ITH THE ORIGINAL SAAS I PROGRAM BY E. L. ~ILSON AND R. M. JONES PNTN 5C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PNTN 6
COMMON/BASIC/NUMNP,NUMEL,NUMPC,NUMSC,ACELI,ANGVEL,TREf,VOL.lfREQ PNTN 7COMMON/NPOA TA/R C1000 I ,CUOE 11000 I ,XR C1000 I ,1l1000 I, Xl (l000 I, T(l000 I PNTN BCOMMCN/ELDATAlIXCI000,51,EP~IIOODI,ALPHACI0001,PSTI1000I PNTN 9
C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• PNTN 10C REAC AND ~~ITE NOOAL PCINT DATA PNTN IIC••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• PNTN 12
100 L=O PNTN 13MPRINT=O PNTN 1'0
110 REA019,10021 N,COOEINI,RCNI,IINI,XRINI,XIINI,TTEMP,PfRES PNTN 15TIIU-HEMP PNTN 16PSTINI-PP~ES PNTN 17NL-L+I PNTN 18lfCL.EQ.OI GO TO 120 PNTN 19IX-N-L PNTN 20OR·I~INI-~ILIIIIX PNTN 21DI-IIINI-IILII/ZX PNTN 22DT-ITINI-TILII/ZX PNTN 23OP=IPSTCNI-PSTILII/ZX PNTN 24
120 L=L+I PNTN 25IfIN-Ll 170,140,130 PNTN 26
130 COOECLI=C. PNTN 27RCLI=~IL-ll+D~ PNTN 2BZILI=ZIL-II+DZ PNTN 29TILl=TIL-II+OT PNTN 30PSTILI=PSTIL-II.DP PNTN 31X~ IL1=0. PNTN 32XZlll=O. PNTN 33GO TO 120 PNTN 3'0
1'00 CO 160 K=NL,N PNTN 35IFC~PRINT.NE.OI bO TO 150 PNTN 36W~ITE16,20051 PNTN 37MPRI NT=4C PNTN 38
150 MPRINT=MPRINT-I PNTN 39IbC WRITE(6,2006J K,CuDE(K),RCKJ,Z(K),XH(KJ,XltK),T(K),PST(K) PNTN 40
IFINUMNP-NI 17C,200,110 PNTN 41170 WRITE16,20071 N PNTN 42
c•••••••••••••••••••••••••••••*****.******••• ** •••********************••PNTN 43C KEAC AND WRITE ELEMENT DATA PNT~ 44c.**•••••••••••**.**•••••• *****.****••****•••** •••••••**••••***** ••*****PNTN 45
200 N=C PNTN 46210 READCS,l0031 M,IIXIM,II.I=I,5I,ALPHAIMI PNTN 47220 N=N+l PNTN 48
IFIM.LE.NI GO TO 230 P~T~ 49IXIN,lI=IXIN-I,lI+1 PNTN 50IXIN,2J=IXIN-I,21+1 PNTN 51lXIN,31=IXCN-I,31'1 PNTN 52IXCN.41=IXIN-l,41+1 PNTN 53IXCN,5J=lXIN-I,5J PNTN 54ALPHAI~)=ALPHAIN-l1 PNTN 55IFIM.GT.Ni GO TO 220 PNTN 56
230 IFINUMEL.GT.N) GO TO 210 PNTN 57RETURN PNTN 58
C PNTN 591002 FORMAT 115,7FIO.01 PNTN 601003 FORMAT 1615,FI0.01 PNTN 612005 FORMAT 1120HINODAL PCI~T TYPE R-ORDINATE I-CROINATE R LOPNTN 62
lAO CR DISPLACEMENT Z LOAD OR DISPLACE~ENT TEMPERATURE PKESSUREPNTN 632 I PNTN 64
2006 fORMAT 1112,FI2.2.2FI2.3,2E24.7,2FI2.31 PNTN 652007 FORMAT 126HCNODAL POINT CARD ERROR N= 151 PNTN 66
END PNTN 67
G-25
SUBROUTINE FLOINIT,R,Z,NUMTC,NUMNP.IMESH,~U~ELI FLONC•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••FlONC THIS SUBROUTINE INPUTS TEMPER_TURE _NO PRESSURE FIELD DATA I~ THE FLONC FORM OF _N _RBITRARY SET OF POINT V_LUES. A TWO-DIMENSIONAL FLONC LINE_R INTERPOL_TION ROUTINE ISUBROUTI~E TEMPI TR_NSFERS FIELD FLONC VALUES TO THE NOO_L POI NT SET. FLONC•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••FlON
C210 TEMINI-ITIIII'TIJJI.TIKK).TILL)1/4.
END
123456189
10111213141516171819202122232425262128293031323334353e313839404142434445464148495051525354555651
FLONFLONFLONFLONFLONFLONFLONFLUNFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONFLONfLONfLONFLONfLONfLON"LONFLONFLONFLONFLONFLONFLONfLONfLONFLONFLONFLONFLONfLONfLONfLONFLON
COMMON/ELOATA/IXII000,51DOUBLE PRECISION X,y,rEMCOMMON/SOLVE/XII100l,YlI100l,TEMI1100l,IOUMI21DIMENSION RII000l,ZII000l,TlI0001
IFINUMTC.GT.OI RH019,10011 IXIII,YIII,TEIIC 1"I-l,NUHCIIFlhUMTC.EQ.-l1 CALL TEMIINUMTCIIFINUMTC.EQ.-21 C_LL TEM21NUMNP,TIIFlhUMTC.EQ.-21 GO TO 200MPRINT-ODO 110 1-I,NUMTCIFIIIPRINT.NE.OI GO TO 100IIRITE16,20011MPRI~T·40
100 MPRINT-MPRINT-l110 IIRITEI6,20021 XllI,Ylll,TEMIII
MPRINT-O00 130 N-l,~UMNP
IFI"PRI~T.NE.OI GO TO 120IIRITEI6,20031MPRI~T=40
120 MPRINT-IIPRINT-lCALL TEMPIRINI,ZIN),TIN),NUMTCI
130 IIRITEI6,20041 N,RINI,ZINI,TINI
DO 300 K=I,NUMEL300 TlKI=TEMIK)
RETURN
200 00 210 N=l,~UMEL
II-I Xl NollJJ-IXIN,21KK=!X1 N, 31LL=IXIN,41
CC TEM IS TEMPORARY STORAGE FOR ELEMENT TEMPERATURESC
C1001 FORMH 13FlO.OI1002 FORMAT 115,11'10.0)2001 FOR~AT IlHl,13X,lHR,14X,lHZ,14X,lHT)2002 FORMAT 13F15.312003 FORMAT 135Hl N R l T)2004 FORMAT II5,2FIO.4,FIO.012022 FORMAT 138Hl NUM8ER CORRESPONOING TO ALTITUDE IS .15)
C
C
CC TRA~SFER NODAL POINT VALUES TO ELEMENT CENTERS BY AVERAGINGC
G-z6
SuBRLUTINE TEMPIR,l,T,~UMTLI TEMP 1c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TEMP 2( THIS SuBROUTINE SOLVES FLR THE NOOAL PCINT TEMPERATURE BY LINEAR TEMP 3( TWO-DIMENSIONAL INTERPOLATION OF THE INPUT TEMPERATURE TEMP 4C DISTRIBUTION TEMP 5C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TEMP 6
DOUBLE PRECISION x,Y,TEM TEMP 7COM"LN/SCLVE/XI17001,Y(1700I,TEMI17001,[CUM(21 TEMP 8DIME~SICN SMALU201,ISMI201 TEMP 9
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TEMP 10C [NIT IALllE TEMP 1lC•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TEMP 12
Jz 1 TEMP 13JMAXz 16 TEMP 14IFI~UMTL.LT.JMAXI JMAX=NUMTC TEMP 15CO Ie 1= I,JMAX TEMP 16SMALL(II~O. TEMP 17
10 ISMIII~O TEMP 18c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TEMP 19C FINe ThE JMAX CLOSEST PGINTS TEMP 20c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TEHP 21
eo 50 1=1.~UMTC TEMP 22OSQ~IXIII-RI ••2+IYIII-II •• 2 TEMP 23[FICS'.GT •• IE-41 GC TC 20 TEMP 24TzTE~.1 [I TEMP 25RE TU~N TEMP 26
20 IFII.EQ.ll SMALLlII=OS' TEMP 27IFII.EO-l1 ISMIll=1 TEMP 28[FII.EQ.11 GG TO 50 TEMP 29[F(SMALLIJI.LE.OSQ.ANO.J.LT.J~AXIS~ALL(J+II=OSO TEMP 30[FISMALLIJI.LE.USO.AND.J.LT.JMAX) [S~IJ+ll=I TEMP 31IFIS~ALLIJI.LE.DSQI GO TO 40 TEMP 3200 30 K= I,J TEMP 33JB=J-K + I TE MP 34IF I JB. EO.OI GO TO 40 TEMP 35SMAllIJB+I/=SMALleJBI TEMP 36ISMIJB+ll=[SMIJBI TEMP 37SMALLIJBI=OSI TEMP 38ISM(JBI=I TEMP 39IFIJB.El.l1 GU TO 40 TEMP 40IFIS~ALLIJB-II.LE.OSQI GO TU 40 TEMP 41
30 CONTINUE TEMP 4240 IfIJ.LT.JMAXl J=J+l TEMP 4350 CONTINUE TEMP 44
c••••••••••••***•••••••**•••••••*******••••••••••• *** ••••••••••******••• TEMP 45C FINC ThE THIRD TEMPERATURt POINT BY ~REA TEST TEMP 46c•••••••••••••••••••••••••••••• ·-•••••••••••••••••••_.*.*** ••** •• ***.***TEMP 47
JCbK=JMAX-2 TEMP 48J=O TEMP 4~
11~ISMIlI TEMP 5012=[SMI21 TEMP 51
60 13=ISMIJ+31 TEMP 52AREA=.5.IYIIIl.XI13I-YlI31OXI[1l+YII3I.XII21-YIIZ1*XII31+ TEMP 53
1 YI121*Xllll-Yllll.XI[21l TEMP 5401=IXI[21-XI[III ••Z+IYI121-YIIIII*.2 TEMP 55
C IF 01 IS APPROXIMATELY O. IT IS ASSUMEC TeAT THERE EX[STS A TEMP 56C CUPLICATIGN OF [NPUT TEMP 57
IFIOI.GT •• lE-31 GO TO 7C TEMP 5812=13 TFMP 5~
J=J+l TEMP 60GC Te 60 TEMP ~1
70 IFIAREA"2.GT •• 1.01.S~.ALLI1l1 GO TG Be TEMP 62J=J+l TEMP ~3
IFIJ.LT.JCHKI GU TO 6e TEMP 64WR[TE16,200CI 1l,l2,I3,J TEMP 65T=TE~llll TEMP 66RETURN TEMP 67
c•••••••••••••••••••••••_•••••••••••••••••••••••••••*••**** ••**•••• *****TEMP 68C FINe TE~PERATUkE INTERCEPT TEMP 6gC••••••••••••••••••••••••*.*••••••••_**_.** •• *•••••*.*******************TEMP 70
G-27
80 GET~=YIIII*ITEMI131-TEMI1211+YI121*ITE~IIII-TEMI1311 TEMP 71I +Y1131*lTEMIl21-TEMlI1l1 TEMP 72CETB=XIIII*ITEMI121-TEMI1311+XI121*ITEMI131-TEMIII11 TEMP 13
I +XI131*ITEMIIII-TEMI1211 TEMP 14DETC=TEMllll*IXI121*YI131-XI131*YI1211+TEMI121*IXI131*YIIII-XIIII*TEMP 75
lYI1311+TEMI131*IXllll*YI121-XI121*YlllI1 TEMP 16T=leET~*R+DETB*l+DETeI/12.*AREA) TEMP 11
C TEMP 78RE TURN TEMP 7q
2000 FORMAT 128H ERROR IN TEMPERATURE INPUT.5~ 11=14,5H 12=14. TEMP 8015H 13=14,4~ J=141 TEMP 81
e TEMP 82ENe TEMP 83
G-28
SueRCUTl~E TEMllhU~TCI TEIII Ic•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TEMl 2C THIS SUBROUTINE IhPUTS A TEMPERATURE CR PRESSURE FIELC FROII AN TEIII 3C EXTERNAL STORAGE CHICE. TEIII ~
C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• rEMl 5DOUBLE PRECISlUh X,V,TEM TEMI 6COIIMCh/SOLVE/XlllDOI,YI1700I,TEMII1001,IDU~121 TEMI 7DIMEhSION Rl5001 ,lI500) ,115001 TEMI BEQUIVALENCE IRlll,XIUOIJI,fllll,V(lZOll),(Tlll,TEIIClZOlll TEll I 9
C TEIII 10REACI9,IGOOI NCASE TEIII 1100 100 I=I,NCASE TEMI 12
100 REACI141 NUMTC,lRIJ),lIJI,TIJI,J=I,NU~TCl TEMI 13CO 200 K=I,hUMTC TEIII I~
XIKI.RIKI TEll I 15VIKl'lIKI TEll I 16
ZOO TEIIIKI=TIKI TEIII 17IoRITEI6,ZOOGI ~U~TC,NCASE TEIII 18RETURN TEMI 19
C TEMI 201000 FORIUT 115 J TEMI 212000 FORMAT 131HI NUMBER OF TEMPERATURE PClhTS=14,7H CASE ,131 TEIII 22
END TEMI 23
G-29
SUBRoUTlhE TEM2IhUMNP.T) TEM2 1C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TEM2 2C THIS SUBROUTINE ASSIG~S A SPECIFIED TEMPERATURE GR PRESSURE TO ALLTEM2 3C NODAL PGINTS. TEM2 "C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TEM2 5
DIMENSION TlhUMNPI TEM2 6C TEM2 1
REACI9.1DOOI TCONST TEM2 800 lCO h=l.NUMNP TEM2 9
100 11 NI=TCChST TEMl 10H TORN TEM2 11
C TEM2 121000 fURMAT (FlC.CJ TEM2 13
END TEM2 1"
G-30
SUBRCUTINE PRESIN(NUMPC.TILTI PRESC••••• •••• • •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PRESC THIS SUBROUTINE READS PRESSURE (TILT-l.I, OR SHEAR (TILT=D.I PRESC BOUNDARY CONDITICNS IN ONE DIMENSICNAL lNUMTC.GT.OI CR TWO PRESC DIMENSIONAL lNUMTC.LT.OI FORM. A LINEAR INTERPOLATION IN THE PRESC I - J PLANE IS CARRIED UUT WHEN PRESSURES DIFfER AT THE ENOS Of PRESC THE IMPLIED LINE SEGMENT. PRESC••••• •••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PRES
~RITE16.20001
I\PRlhT=ONUMP =IABS(NUMPCI00 I~O L=l.NUMPIfINUMPC.GT.OIGO TU 130REAC(~.10001IPI,JPl.IP2.JP2.PRl.PR2
I=IPIJ=JPlP=PRIII NC = aJINC=OIF(IP2.NE.IPlIIINC=IIP2-IPll/IABSIIP2-IPllIflJP2.NE.JPll JINe=(JP2-JPll/IABSIJP2-JPllCI=ABSlfLDATIIPl-IPI))DJ=ABS(FLUAIIJP2-JPlllCIFF=AMAXI(CI.DJIITER=O IFFKAPFA=lIF(IP2.NE.IPI.AND.JP2.NE.JPIIKAPPA=2IFIKAPPA.EQ.2ICIFF=2.*DIFFPINC=IPR2-P_II / DIFF00 110 M=l.ITERIFIITER.EO.CIGO TO liDIFIKAPPA.EO.2IGO TO 10010LD=1JOLO=J1=1'IINeJ=J+JINCCAll PBNDRYtIOLU,JOl~tl,J,P,PINC,TIlT)
GC TC IIG100 lUlO = I
I=HIINCCAll PBNOKY( IOLU,JOLO,I,J,P,PINC,TILT)JCLC=JJ=J'JINCCALL PBNDRY(IOlO,JDLC,I,J.P,PINC.TIlTI
110 CONTINUEIF(KAPPA.EC.IIGO TO 120IOLe= I1=I+IINCCALL PBNCRYI IlJlC,JDLC, I,J,P.PINCd ILlI
120 CONT INUE130 IFINlJMPC.LT.OI GC TC 140
REAC19,l0011 IP.JP.PRCAll PB~ORY(O,O,lP,JP,PR,O.,TILT)
140 CONTINUERETURN
C1000 FORMAT 1415.2FIO.011001 FORMAT (215.FIO.OI2000 FORMAT (21H I J INTENSITY)
END
G-31
PRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRE SPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRESPRES
1234567Bq
101112131415161718l~
2021222324252627282q3031323334353637383q40H~2
~3
4~
454647484~
5051525354555657585q60
SUBROUTINE PBNDRYIIOLO,JOLD,II,JJ,P,PINC,TILTI PBND 1C•••** •••• *•••••*•••••••••••••••••••••••••••••••••••••••••••••••••••••••PBND 2C THiS SUBROUTIN< CONVERTS BOUNDARY PRESSURES AND SHEARS TO PBND 3C EQUIVALENT NODAL POINT FORCES AND STORES THE RESULT IN THE PBND 4C BOUNOARY CONDITION VECTORS. PBND 5C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••P8ND 6
COMMONfNPDATAfRII0001,COUEII0001,XRII0001,1110001,XIII000), PBNO 71 TlI0001 PBND BOOMMONfPLANEfNPP peND 9
c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PBND 10o TILT:l. MEANS PRESSURE peND 11C TI LT=O. MEANS SHEAR PBND 12c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••P~NO 13
IFlIOLD.EQ.OI GO TO 10 peND 14I=NODEIIOLD,JOLOI PBND 15J:NCDEIII,JJI PBND 16PP=IP+PINCf2.lf6. PBND 17P=P+PINC PBND IBGO TO 20 PBNO 19
10 1=11 PBND 20J=JJ PBND 21PP=Pf6. PBND 22
20 PR=PP*6. PBND 23.RITEI6,20Dll I ,J ,PR PBND 24RAD=57.29578 peND 25A=CDDElllfRAD PBND 266=CCDEIJlfRAD P6ND 27PP=Pf6. PBND 28DI=IIIII-IIJII*PP PBND 29DR=IRIJI-RIIII*PP PBND 30RX=2.*RIII+RIJI PBND 31lX=RIII+2.*RIJI PBNO 32IFINPP.NE.OI RX=3.0 PBND 33IFINPP.NE.OI lX=3.0 paND 34
C PBND 35C NODAL POINT FORCE AT POINT I paND 36C PBND 37
SINA=I.-TlLT PBND 38COSA=TILT PBND 39IFICOOEIII.GE.O.1 GO TO 30 PBNO 40SINA=TILT*SINIAI+II.-TILTI*COSIAI paND 41COSA=TILT*COSIAI+II.-TILTI*SINIAI paNO 42
30 IFICOOEIII.NE.I •• ANO.COOEIII.NE.3.1 XRIII=XRIII +RX*ICOSA*OI+SINA*PBNO 431 DRI PBND 44
IFICOOEIII.LT.O.1 GO TO 40 paND 45IFICOOEIII.NE.2•• AND.CODEII).NE.3.1 XIIII=XIIII -RX*ISINA*DI-COSA*PBNO 46
1 DRI paND 47C PBNO 48C NODAL POINT FORCE AT POINT J PBND 49C PBNO 50
40 SH,A=I.-TIlT PBND 51COSA=T IL T PBND 52IFICOOEIJI.GE.O.1 GO TO 50 paND 53SINA=TILT*SINIBI+II.-TILTI*COSla) PBND 54COSA=TILT*COSIBI+II.-TILTI*SINIBI PBNO 55
50 IFICCOEIJI.NE.I ••ANO.COOEIJI.NE.3.1 XRIJ)=XRIJI +IX*ICOSA*OI+SINA*PBNO 561 ORI PBND 57
IFICCOEIJI.LT.O.I GO TO 60 paND 58IFICCOEIJI.NE.2 ••ANO.COOEIJI.NE.3.1 XIIJI=XIIJI -IX*ISINA*OI-COSA*paNO 59
1 ORI PBNO 6060 CONTINUE paND 61
RE TURN paNO 62C PBND 63
2001 FORMAT 1215,FI0.11 paND 64C paND 65
END PBND 66
G-32
SUBRCUTINE ~AllP INUMMAT l MAll IC·.·····························...···· .•...•.•...••...•......•..•......MATL 2C ThE MATERIAL PROPERTIES ARE READ FRC~ FORTRAN UNIT 9 ANC WRITTEN UN MATl 3C ThE OUTPUT FilE MATl 4C••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••MATl 5
CO~~CN/MATP/ROlbl.AOFTSlbl. EI12.22.bl,EEIZII.POROTVI61 MAll bC MAll 7
DO ZOO M=I,HUMMAT MATl 8C MAll 9C REAO ANC WRITE BASIC INFORMATICN FUR A SINolE MATERIAL TYPE MATl 10C MAll 11
REAOl9,I000l MTYPE.NT.ROIMTYPEI,AOFTSIMTYPEI.POROTYIMTYPEI.ISO MATl IZWRITElb.ZOOOI MTYPE,NT,ROIMTYPEI.PORCTYIMTYPEI,ISO MATl 13
C MAll 14C THE MATERIAL PROPERTIES ARE KEAO AS TECHNICAL CONSTANTS IYUUNG'S MATL ISC MOOULI ANC POISSON'S RATIOSl AS A FUNCTION OF TEMPERATURE MATL 16C MAll 17
READl9dOOIl IIEII.J.MTYPEI.J=I.ZZ).I=I.NT) MATL 1800 100 I =I.NT MAll 19
C MAll ZOC IF ISO=O. ALL CONSTANTS MUST BE INPUT MATL 21C IF ISO=I. FIVE CONSTANTS MUST BE INPUT FOR A TRANSVERSELY ISOTROPIC MATL 22C MATER IAl MATl 23C IF ISO=2. TOO CONSTANTS MUST BE INPUT FCR A~ ISCTROPIC MATERIAL MATL 24
IFIISO.EQ.Ol GO TO SO MATl 2SIFIISO.EQ.lI GO TO 40 MAll 2bIFIISO.NE.21 GO T0 SO MATl 27EII.3.MTYPEI= EII.2.MTVPE) MAll Z8EII.4.MTVPE)= EII.2,MTYPEI MATl 29EII.b.MTYPEI= EII.5.MTYPE) MATL 30EII.7.MTYPEI= Ell.5.MTYPEI MATL 31EII.8.MTVPEl= EII,z.MTYPEI MAll 32tll.IO.MTYPEI= EII.9.MTYPEl MATl 33EII.II,MTYPEI= EII.9,MTYPEl MATl 34Ell.13.MTYPE)= EII.12.MTYPEI MATl 35EII,l4.MTYPEI= ElI,lZ,MTYPEI MATl 3bEII.15.MTYPE)= EII.9.MTYPEI MATl 37EII,l7.MTVPEl= EII,lb.MTYPEI MATl 38EII.18.MTYPEI= EII.lb.MTYPEI MATl 39ElI,20.MTYPEI= EIl,l9.MTVPEI MATl 40Ell,2I,MTYPEI= EII.19.MTYPEI MATl 41GO TC 50 MAll 42
40 EII.4.MTYPEI= Etl,2.MTYPEI MATl 43EII.5.MTVPEI= Ell,7.MTYPtl*Ell,2.MTYPEl/EII.3.MTYPEI MATl 44EII.8.MTYPEI= 4.*EII.8,MTYPEI/II.+EII,B,MTYPEI*III.-Z.*EII.5,MTYPEMATL 45
1II1EII.2.MTYPEI+I./EII.3,MTYPEIII MAll 4bIFIEII,l0.MTYPE).LT •• lE-5l GO TC 50 MATl 47EII.II.MTYPEl= EII.9,MTYPEl MATl 48EII.12.MTYPEI= EII.14,MTYPEl*EII.9.MTYPEI/EII.IO.MTYPEl MATl 49EII.IS.MTYPEI= 4.*EII,15,MTYPEI/II.+EII.15,MTYPE)*III.-Z.*EII.12.MMAT1 50
ITYPE)"EII.9.MTYPEI+I./EII,lO.MTYPEIII MAll 5150 IIRITEI6.20011 IEII,J.MTYPEI.J=I.81 MAll 52
WRITElb.20021 1EII,J.MTYPEI,J=9.15l MATl S3IFIAOFTSIMTYPEl.NE.l.1 WRITElb.2003) IEII.J.MTYPEI.J=16,Z21 MATl 54IFIAOFTSIMTYPEI.EQ.I.1 WRITElb.20041 IEII.J.MTYPEI.J=lb.2Zl MATl 55
100 CONT INUE MAll 56C MTl 57C FilL UP THE REMAINDER OF THE E ARRAY FOR ALL POSSIBLE TEMPERATURES MATL 58C MATl 59
DO 200 I=NT,12 HAll 6000 200 J=I.Z2 MATL 61
200 EII.J.MTYPEI=EINT,J.MTYPEI MATL 62RETURN MATL 63
C MATL 641000 FORMAT 12IS,3FID.0.ISI MATL 6S1001 FURHAT 18FIO.0/IOX.7FIO.0/IOX.7FIO.01 MATL 662000 FORMAT IIHI.IIH MATERIAL =.12,SX.SSHNO. OF TEMPERATURES AT WHICH PMATL 67
IROPERTIES ARE SPECIFIED =.IZI5X.14hMASS DENSITY =.EII.4.SX.IOHPOROMATL b82SITY =,EII.4/5X.22HANISCTROPY PARAMETER =,IS) HATL 69
2001 FORMAT 1/7H TEMP=.FS.0/20H TENSILE PROPERTIES.lbX.4HEMT=.FI0.0. MATL 70
G-33
C
IbH ENT=.FIO.O.bH ETT=.FIO.O.8H NUMNT=.F5.3.8H NUMTT=.F5.3.28H NUNTT=.F5.3.bH EPT=.FIO.OI
2002 FORM_T (2~H COMPRESSIVE PROPERTIES.lbX.~HEMC=.FlO.O.bH ENCa.IFIO.O.bH ETC=.FIO.0.8H NUMNC=.F5.3.8H NUMTC=.F5.3.8H NUNTCa.2F5.3.bH EPC=.FIO.OI
2003 FORM_T 130H THERMAL AND YIELO PROPERTIES.lbX.3HAM=.Ell.3.~H AN=.lEll.3.4H _T=.Ell.3.4H YM=.F8.0.4H YN=.F8.0.~H YT=.F8.0.bH PEMRa.2Fb.31
2004 FORMAT130H THERMAL AND YIELD PROPERTIES ,/bX.3HFM=. E1l.3.~H FNa,lEll.3.4H FT=.Ell.3.~H YM=,F8.0.4H YN=.F8.0.4H YT-.F8.0.bH PEMRa.2Fb.31
END
G-34
MATLMAllMAll"All"AllMAll"All"AllMAllMATL"AllMATLMATL
717273n757677787980818283
SGB.CUTINE STIFF STIF IC······ •• •••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••STlf 2C THIS SLBROUTINE Ell~INATES THE CENTER PCINT UNKNO.NS FROM THE STlf 3C CUACRILATERAL ELEMENT STifFNESS MATRIX. ACDS IT TO THE BODY STlff-STlf 4C NESS MATRIX. INCORPCRATES BCUNDARY CDNCITICNS ANU STCRES RESULTS STlf 5C ON AN EXTERNAL STDRA~E DEVICE. STlf 6C.·.·.· ••• • •••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••STIF 7
CC~MCN/BASIC/NUMNP.NUMEL.NUMPC.NUMSC.ACElI.ANGVEL.TREF.VOL.lfREQ STIF 8CC.~~CN/NPDA TA/R 11000 I .CDCE II 000 I. XR IIOCO I .Ill OC~I • Xl I 10001. HI000) STlF 9CC~MCN/ELDATA/IXII000.51.tPRII0001.AlP~AII0C01.PSTIICO01 STlf 10DUUBLE PRECISION CRI.Xl.KK.ll.S.RRR.lll STIF 11DOUBLE PRECISICN CC STIF 12COM~CN/ ARG/ RKR15 I • llll 5 I • KK 14 I • III 4 I • S I 10, 1CI • CRll4 • 4 I • XII 101 • STl F 13
1 PIICI.TT(41.HI60101.HHlt,lCI.ANGltI41.S1GIIOI.EPSllCI.N STIF 14lOUtLE PRECISION A.B STIF 15CUMMCN/SOLVE/RIIOU1.AIIOC.501.NUMBlK.MBANC STIF 16CCMMlN/PlANElNPP STIF 17OI~HSllN LMl41 STIF 18
c••••••••• •••• •••• ·*•••** •••••••••••••••••••••••••••••••••••••••••••••••STIF lqC INIT IAlIlAT ION STIF 20c··.···..······.···.···.··· ·· · STIF 21
RE.INC Z SllF 22NB=Z5 STIF 23NO=ZONB ST IF 24N02=ZONO STIF 25ISTCP=C STIF 2.NU~~lK=O STIF Z7
C STIF 28UG ICO N=I.NC2 STlf 29EINI=O. STIF 30CO lCO ~=I.NO STIF 31
100 AIN.~I=O. STlf 32c.~.*•• *•••••• **•••*•••*.**~**••*****.****•• *••• **.*****••••••••••••••••STIF 33C FCR~ STIFfNESS ~ATR[X IN blCCKS STIF 34c••••••••• ~ ••••*.*••*.*.*.*.*.***••**.***.******••••*•••**••**•••*••••••STIF 35
20U ~u~elK=NU~BlK+l STIF 36NH=NBOINUMBlK+11 STlf 31NM=NH-NB STIF 3BNL=~~-NB+l STIF 39KSHlfT=2.Nl-2 STIF 40
C STIF 4100 340 N=l.NUMcl STIF 42IfllXIN.51.lE.Ul co TO 34U STIF 4300 210 1-1.4 STIF 44IFllXIN.[I.LT.Nll GG TC 210 STIF 45IFIIXIN. [I.LE.NM) GO TC 220 STIF 46
210 CCNTlNUE STIF 47DC TC 340 STIF 48
C STIF 49220 LALl QUAD STIF 50
C STIF 51IFIVCl.GT.O.1 GC TC 230 STIF 52wR[TE(6.20001 N STIF 53(STCP=1 STIF 54
C STlf 55230 IFIIXIN,3I.E~.IXIN.411 GO TG 300 STIF 56
DC 240 [1=1,9 STIF 57CC=SIII,lOI/SllO,lCI STlf 58PlllI=PIII1-CcoPII01 STIF 59CO 240 JJ=1.9 STIF 60
240 SIII.JJl=SIII.JJI-CCOSI10.JJI STIF 61C STIF 62
00 2~0 11=1.8 STIF 63CC=SIII.91/S19.91 STIF 64PIII1=PIIlI-CCOP(91 STIF 6500 2~0 J.J=l.B STIF 66
250 SIII.JJ1=SIII.JJ1-CCOSI9.JJI STIF 61c·•••••••••••• ·.**•••• ** ••**••••••••••••** •••***.*********••• *••••••••••STlf 68C ADe ELEMENT STIFFNESS MATRIX TO BCDY STIFFNESS MATRIX STIF 69c•••••••*.·***.***********.*****.***.*****.***••***********••••*••••••••STIF 10
G-35
300 00 310 1=1,4 STiF 11310 LHIII=2*IXIN,II-2 STIF 72
00 330 1=1,4 STiF 7300 330 K=I,2 STIF 74IlzL~lll+K-KSHIFT STIF 75KK-2*1-2+1< ST IF 7681111-81111+PIKKl STIF 7700 330 J=I,4 STiF 7800 330 L=I.2 STIF 79JJ=L~IJI+L-ll+I-KSHIFT STIF 80LL-2*J-2+L ST IF 81IFIJJ.LE.OI GO TO 330 STIF 82IFINC.GE.JJl GO TO 320 STIF 83~RITE16,20011 N STIF 84(STeP=1 STiF 85"0 TO 340 ST IF 86
C STlF 87320 AIII,JJI'AIII,JJI+SIKK,LLI STIF 88330 CO"TlNUE STIF 89340 WNTINUE STIF 90
C••••••••••••••••• •• • •••••••••••••••••••••••••••••••••••••••••••••• **•••STlf 91C ADC CONOE"TRATEO FORCES STIF 92C • •••••••••••••••••••••••••••••••••••••••••••••••••••••• STIF 93
00 400 N=NL,NH STIF 94IFI".GT.NUHNPI GO TO 500 STIF 95K=2*N-KSHIFT ST IF 96811<1=8IKl+XIINI STIF 97
4008IK-ll=811<-11+XRI"1 STlF 98C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••STIF 99( AOO DISPLACEMENT 80UNDARY (ONDITICNS STIF 100C STIF 101
500 00 ~~O H=NL,"H STIF 102IFIM.GT."U~"PI GO TO 550 STIF 103U=XPIHI STIF 104N=2*~I-KSHIFT STIF 105IFICOOEIHII 540,550,510 STIF 106
510 IFICOOEIHI.EQ.l.1 GO TO 520 STIF 107IFICCDEIHI.EQ.2.1 GO TO 540 STIF 108IFICODEIHI.EQ.3.1 GO TO 530 STIF 109GO TO 540 STiF 110
C STIF 111520 CALL HODIFYIN02''',UI STIF 112
GO Te 550 STiF 113C STiF 114
530 CALL HODIFYIN02,N,Ul STIF 115540 U=XIIHI STIF 116
N·N+l STiF 117OALL HODIFYlND2,N,UI STIF 118
550 CONT INUE STlF 119C•••••••••••••••••••••••••••••••••••••••••••••••••••••**•••***••••••***.STIF 120C kRITE 8LOOK OF EQUATICNS ON FORTR~N UNIT ~NC SHIFT UP LOWER 8LOCK STIF 121C••••**••••••••••••••••*••••••••••••*••••••~•••*•••***••**•••••••••*****STIF 122
kRITE 121 18INI,IAIN,HI,M=I,H8ANDI,N=I,~CI STIF 123( STIF 124
DO tce N=I,ND ST IF 125K=N+NO STIF 12681NI=811<1 STIF 1278IKI=0. STlF 128CO 600 M=I,NO STIF 129AI~,Hl=AIK,HI STIF 130
600 AIK.MI=O. STIF 131c••••••••••••••••••••••••••••••••••••••••••••••••••••********.*********.ST1F 132C CHECK FCR LAST 8LOCK STIF 133c•••••••••••••••••••••••••••••••••••**••••••*•••************************STIF 134
IFINM.LT.NUHNPI GC TO 200 STIF 135IFIISTOP.NE.OI STOP STIF 136RETURN ST IF 137
C STlF 1382000 FORHAT 127H NEGATI~E AREA ELEMENT N(.,141 STIF 1392001 FORHAT 146H 8ANe WICTH EXCEEDS ALLOWAeLE FCP ELEMENT NO.,14) STIF 140
END STIF 141
G-36
SUBROUTINE QUAD QUAC IC•••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••*QUA~ lC THIS SUBROUTINE ASSEMBLES A QUADRILATERAL ELEMENT STIFFNESS MATRIXQUAC 3~ AND LOAD VECTOR BY SUPERFOSITION OF FOUR TRIANGULAR ELoMENTS. QUAD 4C••••••••••• • ••••••••••••••••••••••••••••••••••••••••••••••••••QUAD 5
CO"MCN/BASIC/NUMNP,NUMEL,NUMPC,NUMSC,ACEL1,ANGVEL,TREF,VUL,lFRoQ QUAD 6CCM"CN/NPDATA/RIIDOOI,COCECI000I,XRCIDOOI,111000I,X111OOCI,TlI0001QOAD 7CGMMGN/ELDATA/IXII00C,Sl,EPRlI000l,ALPHAliCCOI,PSTII0001 QUAC 8DOUBLE PRECISION CR1,XI,RR,1Z,S,RRR,111 QUAD 9COMMCN/ARG/RRRCSI,1Z1lSI,RRI41,lZl41,Sl10,1Cl,CRll4,41,XIlI0l, QUAD 10
I PII0I,TTI41,HI6,IOI,HHI6,10I,ANGLEI4I,SIGIIOI,EPSIICI.N QUAD 11CO"KC~/PLANE/NPP QUAD 12
C QUAD 13C INIT !ALl1E QUAD 14C QUAD IS
1-IXI~,l1 QUAD 16J=IXCN,21 QUAD 11K=IXIN,31 QUAD 18L=IXIN,41 QUAD 19
'MTYPE=IXIN,51 QUAD 20IXIN,51=-IXIN,51 QUAD 21
C QUAC 22CALL KPRep QUAU 23
C eUAC 24C·••••• •••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••QUAD 25C fCRK 'UACRILATERAl STifFNESS MATRIX QUAD 26C•••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••QUAO 27
RRRI51=IRIII+RIJI+RIKI+RIlII/4. QUAD 281l1151=11111+1IJI+1IKI+1ILlIl4. eUAD 29CO 110 M=I.4 QUAD 30MM=IXIN,MI QUAe 31IFINPP.NE.OlGO TO 100 QUAD 32IFIRIMMI.EQ.o ••A~C.CUDEIMMI.EU.O.) CGDEIMMI=I. QUAD 33
100 RRRIKI = RIMMI QUAC 34110 1l11KI=11MMI QUAD 35
C ~~ 36DO 130 11=1,10 QUAD 37PI Ill~O. QUAD 3800 120 JJ-l,6 QUAD 39
120 HHIJJ,1I1=0. QUAD 40DO 130 JJ=I,10 QUAD 41
130 SIII,JJl=O. QUAD 4200 140 11=1,4 QUAD 43JJ=IXIN,Ill QUAD 44
140 ANGLEIIII=CODEIJJI/S7.29578 QUAD 45C QUAD 46
VDL=O. QUAD 47IFIK.NE.LI GO TO 150 QUAD 4BCALL TRISTFI1,2,31 QUAD 49RRRI51=IRRRlll+RRRI21+RRRI311/3. QUAD 50111151=1111111+111121+1111311/3. QUAD 51RETURN QUAD 52
( QUAD 53150 (ALL TRISTFI4,I,5) QUAD 54
CALL TRISTFll,2,51 QUAD 55CALL TRISTFl2,3,Sl QUAD 56CALL TRISTFI3,4,51 QUAD 57
( QUAD 5BDO 160 11=1,6 QUAD 59DO 160 JJ=I010 QUAD 60
160 hHIII,JJI=HHIII,JJI/4. QUAD 61C QUAO 62
RETURN QUAD 63END QUAD 64
G-37
SUB~lUTINE MPROP PROP 1C••••• •••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••PROP ZC THIS SUBROUTINE FI~CS ThE STRESS-STRAI~ RELATIONSHIP IN BOOV COOR-PROP 3C CINATES. PROP ~
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PROP 5CGMMC~/BASIC/NUM~P.NUMEL.~UMPC.NUMSC.ACELZ.ANGVEL.TREF.VOL.lfRE' PROP 6COMMGN/MATP/ROlol.AGfTSI61.EIIZ.ZZ.61.EEIZll.PORCTVltl PROP 1COMMCN/NPDATA/RII0001.GODEII0001.XRII0001.Z110001.XZIIOOOI.TII000IPROP 8COMMON/ELDATA/IXII00C.51.EPRII0001.ALPHAII0001.PSTII0001 PROP ·9DOUBLE PRECISION lRZ.XI.RR,ZZ.S.RRR.ZZZ PROP 10COMMCN/ARG/RRRI51.ZlZ151.RRI41.ZZI41.S110.1CI.CRZI4.41.XIII01. PROP 11
1 PllOI.TTI41.Hlb.l01.HhI6,lOI.ANGLEI41.SIGI101.EPSllCI.~ PROP 12COMMCN/CONVRG/IPDuNE.ITCCON.NNLA.NTCA.NTITER.OLDSIGI41 PROP 13COMMON/PLANE/NPP PROP 1~
DOUBLE PRECISION SM~.SPC.GP~.CMN PROP 15DIMENSION CMNI4,41.CPQI4.41.SMNI4.41.SM~TI4.41.SMNCI4.41.SPQI4.4I.PROP 16
ISPCTI4.41.SPQCI4.41.[14.41.CUMMVI4.41 PROP 11EQUIVALENCE ICP~Il.ll.SP\llltlll PRGP 18EQUIVALENCE ISMNll.II.CMNll.111 PROP 19
C••••••• •• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• PROP 20C INITIALIZE PROP 21C••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• PROP 22
MTVPE-IABSIIXIN.511 PROP 23c.·..··· · ···· PROP 2~
C INTERPOLATE MATERIAL PROPERTIES PROP 25c.· ·· ··· PROP 26DO 100 MM=Z.IZ PROP 21M-MM PROP 28IfIEIM.l.MTVPEI.GE.TINII GC TU 110 PROP 29
100 CONTINUE PROP 30110 RATlC=O. P"OP 31
CEN~EIM.I.MTVPEI-EIM-l.1.MTVPEI P~OP 32IfIDEN.NE.O.IRATIO=ITINI-EIM-I.I.MTVPEII/CEN PROP 33DO IZO KK=l. ZI PROP H
lZ0 EEIKKI=EIM-I.KK+I.MTYPEI+~ATIO*IEIM.KK+I.MTVPEI-EIM-l.KK+l.MTVPEIIPROP35C PROP 36[ MODlfV ALL TENSILE MODULI BV MULTIPLVI~G BV ELASTIC/PLASTIC RATIO PROP 31C AS fART [f NCNLI~EAR APPROXIMATIONS PROP 38C PROP 39C PROP 40C SINCE T~E NONLINEAR THEORY CANNOT BE USEO AT THE SAME TIME AS THE PROP 41C MULTIMCDULUS THEORY. USE THE TENSILE PRCPERTIES IN THE NONLINEAR PROP ~2
C APPROACh PROP ~3
C PROP ~~DO 130 M=I.3 PROP 45
130 EEIMI=EEIMI*EPRINI PROP 46DO 140 M=4.6 PROP 41
140 EEIMI=.5-1.5-EEIMII*EPR{NI PROP 4BEE{71=EEI71*EPRINI PROP ~9
IfINTITER.GT.ll GO TO ZOO PROP 50c•••¥$~·••••*••••***••*.*••*•••••**.*.**•••••••••••••*••••••••••••••••••PROP 51C fORM STRESS-STRAIN ~ELATICNSHIP IN M-N-T SYSTEM PROP 52(. -:$.?f.,'::") •••** ••••*********************.*."""**** ••******* * PROP 53C PROP 5~
INITIALIZE STRESS-STRAIN RELATIONS .ITH TENSILE PROPERTIES PROP 55C PROP 56C fORM STRAIN-STRESS ICUMPLIANCEI MATRIX IN PRINCIPAL MATERIAL PROP 51C COORDINATES IM-N-T SYSTHJ PROP 5BC PROP 59
SMNll.ll=I./EElll PROP 60SMNll.21=-EEI41/EElll PROP 61SMNll.31=-EEI51/EE{11 PROP 62SMNll.41=0. PROP 63SMNI2.11=SMNll.Z1 PROP 64SMNI2.21=1./EEIZI PROP 65SMNI2.31=-EEI61/EEIZI PROP 66SMNI2.41=0. PROP 61SMNI3.II=SMNll.31 PROP 6BSMNI3.21=SMN{2.31 PROP 69SMNI3.31=I./EEI31 PROP 10
G-38
SMhI3,41=0. PROP 11SMNI4,ll=0. PROP lZSMNI4,Zl=0. PROP 13SMNI4,31=0. PROP 14SMhI4,41=4./EElll-II./EE111.1./EEIZI-12.*EEI4lI/EE~111 PROP 15
C PRCP 16C PROP 11
CALL SYMIN~ISMN,41 PROP 18C PROP 19
GO TC 500 PROP 80c••••••••• •••••••••••••••••**••••••••••••••••••••••••••••••••••••••••••• PROP 81C FOR~ STRESS-STRAIN RELATICNS IN PRINCIPAL MATERIAL OIRECTIONS PROP 8ZC IM-N-T SYSTEMl PROP 83c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PROP 84C PROP 85C OEFINE CCMPLIANCE MATRIX, SMNT. FOR TENSILE PROPERTIES IN PROP 86C PRlhCIPAL MATERIAL COORDINATES PROP 81C PROP 88
ZOO SMNTlI,lI=I./EEIII PROP 89SMNTll,ZI=-EEI4I/EECll PROP 90SMNTCI,31=-EEC5I/EEI I) PROP 91SMNTll,41=0. PROP 9ZSMNTlZ,II=S~NT(1,2l PROP 93SMNTCZ,ZI=I./EECZ) PROP 94SMNTCZ,31=-EEI6I1EECZI PROP 95SMNTlZ ,41 =0. PROP 96SMNTI3,II=SMNTII,31 PROP 91SMNTI3,21=S~NTIZ,31 PROP 98SMNTI3,31=1./EEI31 PROP 99SMNT13,41=0. PROP 100SMNTl4,lI=0. PROP 101SMNTI4,ZI=O. PROP 10ZSMNTC4,31=0. PROP 103SMNTI4,41=4./EElll-II./EElll+I./EEI2l-IZ.*EEI4lI/EEl11I PROP 104
C PROP 105C OEFINE COMPLIANCE MATRIX, SMNC, FOR COMPRESSIVE PROPERTIES IN PRCP 106C PRINCIPAL MATERIAL CCORDINATES PROP 101C PROP 108
SMNCl1,11=1./EEI81 PROP 109SMNC11,ZI=-EEII11/EEC81 PRCP lieSMNC11,31=-EE1IZI/EEI8) PROP IIISMNCl1,41=O. PROP liZSMNCI2,ll-SMNCl1,ZI PROP 113SMNCIz,ZI-1./EEI91 PROP 114SMNCIZ,31=-EEI131/EEI91 PROP 115SMNC1Z,41-0. PROP 116SMNCI3,II=SMNCl1,31 PROP IIISMNCI3,ZI=SMNCIZ,31 PROP 118SMNCl3,31=1./EEl101 PROP 119SMNCl3,41=0. PROP IZOSMNCl4,11=0. PROP IZISMNC 14,ZI=0. PROP 12ZSMNCI4,31=0. PROP 1Z3SMNCI4,4l=4./EElI41-11./EEI81.1./EEI91-1Z.*EElllII/EEI811 PROP lZ4
c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• PRGP 125C FORM STRESS-STRAIN RElAT1CNS IN PRINCIPAL STRESS DIRECTIONS PROP lZ6C lP-~-T SYSTEMI PROP IZlc•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PROP 128C PROP lZ9
READ 131 OMEGA,SIGP,SIGQ,SIGT,IOlOSIG111,1=I,41 PROP 130C PROP 131C THE OLOSIGIII ARE REAO BECAUSE THEY CCCUR BETWEEN SETS OF CMEGA, PROP 132C SIGP, ETC. AND BECAUSE THEY ARE REQUIRED l~ STRESS lTHEY ARE PRCP 133C PASSEO THROUGH CCMMONI PROP 134C PROP 135
8ETA=OMEGA-AlPHAINI PROP 136C PROP 131C NE~ DIJ MATRIX FOR ROTATION FROM MN TO PQ PROP 138C PRCP 139
CALL ROTATEIO, 8ETAI PROP 140
G-39
C PRGP 1~1
C ROTATE S~NT TO SP"T PROP H2C PROP 1~3
DO 3CO IIEI,4 PROP 1~4
DO 300 JJEI,4 PROP 1~5
OUMMVIII,JJI=O. PROP 1~6
DO 300 KK=I,4 PROP 1~1
300 DU~MVIII,JJIEOUMMVIII,JJI+SMNTlll,KKI*CIJJ,KKI PRCP 1~6
C PROP 1~9
CO 310 JJ=I,4 PROP 150DO 310 11-1,4 PROP lSISP"TIII,JJI=O. PROP 152DO 310 KK=l,4 PROP 153
310 SPOTIII,JJI=SPUIIII,JJI+OIII,KKloDUMMV1KK,JJI PROP 154C PRCP 155C ROTATE S~NC TO sp~c PROP 156C PROP 151
DO 320 11=1,4 PROP 156DC 320 JJ=l,4 PROP 159DUMMVIII,JJI=O. PROP 160DO 320 KK=I,4 PROP 161
320 DUMMYlll.JJI=DUMMVIII,JJI+SMNCIII.KKloOIJJ,KKI PRep 162C ~OP 163
DO 330 11=1,4 PROP 16'0DO 330 JJ-I,4 PROP 165SPOClll,JJI=O. PROP 166DO 330 KK=I,4 PROP 161
330 SPOClll,JJI=SPOCIII,JJI+CIII,KKloDUMMVIKK,JJI PROP 166C PROP 169c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PROP 110C CHOOSE SPO BASEC ON SIGNS AND VALUES Cf P~I~CIPAL STRESSES PROP 171c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PROP 172C PROP 173C DEFINE "EIGHTING FA(.TURS PROP 174C PROP 175
PPO=ABSISIGPI/IABSISIGPI+ABSISIGOII PROP 176OPO=ABSISIGOI/IABSISIGPI+ABSISIG"11 PROP 171PPT=ABSISIGPI/IABSISIGPI+ABSISIGTll PROP 178TPT=ABSISIGTI/IABSISIGPI+ABSISIGTII PROP 179OOT=ABSISIGOI/IABSISIGOI+ABSISIGTII PROP 160TOT=ABSISIGTI/IABSISIG"I+ABSISIGTII PROP 181
C PROP 182SPOll,llESPOCll,l1 PROP 183lFISlGP.GT.O.1 SPOII,II=SPOTll,11 PROP 184
C PRCP 165SPOI2,21=SPOCIZ,ZI PROP IB6IFISIGO.GT.O.I SPOIZ.ZI=SPOTIZ.ZI PROP 187
C PROP 188SPOI3.31=SPOCI3,31 PROP 189IfISIGT.GT.O.I SPOI3,31=SPOTI3.31 PROP 190
C PROP 19lSPOll,ZI=SPOCII,21 PROP 192lF1SIGP.GT.0 •• ANO.SIGO.GT.0.1 SPOII,ZI=SP"TII,ZI PROP 193IFISIGP.GT.O •• ANC.SIGO.LE.O.1 SPOII,ZI=PPO*SPOTII,21+0PO*SPOCII,ZIPROP 194IfISIGP.lE.O •• ANO.SIGO.GT.O.1 SPOIl.ZI=PPO*SPOCII.ZI+OPO*SPOTII,ZIPROP 195
C ~~I%
SPOII,31=SPOCII.31 PROP 197IFISlGP.GT.O•• ANO.SIGT.GT.O.1 SPOIl,31=SPOTII,31 PROP 198IFISIGP.GT.O •• AND.SIGT.LE.O.I SPOII,31=PPl*SPOTII,31+TPT*SPOCII.3IPROP 199lFISIGP.lE.O •• AND.SIGT.GT.O.1 SPOII,31=PPT*SPOCII,31+TPT*SPOTII,3IPROP zoo
(. PROP ZDISPOI2,31=SPOCIZ.3I PROP ZOZIfISlGo.GT.O ••ANO.SIGT.GT.O.1 SPOIZ,31=SPOTIZ,31 PROP Z03IFISIGO.GT.O •• ANO.SIGT.LE.O.1 SPOIZ.31="T*SPOTI2,31+TOT*SPOCIZ,3IPROP Z04lFISIGO.lE.O •• AND.SIGT.GT.O.1 SPO(Z.31="OT*SPOCI2.31+TOT*SPOTI2,3IPROP Z05
C PROP 206SPOII.41=SPOCII,41 PROP Z07IFISIGP.GT.O.I SPOII.41=SPOTII.41 PROP ZOB
C. PROP 209SPOI2,41=SPOCIZ.41 PROP 210
G-40
IFISIGQ.GT.O.1 SPQI2.41=SPQT(2.41 PROP 211C PROP 212
SPQ(3.41=SPQCI3.41 PROP 213IFISIGT.GT.O.I SPQI3.41=SPQT(3.41 PROP 214
C PRCP 215SPQ(4.41=SPQC(4.41 PROP 216IFISIGP.GT.O •• ANC.SIGC.GT.O.1 SPQI4.41=SPCTI4.4) PROP 217IFISIGP.GT.O•• ANC.SIGC.L~.O.I SPQI4.41=PPC*SPQTI4.41+CPQ*SPQC(4.4IPROP 218IFISIGP.LE.O •• ANC.SIGC.GT.G.) SPQ(4.41=PPC*SPQCI4.41+QPQ*SPQTI4.4IPROP 21q
C PROP 220C ENFURCE SYMMETRY OF THE COMPLIANCE MATRIX PROP 221C PRCP 222
SPQ(2.11=SPCll.21 PROP 223SPC(3.11=SPCll.31 PROP 224SPQI3.21=SPCI2.31 PROP 225SPCI4.II=SPQll.41 PROP 226SPQI4.21=SPCI2.41 PROP 227SPCI4.31=SP'(3.41 PROP 228
C PRCP 22qCALL SYMIN~ISPQ.41 PROP 230
C PROP 231C.·.········ •••••••••·••••••••••••••••••••••••••.•.••••.••••..•••••••.••PROP 232C ROTA1E STR~SS-STRAIN RELATIONS FROM PRINCIPAL STRESS DIRECTIONS PROP 233C IP-Q-T SYSTEM 1 TO PRINCIPAL MAT~RIAL DIRECTIONS IM-N-T SYS1EMI PROP 234C··.·········.•.·••••·••··•••••••.•••••••••..•••..•••••••••••••••••.••••PRGP 235C PRCP 236C THIS STEP IS REQOIRED IN ORDER TO PICK UP T~~ THERMAL TERMS THAT PROP 237CARE CEFINED ONLY IN THE PRINCIPAL MATERIAL DIRECTIONS PROP 238C PROP 23qC ROTATE CPC TO CMN PROP 240C PROP 241
CALL ROTAT~IC, BElA) PROP 242C PRCP 243
DO 400 1[=1.4 PROP 244CO 400 JJ=I.4 PROP 245OUMMYIII.JJI=O. PROP 24600400 KK=I.4 PROP 247
400 DDMMllll.JJI=OUMMY[ 1I,JJI+CPQI [[,KKI*OIKKdJI PROP 248C PROP 24q
GO 410 11=1,4 PROP 25000 410 JJ=I,4 PROP 251CMNIII,JJI=O. PROP 25200 410 KK=I.4 PROP 253
410 CMNIII.JJI=CMNIII,JJI+DIKK,III*OUMMYIKK,JJI PROP 254c••••••••••••••••••••••••••••••••••••••••••••••••••••*•••• ***••••••••••• PROP 255C ROT AlE STRESS-STRAIN RELATIONS FROM PRINCIPAL MATERIAL DIRECTIONS PROP 256C (M-N-T SYSTEMI TO BODY DIRECTIONS IR-l-l SYSTEMI PROP 257c••••••• ••••••••••••••••••••••••• *****••** ••••••••** ••*****.**••••****••PROP 258C PROP25qC PROP 260
500 OALPHA:ALPHAINI PROP 261CALL ROTA1EID.OALPHAI PROP 262
C PROP 263C PRep 264C R01AH CMN 10 CRl PROP 265C PROP 266
00 510 11=1,4 PROP 26700 510 JJ=I,4 PROP 268OUMMYI II.JJ 1=0. PRCP 269DO 510 KK=I,4 PROP 270
510 DUMMYlll.JJI=OUMMYIII,JJ1+CMNIII,KKI*0IKK,JJ) PROP 271C PROP 272
00 530 [1=1.4 PROP 273DO 520 JJ=I,4 PRUP 274CRl( 11 dJ 1=0. PRCP 275EO 520 KK=I,4 PRCP 276
520 CRlIII,JJI=CRZlII,JJI+DIKK.III*CUMMYIKK.JJ) PROP 277111111=0. PRCP 278DO 530 M=I.4 PROP 2JqPIMl=IT1NI-1KEF)*ICMNIM,II*EEII51+CMNIM.21*EEI161+CMNIM,31*tEI171IPROP 280
G-41
IFlACfTSlMTVpcl.E~.I./P(~I=CMN(M,II.EEI15/'CMNlM,21.EE1161
I+CM~IM,31·EEI171
530 TTIIII=TTlIIl+OIM,III.PIMIIfl~PP.NE.2IRETURN
CC THE PLANE STRESS MATEklAL PROPERTIES ARE fCUNO FPOM IbE GENERALC STRESS-STRAIN RELATICNSHlb BV SETTING SIGMAZ C AND ELIMINATINGC EPSZ FROM TbE E_CATleNS.C
CRZII,I)=CRZII,II-CRZII,3) ••2/CRlI3,31CR1Il,2)=Ck1ll,2/-CRlI2,3).CR1( 1,31/Chll3,31CR1Il,4/=CR1(I,4/-CR1ll,3)·CRlI3,41/CRlI3,31CklI2,ZI=CR1lZ,ZI-CRZI2,31 ••2/CklI3,3/CRlI2,41=CklI2,41-CRlI2,31·CRlI3,41/CRZI3,31CRlI4,41=CRZI4,41-CklI3,41··2/CRlI3,31TT(II=TTlll-CR11I,31·TT(31/C~113,31
lTI21=11121-CkLI2,31+11131/CklI3,31TTI41=TTI41-CklI4,31+1TI31/CklI3,31cu tce 11=2,4CO 6eo JJ=loIl
600 CR1lll,JJI=CR1IJJ,111C
RETURNEND
G-4Z
PROP 281PRCP 282PROP 283PROP 284PRCP 285PROP 286PRCP 287PRCP 288PROP 289PROP 2gePRCP 291PRCP 292PROP 293PROP 294PROP 295PROP 296Pllep 291PRCP 298PRCP 299PROP 300PRCP 301PROP 302PROP 303PROP 304
S~8RCUTI~E KCTATEI(,T~ETAI RTAT 1L•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••RlAT 2C THE TKA~SFOR~ATIC~ MATRIX 0 I~ CALCULATE( 'CP A KOTATIGN 'THETA' RTAT 3C WHIC. MUST tE EXPRESSED IN RADIA~S RTAT 4C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••RTAT 5
(I~E~SIC~ 014,41 RTAT 6C RUT 7
S2= SI~ITHETAI•• 2 RTAT 8C2= CGSITHETAI •• < RTAT 9SC= Slhl THETAI. CCSlTHETAl RTAT 10
C RTAT 11O(l,II=C2 RTAT 12(ll,21=S2 RTAT 13(ll,3I=C. RUT 14Dll,41=SC RTAT 15(1',1I=S2 RTAT 16C12,21=C2 RTAT 17(12,31=0. RTAT 18012,41=-SC RUT 19(13011=0. RTAT 20(13,21=C. RUT 21(13,31=1. RTAT 22C13,41=C. RTAT 23DI4011=-2 •• SC RTAT 24CI4,21=2 •• SC RTAT 25014,31=C. RTAT 26CI4,41=Cl-S;> RTAT 27
C RUT 28PETlJPh RTAT 29ENe RTAT 30
G-43
SUBRCvTINE TRISTFIII.JJ.KKI TRIS 1C··••• ••••••••••••••••••••*••••••••••••••• ••••••••••••••••••••••••••••••TRIS 2C THIS SUBROUTINE GENERATES A TRIANGULAR ELE~£NT STIFF~ESS MATRIX TRIS 3C ANC U.AC VECTOR. TklS 4C·•••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TRIS 5
LOMMC~/BASIC/NUMNP.NUMEL.NU~PC.NUMSC.ACELI.ANGVEL.TREf.VoL.IFREQ TRIS 6CO~~C~/MATP/ROI61.AOFTSI61.EI1Z.ZZ.61.EEIZll.PURGTYIEI TRIS 7CGM~CN/NPDATA/RIIOOOI.CoOEII0001.XRII0001.111000l.XII10001.TII000lTPIS 8Co~M(N/ELDATA/IXI1000.51.EPRI10001.ALPHAllCCOI.PSTIICC01 TRIS 9DOUBLE PRECISION f.CCMM.D TlUS 10DOUBLE PRECISION CRI.Xl.RR.II.S.RRR.111 TRlS 11CoMMLN/ARG/RRRI51.lll151.RRI41.ll141.SI10.101.CRlI4.4).XIII01. TRIS lZ
1 PI101.TTI41.HI6.1DI.HHI6.1CI.ANGLEI41.SICII01.EPSIICI.N TRIS 13COM~oN/PLANE/NPP TRIS 14OIME~SION 0Io.61,LMI31.0013.31.FI6.101.TPI61 TRIS 15
C.·••••••••••• ••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••• TRIS 16C INiTlALllE TRIS 17C··••••••• •••• ••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••• TRIS 18
LMlll=ll TRIS 19LMlZI=JJ TRIS ZOLM I3I=KK TRI S 21ITR=2 TR I S 22IFIKK.EQ.31 ITR=3 TRIS 23
C TRIS 24RRlll=RRRllll TRIS 25RRlZI=RRRlJJI TRIS 26RRI31=RRRIKKI TRIS Z7lllll=llll III TRIS 28lllZI=llllJJl TRIS Z9ll131=llllKKI TRIS 30
C TRIS 31DO 110 1=1.6 TRIS 3Z00 ICO J=1.10 TRIS 33FII.J)=O. TRIS 34
100 H I.JI=O. TRIS 35DO 110 J=1.6 TRIS 36
110 OII.JI=C. TRIS 37C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••*TRIS 38C FORM INTEGRAL lGIT*ICRZI*IGI TRIS 39C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••*.*••••••rRIS 40
CALL I~TER TRIS 41C TRIS 4Z
VCL=VCL+Xlll) TRIS 43IFl~PP.EQ.OIGO TO ZOO TRIS 44
C TRIS 45C FOR PLANE PROBLEMS TRIS 46C TRIS 41
OIZ.ZI=Xllll*CRlll.l) TRIS 480IZ.31=Xllll*CRlll.41 TRIS 490IZ.~I=OIZ.3) TRIS 50CIZ.6)=XIIII*CRlll.ZI TRIS 51013.3)=XII1l*CR1l4.41 TRIS 5Z013.51=013.3) TRIS 53013.E)=XIIII*CRlIZ.41 TRIS 54015.51=013.31 TRIS 55015.61=013.61 TRIS 56016.61=Xllll*CRllZ.Zl TRIS 57GO TC 210 TRIS 58
C TRIS 59C FOR AXISYMMETRIC PROeLEMS TRIS 60C TRIS 61
ZOO (1101) = XI131 * CRll3.31 TRIS 6Z011.21=XIIZl*ICRlll.31+CRlI3.31) TRIS 63011.3)=XI151*CRlI3.31+XIIZI*CRlI3.4) TRIS 64OII.51=XIIZI*CRZI3.41 TRIS 650II.EI=XIIZ)*CRlIZ.31 TRIS 660IZ.2)=XIIII*ICRlII.11+Z.*CRlll.31+CRI13.31) TRIS 670IZ.31=XI141*ICRlll.31+CRlI3.311+XI111*ICRlll.4)+CRlI3.411 TRIS 68CIZ.5)=XIIII*ICRlI1.41+CRlI3.41 I TRIS 690IZ.61=Xllll*ICRll I.Zl+CR1l2.31) TRIS 70
G-44
CI3.31=Xllbl.CKZI3.31+XIIII.CRll~.41'2 ••XI141.CKlI3.41 TRIS 71013.51=XIIII.CKll~.~I+XI141.CRZC3.41 TRIS 72013.bl-XI141.CRllZ.31+XIIII.CRlIZ.~1 TRIS 73015,51=XI1II.CRll~,~1 TRIS 74015,61=XICII.CRllZ.~1 Tns 15CC6,bl=XIIII.CRlIZ.ZI TRIS 7b
C HIS 77210 00 Z20 1=I.b TRIS 78
[0 22C J=I.b HIS 7~
22001J,II=Otl.JI TRIS 80C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• TRIS 81C fOR~ COEffICIENT-DISPLAC~MENT TRANSfCR~ATIC~ MATRIX TRIS 8ZC•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TRIS 83
CUMM=RRIZI.lllI31-lZ1111+RRlll.IZZI21-ZZ1311+RRI31.111CII-ZZlZ) I TRIS 84C01I.II-IRR1ZI·ZZI31-RRI31.ZZIZII/COMM TRIS 8500(I,21=IRRI31.IZCI)-RR1II.ZZI311/CC~M TRIS 8b00(1.31=IRRIII.ZZCZI-RR(ZI.ZZ(II)/CC~M TRIS 87CDI2,II=CZZlZI-ZZI311/CUMM TRIS 88DDI2.Z)=IZZI31-ZZ1111/COMM TRIS 8~
COI2.31=IZZIII-ZZllII/CUMM TRIS 90OOI3.11=IRRI31-RR1ZII/COMM TRIS 91COI3.ll=CRRIII-KRI311/COMM TRIS 92COI3.31=IRRI21-RRIIII/COMM TRIS 93
C TRIS 94Du 300 1=1.3 TRIS 95J-2.LMIII-I TRIS 9bHll.JI=DDlltil TRIS 97HIZ,J)=ODll.11 TRIS 98HI3.J)=DOI3.11 TRIS 99~14,J+1I=Oull.11 TRIS 100HI5.J+II=00IZ.11 TRIS 101
300 HI6.J+II=0013.11 TRIS 10Zc••••••••••••••••••••••••••••••••••••••••••••• $ •••••••••••••••••••••••••T~IS 103C ROTATE UNKNC.NS If REQUIRED TRIS 104C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TRIS l05
CC 410 J=I.ITR TRIS lObI=L~IJI TRIS 101If(A~GCEIII.Dt.O.1 DC TO 41v TRIS 108;INA=,INIANDLEIIII TRIS 109COSA=COSIANGLUIII TRIS 110IJ=l·1 TRIS IIIGO 4DC K=I.e TRIS liZTE~.=f(K.IJ-II IRIS 113t-IK ..J-II=TEM.CGSA+H(K,IJI·~INA HIS 114
400 HIK.IJI=-Tl~.SINA+HIK,IJI*CCSA TRIS 115410 CO~T1NOE TRIS lib
C**••••** ••*** •• *••***.****************.********************************TRIS 117C FOK~ ELEMENT STifFNESS MATRIX IHI1.101*IHI TRIS 118C•••••••••••••••••_.***.*••• *••••_*.**** •••• __ ••• **********.******••••••TRIS 119
DO 510 J:ldO TRIS IZOCO SID K~I.o TRIS 121IFlhIK.JI.EI;.O.1 vG TU >10 TRIS 12lCO 500 1=1.0 IRIS 123
500 FII,JI=FII.JI+OIl.KI*tllK.JI IRIS Il4510 CCNTINUE IRIS Il5
00 530 1=lde IRIS IlbDU 530 K=I.6 IRIS 127IfIHK.II.l<;.O.1 DC Tr. 53(' HIS 128OU 5lC J=I.IO TRIS 12~
5Z0 SII.J)=SII.JI+HIK,!I*FIK.JI TRIS 130530 CO~TINUE TRIS 131
c••••••••*••**.**••••*.*** •••***.*********.********~******•••***********THIS 132C FOR~ ecuy FCRCE MATRIX TRIS 133c••••••••••••••** ••••********************************************•••• ***TRIS 134
~TYPE=IABSIIXIN.511 TRIS 135IF(~PP.~£.CI GC TO bOU IRIS 13b
C TRIS 137C FUR AXISYMMETRIC PRGeLL~S TRIS 138C TRIS 13~
CU~~=RCIMTYPEI*ANGVEL*'2 TRIS 140
G-45
TPlllzCCMM*XI171 + X1121*TT131 - XI121*PSTlhl TRIS 1~1
TPI2IzCO~M*XI191 + Xllll*ITTlll+TTI311 - Xllll*Z.*PSTINI TRIS 1~2
TPI31-CCMM*XIII01+ XII~I*TTI31 + Xllll*Tll41 - XI141+PSTINI TRIS 1~3
CO~Mz-ROIMTYPEI*ACELl TRIS 1~~
TPI41-COMM*Xllll TRIS 1~5
TPI51-COMM*XI171+Xllll*TTI41 TRIS l~b
TPlbl zCOMM*XI181 + Xllll*TTIZI - Xllll*PSTlhl TRIS 147GO TC blO TRI S 148
C TRIS 149C FOR PLAhE PROBLEMS TRIS 150C TRIS 151
bOO ACELR z ANGYEL TRIS 152COMM z -ROIMTYHI * ACELR TRIS 153TPIII - COM~ • XlIII TRIS 154TPI21 - CUMM * XII71 + XliII. HIli - XlIII * PSTlNI TRIS 155TPI31 z COMM. XI181 + XlIII. HI41 TRIS 15bCOMM - -ROIMTYPEI • ACEll TRIS 157TPI41 = COMM • XliII TRIS 158TPI51 = COMM • XI171 + Xliii. TTI41 TRIS 159IPlbl = COMM • Xl181 + XlIII. III21 - XlIII + PSTlh) IRIS IbO
610 00 620 I = 1,10 TRIS IblDO 620 K=l,b TRIS Ib2
620 PITlzPIII+HIK,II*TPIKI TRIS Ib3C TRIS 164C FORM STRAIN TRANSFORMATION MATRIX TRIS Ib5C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••TRIS 166
DO 7CD 1-1,6 TRIS Ib700 100 J=I,IO TRIS Ib8
100 HHII,JI=HHII,JI+HII,JI TRIS Ib9C TRIS 170
RETURN TR lSI 7IEND HIS 112
G-46
SUBROUTIhE IhTE~ IhTE 1C.· ••••••••••••••••• • ••••••• •••••••••••••••••••••••••••••••••••••••••••• [NTE 2~ THIS S~BROUTINt PERfCRMS A ~UMERICAL I~TEGRATION OF NECESSARY INTE 3C FU~CTIGhS THROUGH THE VOLUME OF THE TRIANGULAR ELEMEhT INTE 4C.····.······ · ·.·...•...........................•........... 1NTE 5
COUBLE PRE~ISION XM,R.AREA.Z,XX INTE 0COUBLE PRE~ISION ~RZ.XI.RR.ZZ.S.RRR.ZZZ INTE 1COMM(N/ARG/RRRI51.ZZZI51.RR(41.ZZ141.SIIO.IGI.~RZI4.41.XIIIOI. INTE B
I PIIOI.TT(41.Hlb.IOI.HH(0.ICI.ANGLEI41.SIGI101.EPSllCI.N INTE 9COM~CN/PLANElNPP INTE ICDIMEhSION XN/ll.RI71.Z171.XXI91 INTE 11DATA XX/3*.125,j91805448.3*.1323941521B84 •• 22500000CCOOO. INTE 12l.b90140418028.-.4104.01~2314/ INTE 13
C IhTE 14Rlll=(RRlll+RRI21+RRI311/3. INTE I~
1111-IZ111l+ZZ121+ZZ13II/3. INTE 10AREA=.5*(RRIII*IZZI21-ZZ1311+RRI21*IZZI31-ZZIIII+RR(31*IZZIII-ZZI2INTE II
1111 INTE 18C INTE 19
IFI~PP.NE.OIGO T~ 600 INTE 20C INTE 21
DO ICO 1=1.3 INTE 22J=I+3 INTE 23RIII=XXI81*RRIII+II.-XXI811*PI11 INTE 24PIJI=XXI91*RRIII+II.-XXI911*PI11 INTE 25Z(II=XXI81*ZZIII+II.-XX(811*Zll) INTt 20
100 ZlJI=XX(91*ZlIll+Il.-XXI911*ZI11 INTE 21C INH 28
DO <00 1=1.1 INTE 29200 XMIII=XXIII*RIII I~TE 3C
C INTE 31DC 300 1=1,10 IhTE 32
300 XI III=C. INTE 33~ INTE 34
CO 4CC 1=1.1 INTE 35XIIll=XIIll+XMIII INTE 30XII21=XIl21+XMIIIIRIII INTE 31XII31=XIl31+XMIII/IRIII**ZI INTE 38XI141=XIl41+XMIII*ZIII/RIIl INTE 3,XI151=XIl51+XMIII*ZIIIlIR(II**ZI INTE 40XI161=XIl61+XMIII*IZIII**21IlRIlI**21 INTE 41XII1I=XIIlI+XMIII*RIlI INTE 42XI181=XI181+XMIII*ZIII INTE 43XI(SI=XIl9l+XMIII*IRII)**21 INTE 44
400 XIIlCI=XIIlO)+XMIII*RII)*ZIII INTE 45C INH 46
DO 500 1=1.10 INTE 41500 XIIl'=Xlll'*AREA INTE 48
~ INTC 49C INTE 50
RETURN INTE 51600 XlIII AREA INTE 52
XI111 = ~111 * AREA INH 53XI181 = Zlll * AREA INTE 54RETURN INT. 55END INTE 50
G-47
SUB~CUTlhE MODIFYlhEQ,h,UI MODI 1, •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••"001 2C THIS SUB~OUTINE INCORPORATES DISPLACEMENT BOUNDARY CONDITIONS BY MODI 3C SYMMET~ICAL MODIFICATION OF THE BODY STIFFNESS MATRIX AND LOAD MODI 4C VECTOR. MODI 5C MODI 6
DOUBLE PRECISION A,B MODI 7CDMMCN/SOlVE/BIIOOI,A1I00,501,NUMBLK,MBAND IIODI B
C 11001 9DO 10 II-Z,MBAND 11001 10K-I\-II+1 11001 11IF1K.LE.01 GO TO 5 MODI 12B1KI-BIKI-AIK,III*U MODI 13
,AI K.II)=O. 11001 145 K-"+"-I MODI 15
IF1NEQ.LT.KI GO TO 10 11001 16BIKI-BIKI-AIN,III*U MODI 17AIN,III"O. 11001 IB
10 CONTINUE 11001 19C 11001 20
AIN,lI=I. 11001 2181"'-U MODI 22
C 11001 23RETURN MODI 24END MODI 25
G-48
S~BRC~TINE SYMINvIA,NMAXI SYMI Ic ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 5y"[ 2C THIS S~BROUTINE INVERTS A SYMMETRIC ~ATRIX (F CRoER ~~AX SYMI 3c ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• Sy"1 4
COUBLE PRECISION A,O SY'Ol 5DIMEhSICh AINMAX,NMAXI SYMI c
C SYI'I 1DO 3CO N:I,NMAX SYMI 8
C SYMI 9D=AIN,NI SYMI 10DO lOa J=I,hMAX SYMI 11
100 AIN,JI=-AIN,JI/D SYMI 12C SYMI 13
CO 210 Isl,NMAX SYMI 14IFlh.EQ.II GG TO 210 SYMI 15CO 200 JsI,NMAX SYMI 16IFIN.NE.J' AII,JI=AII.JI+AII,NloAIN.JI SYMI 11
200 CONTINUE SYMI 18210 AI I,NI=Al I,NI/O SYMI 19
C SYMI 20300 AlN,~I:l./O SYMI 21
C SYMI 22RET~RN SYMI 23END SYMI 24
G-49
SUBR(UTI~E SOLV SOLV 1C••••••••••••***•••••••***.**••••••****.**•••••***••**••••••••••••••••••SOLV 2C THIS SUBROUTINE SCLVES A SET OF BANDEC LihEAR SIMULTANEOUS SOlV 3C EQUATIONS ~y GAUSS ELIMINATION. SOL V ~
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••SOlV 5COMMCN/BASIC/NU~~P.NU~EL.NU~PC.hUMSC.ACELZ.ANGVEL.1RBF.VOl.IFREQ SOlV 6COUBLE PRECISION A.B.C SOL V 7COM~CNISCLVE/BIIOOI.AII00.501.NUMBLK.MBAhD SOlV B
C SOlV 9MM=I'BAND SOL V 10NN=50 SOLV 11hL=Nh+ 1 SOL V 12hH=M+hN SOL V 13RE~ INO 1 SOlV HRE~aD 2 SOLV 15NB=O SOLV 16GO TC 120 SOLV iT
C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• SOlV 18C REDUCE EQUATIONS ~y BLCCKS SOlV 19C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••SOLV 20C SOlV 21C 1. SHIFT ~LCCK OF EQlATIONS SOlV 22C SClV 23
100 NB=~B+l SOL V 2~
CO 110 h=I. h~ SOLV 25NM=~h+h SOL V 26BI~I=BlhMI SOLV 218INMI=0. SOlV 2800 110 ~=I.~M SOLV 29AI~.~I=A(NM.MI SOLV 30
110 AINM.MI=C. SOL V 31C SOlV 32C 2. READ NEXT BLOCK OF EQlATIONS INTC CCRE SOLV 33C SOLV 3~
IflhUMBLK.U;.NBI GU TO 130 SOLV 35120 REACIZI IBINI,(AIN.,41.M=I.MMI.N=NL.NHI SOLV 36
IFINB.EQ.OI GO TO 100 SOLV 37C SOL V 38C 3. REDUCE BLOCK OF EQUATIONS SOLV 39C SOLV 40
130 00 160 N=I.NN SOLV 41IfIAIN.l'.EQ.O.1 GO TO 160 SOLV 42Blhl=BINI/AIN.11 SOLV 4300 150 L=2.MM SCLV 44IFIAIN.LI.EQ.O.I GO TO 150 SULV 45C=AIN.LI/AIN.11 SOLV 46I=N+L-l SOLV 47J=O SOL V 48DO 140 K=l.I'M SUL V 49J=J+l SOL V 50
140 AII.JI=AII.JI-C*AIN.KI SOLV 51BIII=Bll'-AIN.LI.BINI SOLV 52AIN.L)=C SCLV 53
150 COhTINlJE SOLV 54160 CChTiNUE SOLV 55
C SOLV 56C 4. ~RITE BLCCK OF HECUCEC EQUATIONS CN FORTRAN UNIT 1 SOLV 57C SGLV 58
Ifl~UMBLK.EQ.NBI GO TO 200 SOLV 59WRITE III IBINI.IAIN.MI.M=2.MMI.N=I.Nhl SCLV 60GO TC lOG SOLV d
C•••••••••••••••••••••••••••••••••••••••••••••••••****•••••************.SOlV 62C BACK-SUeSliTUTION SOLV 63C••••••••••••••••••••••••••••••••••**•••••••••••• ***••**** •••*•••***••**SOlV 64
200 DO 22G M=I.NN SOLV 1>5N=hN+I-M SOLV 66DO 210 K=2.MM SOLV 67L=N+K-l SOLV 68
210 BINI=BINI-AIN.KI*BILI SOL V 69N~=N+NN SUL V 70
G-50
BI~M)=BI~) SOlY 11220 ftINM,~BI=B(~1 SOlY 72
~B=~B-l SOlY 73IFINB.EQ.OI GU TO 30e SOlY 7"BftCKSPACE I SOlY 75READ III (B(N),(ftIN,M),M=Z,MMI,N=I,~~1 SOlY 7bBACKSPACE I SOlY 77GO TG 200 SOlY 7B
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••SOlV 19t CRCEP FtRMER UNKNl.~S IN B ftRRAy SOlY 80C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••SOlV 81
300 K=C SOlY 8200 310 NB= I, NUMBlK SGl Y 8300 310 N=l,~~ SOlY B"NM=N+NN SOlY 85K=HI SOlY 8b
310 8IK/=AINM,N6) SOlY 87C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••SOlV 88C ~RITE SOlUTlCN SOlY 8~
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••SOlV 90MPRINT=O SOlY ~l
00 410 N=I,NUMNP SOlY ~2
IFIMPRINT.NE.O) GO TC 400 SOlY ~3
~RITE Ib,2000) SOlY ~4
MPRINT=40 SOlY 95"00 MPRINT=MPRINT-I SOlV 9b410 NRITE Ib,20011 N,BIZ*N-ll,BI2*NI SOlY 91
RE TURN SOL V ~8
C SOlY 992000 FORMAT 113Hl NODAL PCI~T,lBX,2HlR,IBX,2hlZ) SOlV 1002001 FORMftT 1113,2E20.7) SOlY 101
ENC SOlV 102
G-51
SUBRGUTI~E STRESS STRE IC•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••STRE 2C THIS SLBROUTINE SeLVES FCR ELEMENT STRESSES ANO STRAINS ANQ STRE 3C WTPUTS THESE RESULTS STI<E 4c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••STRE 5
COHMeN/eASIC/NUMNP.~UHEL.NUHPC.~UHSC.ACELl.ANGVEL.TREF.VOL.IFREQ STRE 6COHHeN/HATP/ROI61.AGFTSI61.EI12.22.61.EEI211.POROTYI61 STRE 7COHHCN/NPDATAIRI100DI.CGeEIIOOOI.XRII0001.111DDOI.XlI1OOOI.TIIOOOISTRE BCOMMON/ELOATA/IXIIOOO.51.EPRI10001.ALPHAIIOCOI.PSTIICO01 STRE 9DOUBLE PRECISION A.B SIRE 10DOUBLE PRECISION CRl.XI.RR.ll.S.RRR,lll STRE IIDOUBLE PRECISION RO.XI.X2.X3.YI.Y2.Y3.Y4.SUH.UI.VI.SI~A.CGSA.U.V. STRE 12
I RI.ll.C STRE 13COMMCN/ARG/RRRI51.11l151.RRI41.1l141.SII0.IOI.CRlI4.41.XIIIOI. STRE 14
I PIIOI.TTI41.HI6.IOI.HHI6.IOI.ANGLEI41.SIGII0I,EPSIICI.N sTRE 15COMMCN/SOLVE/BIIOOI.AIIOC.501.NUHBLK.HBANO STRE 16COHHCN/CONVRG/IPQONE.ITCOC~.NNLA.NT(A.NTITEP.OLOSIGI41 sTRE 17COMHCN/PLANE/NPP STRE 18OIHENSICN TPl61 sTRE 19
c STRE 20C INIT IALilE STRE 21c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• STRE 22
XPE=C. STRE 23XKE=C. STRE 24HPRINT=Q SIRE 25ERRCR=.005 STRE 26IPoeNE=1 STRE 27HcceN=1 STRE 28REklNO 3 STRE 29CALL RESTII) STRE 30
c••••••••••••••••••••••••••••••••••••_••••••••••••••••••••••••••••••••••STRE 31C CALCULATE ELEMENT STRESSES STRE 32c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••***•••• STRE 33
CO 2CO N=l.NU~El STRE 3~
C STRE 35IXl~.51=IABS(IX(N.511 STRE 36
C STRE 37If.IIFREQ.EQ.OI CAll ~PROP STRE 38IFlIFREQ.EQ.1) CALL QUAD SIRE 39
C STRE 40MTYPE=IABSIIXIN.511 STRE 41IXIN.51=IABSIIX1N,511 STRE 42IfIIFREQ.EQ.II GC TO 30 sTRE 43
C STRE 441=IXlN.1) STRE 45J=IXI~.21 STRE ~6
K=IXIN.31 sTRE 47L=IX1N.41 STRE 48RRRllI = R( II STRE 49RRRI21 RIJI STRE 50RRR(3) RIKI STRE 51RRR(4) = RIll STRE 52111111 = lill STRE 53111121 IIJI STRE 54111131 = llKI STRE 55111(4) = llli STRE 56RRRI51 IRRRl11 + RRRl21 + RRRl31 + RRR(4)1 I 4. sTRE 57lZZISI = IZZllll + ZZll21 + lZZDI + ZZl(411 14. STRE 58
C STRE 59RO = O. STRE 60Xl = O. STRE 61X2 = O. SIRE 62X3 = O. STRE 63Yl = O. STRE 64Y2 = O. STRE 65Y3 = O. sTRE 66Y~ = D. sTRE 67SUM = O. STRE 68
C SIRE 6900 2e 1=1.4 STRE 70
G-52
CALCULATE ENERGY TERMS FOR FREQUENCY APPROXIMATION
CO 50 1=1.2RRIIl=PlI*BIDG 50 K=I.8
50 RRlIl=RRII)-SII+8,KI*PIKI
RRIlI=HIZIRRlZI=TPI61RRI31=ITPIII+TPIZI*RRRI51*TPI31*ZZZI511/RRRI51RRI41=TPI31+TPI51
30 00 40 1=1.4II=Z'IJJ=Z*I XI N.llPIII-II=BIJJ-II
40 Pllll=BIJJI
STRE 71STRE 12STRE 13STRE 14STRE 15STRE 16STRE 11STRE 18STRE 79STRE 80STRE 81STRE 8ZSTRE 83SIRE 84STRE B5SIRE 86SIRE 81STRE 88STRE 89STRE 90STRE 91STRE 92STRE 93STRE 94STRE 95SIRE 96STRE 97STRE 9BSTRE 99STRE 100SIRE 101SIRE 10ZSTRE 103STRE 10ltSTRE lOSSTRE 106STRE 107STRE 108STRE 109STRE 110STRE IUSTRE I1ZSIRE U3STRE U4STRE USSTRE U6STRE U1STRE UBSTRE 119SIRE lZOSTRE lZISTRE 122STRE 123STRE 124STRE 125STRE 126STRE 121STRE 128STRE 129STRE 130STRE 131STRE 132STRE 133STRE 134STRE 135STRE 136STRE 131STRE 138STRE 139STRE 140
* RRIZl*CRZIZ,31 * RR141*o.
= -IRRlll*CRZII,31I CRZl3,3)
IFlhPP.NE.ZI GO TO 110CI3=CRZII,31CZ 3=CRZl Z,31
D = Xl * x3 - X2*.2RR I 11 = (X3 • Yl - X2 • Y21 I URR(.21 = IXI • Y3 - X2 • Y41 I 0RRl31 = SUM I RURRI41 = IXI • Y, • X3 • Y4 - X2 • IVl * nil I 0GO TC 100
11 = IXI~oIl
UI=BIZ*II-I IVI=BIZ*IIIIFICCOEI III.GE.O.l GO TO 10SlhA=SlhICGDEIIII/,7.2957HICOSA=CCSICODEIIII/57.2957HIO=UI*CGSA-VI*SINAV=VI*COSA*UI*SINAUI:l::UVI=V
10RO=RC'RIIIISUM = SUM * UIRI = RRRll1 - RRRl51ZI = ZZZIII - ZZZI;1Xl=Xl+RI··2XZ=XZ*RI'ZIX3=X3* Z1**2YI=Yl*RI*UIYZ=YZ*ZI*Uln-Y3*z I*V I
ZO Y4=Y4+RI*VI
COMM=SI9,91*SIIO,101-SI9,101*SIIO,91IFICOMM.EQ.O.IGO TO 60PI91=ISIIO,101*RRIII-SI9,101*RRIZII/CC~M
PIIOI=I-SIIG,91*RRI11*SI9,91*RRIZJI/CC~~
60 00 70 1=1,6TPIII=O.DO 70 K=I,IO
10 TPIII=TPIII*HHII,KI*PIKI
DO 90 1=1,10COMM~O.
00 BO K=I,lO80 COMM=COMM+SII,Kl*PIKI90 XPE=XPE*COMM*Plll
XKE=XKE*VOL*ROIMTYPEl*IPI91**Z*PI101**21
100 IfINFP.EQ.II RRI31IFINPP.EQ.ZI RRI31
1 CRZ13.41 - TT(31)
C
C
C
C
C
CCC
C
G-53
CCC
CCCCC
CCC
CCC
CCC
C43=CR1I4,31C33=CR1I3,31
110 00 130 1=1,4SIGI II=-TTI IIIFINPP.EQ.2.ANO.I.EQ.31 SIGlll=O.IFINPP.EQ.2.ANO.I.EQ.3) GO TO 130DO 120 K=I,4IFINPP.EQ.2.ANU.K.EQ.31 CRZII,KI=O.CRZIK,II=CRZII,KI
120 SIGlll=SIGII)+CRlll,KI*RRIKI130 CONTINUE
IFINPP.EQ.II CRlI3,31=1.
CALCULATE P~INCIPAL STRESSES
CC=ISIGlll+SIGI211/2.66=ISIGlll-SIGI211/2.CR=SQRTI88**2+S1GI41**21SIGI51=CC+C~
SIGI61=CC-CRSIGI11=2B.64789*ATAN21SIGI4I,B81IFIA6S1S1GI411.LE •• IE-61 SlGI11=0.CALCULATE PRINCIPAL STRESSES AT ANGLE C~EGA TO SIGP
ITH1S STEP IS REUU1REO BECAUSE C~EGA IS NCT NECESSARILY THEANGLE TO SIG(5)1
CMEGA=SIGI71/51.29578SIGP=CC+B8*COSI2.*OMEGAI+SIGI41*SINI2.*C~EGAI
SIGQ=CC-8B*COSI2.*C~EGAl-SIGI41*SINI2.*C~EGAI
SIGT=SIGI31
CALCULATE STKESSES I~ M-N SYSTEM
SIGI81=SIGlll*COSlALPHAlNll**2+SIGI21*SINIALPHAINII**21+2.*SIGI41*SINlALPHAlNII*COSlALPHAINII
SIGI91=SIGlll*SINlALPHAINll**2+SIGI21*(CSIALPHAINII**21-2.*SIGI41*SINIALPHAINII*COSIALPHAINII
SIGI 101=ISIGI21-SIGllll*SINIALPHAINII*(OSIALPHAINI I1+SIGI41*ICOSIALPHAINII**2-SINIALPHAINII**21
TEST fCR CONVERGENCE OF TENSION - COMPRESSION APPROXIMATIONS
IFINTCA.LE.ll GO TO 160IFINTITER.LE.11 ITCOON=OIFINTITER.LE.11 GO TC 16000 150 1=1,4SIGKK=O.00 140 J=l,4
140 SIGKK=SIGKK+ABSISIGIJIIIflABSISIGIII/SIGkKI.GT •• lE-3.ANO.ABSIISIGIII-OLOSIGIIIi/ISIGIII+
I(LOSIGIIIII.GT •• IE-ll ITCOON=O150 CONTINUE
CALCULATE PLASTIC MODULUS RATIO
160 IfINNLA.LE.l1 GO TO 110IFIEEI181*EEI191*EEI201.EQ.0.I GO TO 110ANG-ISIGI11/51.295181-ALPHAINIS2-SINIANGI**2C2-CCSIANGI**2Rl=SIGI51/IEEI181*C2+EEI191*S21R2=SIGI61/IEEI181*S2+EEI191*C21R3-SIGI31/EEI201ESIG=SQRTI(IRI-R21**2+IRI-R31**2+IR2-R31**2)/2.1CLOEFR=EPRINISTRAIN-ESIG/EPRINIEPRINI=I.IfISTRAIN.GT.l.1 EPRINl=11.+EEI211*ISTRAI~-1.11/STRAIN
CCNVER=EPRINI/OLOEPR
G-54
STRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESIRESTRESTRESTRESTRESTRESTRESTRESTRESIRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRESTRE
14114214314"H51"61"114814915015115215315"15515615115815916016116216316"165166161168169110111112113111011511617111817918018118218318418518618118818919019119219319419519619719819920020120220320420~
206207208209210
IFIABSICGNVER-I.I.~T.ERRL~1 IPOCNE-C STRE 211C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• STRE 212C ~RITE STR"SSES STRE 213C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• STRE 214
170 IFI~PRINT.NE.O) Gu TO lac STRE 215.RITE 16.2000) STRE 216~PRINT-40 SHE 217
180 ~PRINT-~PRINT-I STRE .218.RITEI6.20011 N.R~RI51.ZZZI51.ISIGlll.I.I.101 STRE 219.RlTEIIOI RRRI51.lZl151.1SIGIII .1-1.101 .C~EGA.SIGP.SH;.l.SIGT STRE 220
c••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• STRE 221C CALCuLATE ~ECHANICAL STkAINS STRE 222c•••••••••••••••••••••••••••••••••••••••••••• 9 ••••••••••••••••••••••••••STRE 223
CALL SY~INVICRl.41 STRE 224C STRE 225C CALC~LATE STRAINS IN R-Z SYSTEM STRE 226C STRE 227
DO I~O 1'1.4 STRE 228EPSIIl~O. STRE 229DO I~O K=I.4 STRE 230
190 EPSlll=EPSlll+CRZII.KI*SIGIKI STRE 231IFINPP.EQ.21 EPSI31--ICI3*EPSlll+C23*EPSI21+C43*EPS(4)I/C33 STRE 232
C STRE 233C CALCULATE PRINCIPAL STRAINS STRE 234C STRE 235
EPS~2~EPS(4)/2. STRE 236CC~IEPSI1I+EPSI211/2. STRE 237BB~IEPSll1-EPSI2)1/2. STRE 23BCR~SQRTl BB"2+EPS42**2 I STRE 239EPSI51-CC+CR STRE 240EPSI61=CC-CR STRE 241EPSI71=2B.647B9*ATAN2IEPS42.BBI STRE 242IFIABSIEPS421.LE •• IE-61 EPSI71=0. STRE 243
C STRE 244C CALCULATE STRAI~S IN N-S SYSTEM STRE 245C STRE 246
EPSI81-EPSII1*COSIALPHAINII**2+EPSI21*SINIALPHAINlloo2 STRE 247I+EPSI4IoSINIALPHAINII*COSIALPHAINII STRE 248EPSl~)=EPSll)*SINIALPHAIN11*02+EPSI210CCSIALPHAINIIO*2 STRE 249
l-EPSI41*SINIALPHAINII*COSIALPHAINII STRE 25CEPSII01=2.*IEPSI21-EPSIIII*SINIALPHAINI1*COSIALPHAINII STRE 251
I+EPSI4IoICOSlALPHAINII002-SINIALPHAINII*021 STRE 252200 .RITEII01 IEPSlll.I=I.101 STRE 253
c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••*.**STRE 254C .RITE STRAINS IN PERCENTAGE FOR~ STRE 255c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• *STRE 256
RE.INC 3 STRE 257CALL RESTl41 STRE 258~PRI~T=O STRE 259DO 320 N-l.NUMEL STRE 260IFl~PRINT.NE.OI GO Te 300 STRE 261.RITEI6.20021 STRE 262MPRI~T=4C STRE 263
300 ~PRI~T-MPRINT-l STRE 264REAOll01 RRRI5).ZII151.ISIGI11.1=1.101.C~EGA.SIGP.SIGQ.SIGT STRE 265REACIIOI IEPSIII.I=I.I01 STRE 26600 310 J-l.l0 STRE 267IFIJ.~E.71 EPSIJI-l00.*EPSIJI STRE 268
310 CONT INUE STRE 269.RITEI31 OMEGA.SIGP.SIGQ.SIGT.ISIGIII.I=I.41 STRE 270
320 .RITEI6.20031 N.IEPSIII.I=I.IOI.TINI STRE 271RE.no 3 STRE 272
c•••••••••••••••••••••••*••••••••••*••••••••• *•••••••••*•••*••••********STRE 273C CALCLLATE A~O .RITE APPROXI~ATE FU~OAME~TAL FRE'UE~CY STRE 274c•••••••••••••••••••••••••••••••••••••••••••••••••••*.*••••**.*.*.******STRE 215
IfIIFREQ.EQ.OI RETURN STRE 270.=S'RTIXPE/XKEI STRE 277~RITEI6.20041 • STRE 278RETURN STRe 279
C SIRE 28C
G-55
2000 FCR~AT IlC9hl tL " I SI(MAR SIGMAI SIGHAT SIGHSTRE 181IARZ SIGMA~AX SIGMAMIN ANGLE SIGMAM SIGMAN SIGMAMNI STRE 282
2001 FORMAT Ilb,lX,lF7.2,t'9.0,F7.2,3F9.01 STRE 2832002 FORMAT (98HI tL EPSR EPSI EPST EPSRI EPSHAX EPSMIN STRE 28~
I ANGLE tPSM EPSN EPSMN TEMPERATUREl STRE 2852003 feRMAT IIS,bF8.3,Fd.2,3FB.3,FI3.01 STRE 286200~ FOR~AT 136HCAPPRJXIMATt FUNDAMENTAL fREQUENCY =EI2.Sl STRE 287
ENe STRE 288
G-56
SUBROUTI~E PlTMIITYPEI PlTM IC•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••**••PlTM 2C THIS SUBROUTINE PlCTS ThE FINITE ElEME~T MES~ PllM 3C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PlTM 4
COMMCN/BASIC/NUMNP.NUMEl.NUMPC.NUMSC.ACEll.A~GVEl.TREF.VOl.lfREQ PlTM 5CO~MON/NPDATA/RIIDOOI.CODEI10001.XRII000l.ZII0001.XZIIOOOI.TIIOOCIPLTM 6COMMCN/ElOATA/IXIIOOO.51.EPRI10001.ALPHAI10001.PSTI1CO01 PLTM 7COMMCN/PTT/IPlOT.TITlEI201.RMIN.ININ.CElP.TllT.FACT.ICEF PlTM 8CIME~SION XI51.YI51.0UMI51.0UMI151 PLTM q
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PlTM IeC PLOT AEROSPACE HEADING AND TITLE PlTM IIc•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PlTM 12
DATA DUM/2 ••4 •• 6 •• 8 •• 10.I.DUNl/5.0.1 PLTM 13IFIITYPE.EQ.21 GO TO 50 PLTM 14CAll PLTfl,15,O.,2.,CUM,O.,1,O.,2.,8,O,CUMl,O,O,O.o.O,OJ PlYM 15CALL PlTI2.0.11 PlTM 16CAll PlTI3.5C.200 •• 175.80.TITlEI PLIM 17CAll PlTI4.500.500.3.1 PlTM 18CAll PlTI5.10.PlOTIMI PlTM Iq
, ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• PlTM 20C OBTAIN MAXIMUM AND MINIMUM RAND Z PlTM 21c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PlTM 22
50 IFtDElP.E~.O.1 GO TO 100 PlTM 23ZMAX=ZMI~ Pl TM 24RMAX=~MIN PllM 25GO TO 110 Pl TM 26
100 lMlh=ZIlI PUM 27ZMAX=ZMlh PLTM 28RMIN=~(l1 PlTM 2qRMAX=RMIN PLTM 30
C PUM 31110 DO 120 I=I.NUMNP PlTM 32
IFIZIII.lT.ZMINI ZMIN=ZIII PlTM 33Ifll~AX.ll.lI111 ZMAX=lI11 PUM 34IFIRIIi.LT.RMINI RMIN=Rlll PlTM 35IFIRMAX.ll.Rllll RMAX=Rlll PUM 36
120 CONTINUE PLT M 37C PUM 38
RA=UMAX-RMINlIllMAX-ZMINI PLTM 3qIFIiPLOT.EIl.ll RA=I./RA PlTM 40IFIRA.lT.I.1 GO TO 150 PlTM 41TIlT=I. PUM 42
c ~ •••••••••••••• PlTM ~3
C I-AXIS VERTICAL ANO R-AXIS HCRIIC~TAl PlTM 44c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PLTM 45
IFICElP.E~.O.IDElP=IZMAX-lMINl/IO. PlTM 46CAll PlTII.15.RMIN.OElP.X.0 •• I.ZMIN.OElP.0.I.Y.12.12~ElEMENTPlOT.PlTM 47
16.6hR-AXIS.6.6Hl-AXISI PlTM 48DO 140 N-I.NUMEl PlTM 4900 130 M=I.5 Pl TM 50I=IXIN.MI PlTM 51IFIM.EQ.51 1=IXI",Il PlTM 52XIMI=RIIl PlTM 53
130 YIMI-lIll PUM 54140 CAll PlTIZ.5.11 PlTM 55
GO TO 180 PUM 56c•••••••••••••••••••••••••••••••••••••••••••••••••••*•••••••••••••••••••PLTM 57C R-AXIS VERTICAL ANO Z-AXIS HORIZONTAL PlTM 58c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••PlTM 59
150 IFIOElP.EQ.O.1 OElP=IRNAX-RMINI/IO. PlTM 60TlLT-O. Pl TM 61CAll 'lTII.15.ZNIN.OElP.X.0 •• I.RMIN.OElP.0.I.Y. 12.12htlEMENT PlOT.PlTM 62
16.6HZ-AXlS.6.6HR-AXISI PlTM 63C PlTM 64
00 170 N= I. NUMEl PLTM 65DO 160 M=I.5 Pl TM b6I-IXIN.MI PLTM 67IFIM.EIl.51 1=IXI",I1 PlTM 68XIMI=ZIII PLTM 6q
160 YIMI=Rlll PLTM 70
G-57
170 CAll PLT12,5,l1 PlTM 11C PUM 12
180 CAll PlTC5.30,PLLTI~) PUM 13C PLTM 14
RETWRN PLTM 15END PUM 16
G-58
S~HRCUTINE CUNTRIIST.RTI CONT Ic·········.··.......•....•...........•.....••.....•.••..•.•••••....•...•CONT 2C THIS S~8ROUTINE SETS UP THE DATA FOR ORA"I~C CONTOUR PlCTS OF CO NT 3C FUNCTION VALUES. CaNT 4c······· · CONT 5
CO~~CN/8ASIC/NU~NP.NUMEL.NU~PC.NUMSC.ACELl.ANGVEL.TREF.VOL.IFREQ CONT 6CO~~CN/NPOATA/RII0001.CODEIIOOOI.XRI10001.lI10001.XtI1OOOI.TIIOOOICUNT 7COMMON/ELOATA/IXIIOOO.SI,EPRII0001.ALPHAIICCOI,PSTIICO01 CO~T 8CUM~ON/PTT/IPLOT.TITLEI201.RMIN.IMIN.DELP.TllT.FACT.ICEF CaNT 9CO~MON/IU/IMINIIOOI.IMAXIICOI,JMINI2SI.JMAXI2SI.MAXI.MAXJ.NMTL.N8CCONI 10DOu8LE PRECISION CRl.XI.RR.lI.S.RRR.llZ CaNT IICO~MCN/ARG/RRRI51.ZIZISI.RRI41.IZ141.SIIO,ICI.CRZI4.41,XIIIOI. CONT 12
1 PIIDI.TTl4I,HI6.101.HHI6.1CI.ANGLEI4I,SIGII0I,EPSII0I,N CONI 13OOU8LE PRECISION .NS CaNT 14COMMGN/SOLVE/ANSI20001."12S.1001 CaNT 15OIME~SIGN XTI201.VTI201.CISOI.A813I,CI12SCI.COI2S01 CaNT 16CIMENSIGN HE.OIS.221.TlISI.T215I,T3ISI.T4151.TSISI.T6ISI.T7151. CO NT 17
IT8ISI.T9151.110ISI.Tl1ISI,TI2ISI.T13ISI.TI4ISI.T15151.T16151. CaNT 182TI71SI,T181SI.TI9ISI.T20ISI.T21151.T22151 CONT 19DIME~510N XIZS.IOOI,VIZS.I00I,DUMI50001 CaNT 20EQUIVALENCE IIXll.ll.DUMllll.IOUMlll.Xll.11I,IDUMI2S011.Vll.111 CaNT 21EQUIVALENCE IHEADll.11.TlI11I,IHEADll.21.T21111.IHEACll.31.T31111.CaNT Z2
IIHEADll.4I,I41111.IHEADll.51.TSllll,IHEADll.61.T61111.IHEAOll.71. CO NT 232T71111.IHEAOll,8I,T81111.IHEADll.91.T91111.IHEAOll.ICI.TIOllll. CaNT 2431HEADll.111.TllI111.IHEAOll.1ZI.TlZllll.IHEAOll.131.T131111, CaNT 254IHEADll.141.T141111.IHEAOll.151.T151111.IHEAOll.161.T101111. CONT 20SIHEAOll.171.TI71111.IHEAOll.181.TI8111I,IHEADll,191.T191111.IHEADICONT 2161.201.TZOllll.IHEADll.Z11.T211111.IHEADll.2ZI.T221111 CDNT 28
GATA XT.YTl40 04H I CaNT 29DATA Tl 14HSIGM.4HA R .4H .4H .41- I CONT 30DATA T2 14HSIGM.4HA I .4H .4H .4H I CONT 31DATA 13 14HSIGM.4HA T .4H .4H .41- I CaNT 32DATA 14 14HSIGM.4HA RZ.4H .4H .4H I CaNT 33DATA Ts 14HSIGM.4HA ~A,4HX .4H .4H I CaNT 34DATA T6 14HSIGM.4HA ~1.4HN ,4H .4H I CaNT 35DATA 11 14HANGL.4H£ TO.4HSIGH,4HA ~A.41'1~ I COtH 36DATA T8 14HSIG~,4HA N .4H ,4H .41- I CaNT 37DATA T9 14HSIGM.4HA S .4H .4H ,41'1 I DO NT 38DATA TI0/4HSIGM,4HA NS.4H .4H .4H I CaNT 39DATA T11/4HEPSI.4HlON ,4HR .4H .4H I CaNT 40DATA T12/4HEPSI.4HlON .4HZ .4H ,41'1 I CONT 41DAT. TI3/4HEPSI.4HLON .4HT .4H ,4H I CONT 42DATA TI4/4H£PS1,4HLON ,4HRZ ,4H .4H I CONT 43DATA TI5/4HEPSI.4HLON ,4HMAX .4H .4H I CaNT 44DATA T16/4HEPSI.4HLON ,4HMIN .4H .41'1 I CONI 45DATA T17/4HANGl.4HE lG.4HEPSI.4HLON ,4H~AX I CaNT 46DATA TI8/4hEPSI.4HLON ,4HN .4H ,41'1 I CaNT 47DATA TI9/4HEPSI.4HlON .4HS .4H .41- I CONT 48GATA T20/4I'1EPSI,4HLON ,4HNS ,4H ,41'1 I CONT 49DATA T21/4HTEMP.4HERAT.4HURE .4H .4H I CONT 50DATA TZ2/4HPORE.4H PRE,4HSSUR.4HE .4H I CaNT 51IFIISTART.EQ.21 CALL RESTI21 CDNT 52IFIISTART.NE.21 CALl RESTl41 CaNT 53
c••••••••••••••**••••••••••••••••••••••••••••••••••*•••••••••••••••** •••CONT 54C PLOT THE DEFOR~ED MESH CONT 55C4~••• $ ••••••••*** ••*••**.**.*•••••••••••••••••••••*••*••••••*••••••••••CO~T 56
IFIIDEF.NE.11 GO TO 200 CONT 57K=D CONT 5800 100 1=I.NUMNP CONT 59K=K+l COH 60RIII=RIII+ANSIKIOFACT CaNT 61K=K+ 1 CONI 62
100 ZIII=ZlI I+ANSIK'*FACT CONT 63CALL PLTMIZI CONI 64c···· · *••••••••••••CCNT 65
C READ CONTROL INfORMATION AND TRANSFER STRESSES AND STRAINS FRG~ CONT 66C FORTRAN UNIT 10 TC FCRTRAN UNIT 1 CONT 67c··.··..·..· CONT 68
200 READI9.1COOI N~~Pl CaNT 69IFIN~~PL.EQ.OI RETUR~ CONT 70
G-59
Kt~I~O I CONT 11CC 210 N<I,~UMEL CUNT 72RtACllOI RKKI5),lllIS),(SIGIII,I<I,101 CUNT 73RUOllO) IHSIII,I<I,JO) CONT 74IF! J[cF.t.c.1) ~c Te 210 CONT 75~1<IXI~.lJ CONT 76N2<IXIN,21 CONT 77~3<1XIN. Jl CUNT 78f\4=IX(f\,4} CGNT 19~RR(:)=(H(Nl)+k(~L}+k(~3)+~(~4)J/4. CO~T 80lllISI"llU'.[)+lINd+llNjJ+[IN411/4. CONT BI
210 .. KilL« 1) t<,kQ(5l ,LlL(S), ISiGlll d=l,lC) ,(l::P~{I) ,l=l,lC) CCt\T 82c••••••••••·****.·.·*•• * ••••••••••••••••••••••••••••••••••••••••••••••••CUNT 83C BEGIN OL LOCP Te PLUT FUNCTIUNS CONT 84( •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••CGNT 85
00 BCO LLL<I,'U~fL CUNT 86KtolNC I CONT 87
C••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••CONT 8~
C REAC CONTROL INFORMATIUN FCR A FU~CTlC~ CCNT 89C••••••••••• •••••••••• • •••••••••••••••••••••••••••••••••••••••••• *••*•••CUNT 9C
REJCI9dOOOIICNT,fCM,ICIII,I<I,ICMI CO NT 91.RllE 16,z00CI ICNT CONT 92
c••••••••••••••••••*••*•••••*••••••••••••••••• ~ ••••••••••••••* ••••••••••CUNT 93C LABEL CUNT00RS IF ICNT IS ~EGATIVE CONT 94C·••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••CONT 95
1'< 1 CCNT 96IFIICNT.LT.Gl M<O CONT 97IFIICNT.LT.01ICNT=-llNT CONT 98
c••••••••••••••• * ••••••••••••••••••• *•••••••••••••••••••••••••••••••••••CONT 99C CRCEP CCNTCURS CUNT ICCc••••••••••••••••••••••••••••••••• *••** •••••••••••••••••••••••••••••••••CONT 101
DO 3CO l<l,llM CONT 102au 3CO J<I, ICM CONT 103IFICIJI.GE.Cllll GU TC 300 CONT 104TE~FRY=ClI) CONT IDSCII1<CIJI CONT 106CIJI<TtMPRY CONT 107
300 CONT (NUE CONT 108C CONT IC9c •••••••••••••••••••••••••••••**•••*•••* •••*** •••••*•••••••••**•••••••••CONT 110C PuT FUNCTION VALUES IN • ARRAY CONT IIIc••••••••••••~.***.*••**••••••***••••••••••*•••*••***.*•••••****•••••••*Cu~T 112
NU~J= MAXJ-I CONT 113NUI'I< MAXI-I CUNT 114N<I CONT 115DO 400 J=I,NUMJ CONT 116IST~RT= MAXOIIMINIJI,IMINIJ+111 CONT 117ISTOP< I'INOIIMAXIJ1,lMHIJ+1)J -1 CONT 11800 400 I<!START,ISTOP CONT 119READfl1 RRk(51,ZZZI51,ISIGIKI,K=I,10),IEPSIKI,K<I,101 CONT 120XII.JI= TILT*RRRI51 + 1l.-TILTI*Z11151 CONT 121YII.JI= TILT*ZZl(~1 + 11.-TILTI*RRRI51 CCNT 122IFIlCNT.GE.I.ANO.ICNT.LE.I01 .11,JI< SIGIICNTI CONT 123IFIICNT.GC.1l.ANO.ICNT.LE.201 WII,JI< EPSIICNT-IOI CONT 124IFIIC~T.EC.211 WII,JI< TlNI CONT 125IFIICNT.EQ.22J WII,JI= PSTlNI CONT 126
400 N= N+I CONT 127DO 510 J~I,NUMJ CONT 128IST~RT~ M~XOIIMINIJI,IMINIJ+111 CONT 129IFI1START.LE.II GC TO 510 CONT 130IMI< ISURT-I CONT 131DO 500 1<I,1Ml CONT 132XfI,JI~ XIISTART.J) CONT 133YII,JI< YIISTART,JI CONT 134
500 .II,JI= WIISTART,JI CONT 135510 CONTINUE CONT 136
00 530 J<I,NUMJ CONT 137ISTOP< MINOIIMAXIJI,lMAXIJ+111 CONT 138IFIISTGP.GE.MAXII GC TO 530 CONT 13900 520 1<ISTUP,NUMI CaNT 140
G-60
XII,JI= XIISTOP-l,JI CuNT 141Yll.JI= yIISTOP-I,Jl CONT 142
520 .II,JI= .IISTOP-l,Jl CONT 143530 CONT INUE ceNT 144
C CONT 145XMIN=TIlToKMIN.ll.-TllTlolMIN CaNT 14~
YMIN=TIlTolMIN'II.-TllTIOK~IN CaNT 147c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••CONT 148C lABH FUNe TION PlUTS CaNT 149c·••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••CUNT 150
CAll PlTll,NIT.XMIN,OElP.CI.0.1,YMIN.DElP.C.1,CO,2D,~tAOI1.leNTlCUNT 1511.BO.XT,ec,YTI CUNT 152
c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••CONT 153C CRA" CONTOURS CCNT 154c•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••CONT 155
CAll DRA.IX.y,25,C.ICM.CI,CO.N~MI,NUMJ,~,~MIN,OElP.Y~IN,UElP,WI CUNT 15~
11X=50 CaNT 157IIY=1025 CaNT 15HCAll PlTI3,11X,1IY,.12~,16,16HCCNTC~KSPlCTTEOI CUNT 159co ~OO 1=1. ICM CONT 1601IY=llY-25 CGNT 161CAll BOFICIII,Ael CaNT 162
6UO CAll PlTI3,IIX,IIY,.125,12,Abl ceNT 1~3
l=O CONT 164c· ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••tONT 165CORA. BCUNOARY CONT 166c •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••CONT 167
JM=MAXJ CC~T 1beDC 100 K=1.25 CONT 1~9
100 JM=~INOIJMI~(KI,JM) CO~T 1101=1~1~IJMI CONT 111J=J~ CONT 112
710 l=l'l CONT 113lFll.GT.2501 WRITE16.20021 CUNT 174Ifll.GT.2501 GO HJ 15G ceNT 115NP'NCCE II, J I CONT 176COIll=TIlToZINPI.11.-TllTIORINPI CONT 171CIIll=TIlToRINPI'11.-TILTloZINPI CaNT 118!FIJ.NE.JMI GO TO 120 CUNT 119Ifll.EQ.IMINIJMI.AND.l.GT.11 GO TO 150 CO NT 180
720 IACC=O CONT 1B1JAOC'O CUNT IB2IfIJ.EQ.JMINIII.ANO.I.NE.1MAXIJII IAOC='l CONT 183IFIJ.EQ.JMAXIII.ANO.I.NE.IMINIJII IAOO=-l ceNT 1B4IFII.EQ.1MINIJI.ANO.J.NE.JMINIIIJ JACC=-l CUNT 185IFII.EQ.IMAXIJI.ANO.J.NE.JMAXIIII JAOO='l CaNT 1B6IFIIAOO.NE.O.OR.JAO~.NE.OI GO TO 130 CUNT 1B1IFII.EQ.IMAXIJ-lIl IADO=9! CUNT 1B8IFll.EQ.1MINIJ.111 lAOO=-1 CUNT 189!FIJ.EQ.JMAXll'1I1 JACC=.l CONT 1~0
jfIJ.EQ.JMINII-lll JAOO=-l CUNT 191730 IFlIADO.EQ.O.AND.JADO.Et.OI .RITEI6,2001l I,J CONT 192
I=I+IAOO CGNT 193J=J'JAOO C<JNT 194GO TC 110 CCNT 195
750 CAll PlTI2,L,ll CUNT 196800 CAll PLTI5,Il,Pll~1 CONT 191
C ceNT 19BRETURN CUNT 199
C CCNT 2001000 FORMAT 1216,10F6.01 CUNT 2012000 FORMAT 124H BEGIN PlCT GF FUNCTION 131 CONT 2022001 FORMAT 121H PERIMETER PLGT ERROR AT 1=.15,3H,J=,15J CONT 2032002 FORMAT 154H PERIMETER CONTAINS MORE THAN 250 POINTS, PLOT STOPPEOICUNT 204
END ceNT 205
G-6!
SUB~(UTIN£ CRAw (X,Y,~l,C,NP,CI,CC,~,N,Nle,xs,XIt¥St~I,IJ DRAW 1C••*.*.···.**··.·*~*.*•••·*••••******.*••••*•••••••••••••••••••*••••*••• CRA~ 2C DRA. IS A STAN0ARC AEROSPACE CLFPCRATIC~ SLepOUTINE DRAW 3C HlP CCf\tTCLJf' 1"( L T LI-lA ..... ING CRAW 4C••*.*··*~·**·*··***·*··*****•• • •••••***•••••••••••••••••••••••••••••••• ORAW 5
CI~E~SIC~ .1~1.ll.ll~I.II.Clll.CIII).CCII'.TI151.TII~I.TI151. ORA. 6leCCI71.Z1~I,ll ORAW 7
(AlA C£l/1 .. Cf:-6/ DRAW 8C••••• ·.****·••***.*.·.·•• ••••• • •••••••••••••••••••••••••••• ~ •• * ••• o.*~*CRAW 9C TEST CRUER cr cc"e",c, ORA. 10( ••••••••••••••••••• * ••••••••••• * ••••••••••••••••• *•••o.*••••• *•••••••••D~AW 11
~P~I=NP-l DRAW 12DC lCO K=l,~PMl DRAM 13kPI=k+l CRAW 14IFIClkPU·LF.enll .RIIEI<,,10001 KP,I.k DRAW 15
100 CCNT I~UE DRAW 16KS~~P+1 ORA~ 11ICh= -I DRAW 18N'H=~-I DRAW 19~~I=~-I DRAW 20IFIMB.U•• I.CR.~P.H.. lI GC TC JOG DRAW 2110IF=L(2)-CIII DRAW 22IFI~P.Ew.21 GLJ TO 210 DRAW 23
c**••••• ••• * •••****· •• o•••••••••••••••• **o •• o•• *••••• *•••••••*••••••••••DRAW 24C IF SUCCESSIVe CONTOUR VALUES DIFFER AI A CL~STANT. T~E PLeT IS ORA. 25C ANNGTATEO '1 TH THE INCKE~E~I. TeIS IS A~ LPTION DRAW 26C•••• ••••••••••••••••••••••••• *•••••• ** •••••••• **.* ••• **.*.*~.*•• ·.*.*•• DRAW 27
IDfF=-l.CEJe DRAW 2800 2CO 1=3.~P DRAW 29ZOCf=l.-IDIF/IC( II-Cll-ill DRAW 30IFIABSIIOOFl.GT.lUEFI ZOEF=ZUCF ORA. Jl
200 CG~T lNUE ORA. 3lIFIABSIlDEFI.~T.DELI GC TU 300 ORb 33
210 CALL BDFIZCIF.8CCI1'l OKA. 34CAll PLT (3 .. 50,6C,O.l,12 .. I.:H..L;(l)) i)RA~ 35
C•••*••••• *•• *.**••**.***.** ••••••***••• ~ ••••• *••••*.**••••••**.*•••**.*DRA~ 36C PARA'EIER 1 LETERMINES ThE C1RtCTILN CF TRAVEL LF THE PLCTTER PE~ CRAW 37C PARAMETER IS ALTERNATES THE ENC POI~TSCF ITS TRAVEL DNA. 38C I.E. FeN SUBSCRIPT J = AN COJ INTEGER. INAVEL IS 8ET.EEN 1 A~D M-1DRAW 39C FOR SUBSCRIFI J = A~ EVE~ INTE~ER TRAVEL IS BET.Et~ M ANO 2 DRAW 40t ••••••••••••••*·.** ••••• *•••••• *••••••••••••••• ***.** •• *••••*.* •• *•• *•• D~AW 41C LGOP Te INLREME~T I DRAW 42C••••••••••••••• - ••***••••••••••••• *•••• *••••••••••••• • *.******.***.**••ORA~ ~3
300 CO 390 J=l.NMl DRAW 44IFI~CDIJ.<I.E~.OI GO TU 310 ORA. 45IS=1 DRAW 46I ST=C ORA. 47GC H 320 DRAW 48
310 IS=-I DRAW 49IST=" DRAW 50
C•••••• • •••••*****·.********.********•• ** ••******************.*********.DRAw 51C LOCP TO INCREME~T I OPA. 52C*•••••••• ·.***·**.*••••••**·*********.**.*** •• **.********.********.**.*DRA~ 53
320 co 3l?O IN=l,"'''ll CRAW 541=1~*IS+1ST DRA. 55TZIll=III.JI DRAW ~6
TZC2I=III+l.JI OPAW 57TZ(3)=III+!.J+ll DRAW 5BTl(4)=III.J+1I DRAW 5STlI51=TIIII DRAW 60
C DRAW 61C BA = ~AxIMU~ FUNCTICN VALUE IN THIS GRIO DRAW 62C AA MINIMU~ FU~CTIO' VALUE I~ IHIS GRIC DRAW 63C ORA. 64
eA=AMI~[(Tllll.TlI21.rI131.TI141) DRAW 65AhA~AX11TlI1l.TlI2l.TI131.TI14}' DRAW 66IFICIII.GT.AA.UN.CINPI.LT.BAI ~O Te 3&C DRAW 67TXlll=l( I,J) DRAW 68TXI21=XII+l.JI ORA>. 69TI(3)=III+l.J+ll DRAW 70
G-62
TXI41=XlI,J+II DRAW 71TXI51=TXlII DRAW 72TYlll=Yll,JI DRAW 73TYI21=Yll+I,JI DRAW 74TY13I=Yll+l,J+ll DRAW 75TYI41=Yll,J+1I DRAW 76TYI51=TYlll DRAW 77ISTCf-O DRAW 78
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••ORAW 19C FlhC ALL CChTOUR LlhES IN A GRIC ELEME~T DRAW 80C••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ORAW 81
00 31C ~-I, hP DRAW 82IFlCl~I.GT.AA.OR.Cl~I.LT.~AI GO TO 36C DRAW 83IFL=G DRAW 84JMffLG=O DRAW 85
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••ORAW 86C JMPfLG = 1 CAUSES THE NEXT SIDE IN TrE SEARC~ING SE~LENCE TO 8E DRAW 81C S~IPPED. THIS OCCURS WHEN A CONTOUR PASSES THRUUGH IrE CORhER DRAW 88C COMMCN TO THE PRESENT SIDE AND THE NEXT SICE 1U BE EXAMINEC DRAW 89C • •••••••••••••••••••••••••••••••••••••••••••••••• OkAW 90C CHEC~ ALL fCUR SIDES UF EAC~ GRIC DRAW 91C•••••••••••••••••••••••••••••••• ••••••••••••••••••• ~·•••••••••••••••••• ORAM 92
CO 3~C L=I,4 DRAW 93IF I J~fFLG. NE .11 GG TC 330 DRAW 94J"PFlG=O DRAW 95GO TC 350 DRAW 96
330 WHERE=TIIL+ll-CI~1 DRAW 97IFIWHERE.EQ.O.1 JMPFLG=1 DRAW ~8
WHERE=WHERE*IC(KI-TIILII DRAW 99IFIWrERE.lT.O.1 GU TG 350 DRAW 100IFl=IFL+I DRAw 101QzICI~I-TIILII/ITIIL+II-TIILII DRAW 102CIIIFLI=TXlll+O*ITXIL+II-TXlLII DRAW 103CDIIFLI=TYILI+,*ITYll+II-TYlLII DRAW 104IFIIFL.NE.21 GO TC 350 DRAW 105IflICH.EC.l.UR.NL8.E'.11 GO To 340 ORA. 106IFl~.NE.I.ANO.K.NE.NPI GU TO 3~0 DRAW 107
C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••OKAW 108C LABEL SMALLEST AND LARGEST VALUED CCNTCURS IF ANNOTATICh CPTION DP.AW 109C USED I.E. NL8 NOT EQUAL TO I. DRAW 110c••••••••••••••••••••••••••••••••••••••••••••••••••••••*••**••*••••••••*O~AW 111
IFl~s.EQ.KI GO TG 340 ORA. 112KS=~ DRAW 113ICH=ICH+I DRAW 114IX=ICIIIFll-XSI*IOO./XI-5. DRAW 115IY=(CO(IFLI-YSI*100./YI+45. DRAW 116CAll BOF lCIKl,BCOlllI DRAW 11?CALL PIT 13,IX,IY,O.I,l2,8COI1l1 ORA. 118
3~0 CONTINUE DRAW 119CAll Pl1 12,2,11 DRAW 120ISTCP=I ORA. 121GO TC 370 DRA. 122
350 CONT INUE DRAW 123360 IFlISTOP.EQ.lI GC TO 380 DRAW 124370 CGhTINUE ORA. 125380 CONTINUE ORA. IZl>390 CONT INUE DRAW 127
RETURN DRAW 128C DRAW 129
1000 FORMAT 15H THE 12,28HTH CCNTOUR IS LESS THAN THE 12,10HT~ CCNTUJRIORAW 130ENG DRA. 131
G-63
SUER[UTINE t"Flo,ABI 8UFI
C*··***·**~··***~#****·**********************~.********.*.~.**********.*BuflC BUf IS A SfANCARC AEkGSPALl CURPGKATICN I"~ ,6lll~V1I ,UeRLLflNE bUFIC hJR LeNTCUk PLGf ANNLTATIU, eUF!t··****···········**···********·.·.*****.** •••• ******* ••••• ***** ••• *.***BGFt
[
LI~E""SILr.. Ae(3J,do(31,l£kC(3J,lL.JIl>(lC),Jt:(61CAlA IGIGIlCLFCCOOU,lOCFIOOOO,ZUOF2C00C,lCCF300CC,lCCf4000C,lLOGf5uCCCtZLOFbOCCC,lOC~7CCCC,lGCr8DCCttZCCF900~CI
DATA edlluOCC4dCC,luOU000vU,IC~UOOOOC/
CAl A Idl,...K,I'lIt>lU,S,ll:RL/Zl,C4JuCCG,I.UC6CCC,,1.,4 ..... ...;. ,4h ,4H I
( = /J.1i~(eJ
IFIC .E~. C.I UL TL 4CLlOP = aK = a
10 IFI( .LT. I.IG[ TL ICu20 IFIC .GE. IG.IU[, Tu 2UU30 I = [
t< = K + 1IBIKI = ILIUII+l)IfIK.EC.6IGC T[ 300(= I (- FLCAT I I ) I *1 a•GO T( 30
100 lEn = lOP - I( = ( * 10.UC 1 ( 10
200 IEXP = IEXP + I(=(/IU.GG T( 20
300 IFllEXP.LT.CIABI JI Cl<lecI31.MINLJSIIFIIEXP.CE.llIAbI31 URlbdI31.1eLNKIlOP = IABSIIEXPII=IEXP/IOJ = Iu IGI I+ll 1 256.A8(3) ::: CPfAl:H3),J)I = IDP - 10 * IJ = IOIGII+1l 1 65536AS!31 = CRIAeI3l.JIIFlE.LT. 0.11 = ~INUS * 256IFIE.GT. 0.1 I = 18L~K * 2S6ABIII = I1RIBB!!),!)ABlll = [RIAEIII.IBIIII1 = WI21 1 65536A8111 = URIABII).III = IBDl * 256ABUI = GRIBBI21,I1ABI21 = GRIASI21,IBI411I = lel51 1 256Ael21 = LRIABI21.1)J = WI61 1 65536ABI21 = [RIABI21,! IRE TURN
400 OG 4:0 r = 1,3450 AB (j' = zoe III
RE TLRNENe
G-64
BeFleUF IBCFIeOFIeeFI88 f 1t:H;F 1BOfl8L'r 180HtWF 1ilCFlBLFIeOFlfiNIeUf!ECFIBOFIBUFIBLFl"Df IBOFIeUfleOFIBOFlBOFIbCflBuF IBOFI80F!BOFIBOFIeOfleOflSOFI8eFIBOFISDFIBUfIBOFIBeF IBOflBeH80FlBOFIBOFI8eF!BOHBeFI80FIBDF I
I2J4
567,1
10II121314I')161718I"2u21a232425262728293031323334353637383S4041424344454641484950515253545'>56
G.4 UNIVAC 1108 FORTRAN IV COMPUTER PROGRAM LISTING
Certain program modifications must be made in order to run the SAAS
III program On the UNIVAC 1108. These modifications affect Subroutines
MAIN, REST, INTER, PLTM, CONTR, DRAW, and BDF. Since most of the
changes are related to element and stress and strain contour plotting, the
program can be converted quite easily if plotting is suppressed. Then, only
the change in INTER is required. (Even this change is somewhat anomalous
since the data statement should have worked on a UNIVAC 1108. Evidently, the
change is necessary only on the machine at Southern Methodist University.)
The plotting features are actuated by performing the remaining (significant)
changes that are compatible with standard CALCOMP plotting instructions
as given in Ref. 26.
1. MAIN Changes
a. Change TIT LE(20) to TIT LE(l4) and add XAXLEN to
COMMON!PTT! ...
b. Add after MAIN 20
DIMENSION IBUF(lOOO)
DATA ISW!O!, NLOC!lOOO!
c. Replace statement 400 by
400 IF(IPLOT.NE.l.AND.IPLOT.NE.2) GO TO 500
C IF ISW = 0, INITIALIZE PLOT TAPE
IF(ISW. EQ. 0) CALL PLOTS(IBUF, NLOC, 4)
IF(ISW. EQ. 0) CALL PLOT(O., 0., -3)
ISW = 1
CALL PLTM(l)
G-65
d. Replace CALL PLT(6) by
C PUT END OF FILE MARK ON PLOT TAPE
CALL PLOT (0., 0.,999)
e. Change 20A4 in FORMA TS 1000 and 2000 to 13A 6, A2
2. REST Change
Change F(27) to F(22) in COMMON/PTT/ ...
3. INTER CHANGE
Change the data statement to nine explicit assignments, e. g.,
XX(I) = .1259391805448, etc.
4. PLTM Changes
See the subroutine listing on the following pages.
5. CONTR Changes
a. Change TITLE(20) to TITLE(14) and add XAXLEN to
COMMON /PTT / ...
b. Change the dimension 5 of HEAD and T1 through T21 to 3
c. Change the data statement for T1 through T21 to 18H formats
such as 18HANGLE TO EPS MAX
d. Replace CALL PLT( 1 - - (now CONT 151 and 152) by
CALL SYMBOL(O., 10. 5, .21, HEAD(l, ICNT). 0.,18)
IF(TILT.EO.1.) CALL AXIS(O. , 0., 6HZ-AXIS, 6, 10.,
90., ZMIN, DELP)
IF(TILT. EO. 1. ) CALL AXIS(O., 0., 6HR-AXIS, -6,
XAXLEN, 0., RMIN, DELP)
G-66
IF(TILT. EQ. 0.) CALL AXIS(O., 0., 6HR-AXIS, 6,
10. , 90. , RMIN, DELP)
IF(TILT. EQ. 0.) CALL AXIS(O., 0., 6HZ-AXIS, -6,
XAXLEN, 0., ZMIN, DELP)
e. Change IIX=50 and IIY=1025 in CONT 157 and 158 to XDIST=. 5
and YDIST=l 0.25 respectively
f. Change CALL PLT(3 .•• ), now CONT 159 to
CALL SYMBOL(XDIST, YDIST, .14, 19H
CONTOURS REQUESTED, O. , 19)
g. Change IIY = IIY-25 in CONT 161 to YDIST=YDIST-.25
h. Change statement 600 in CONT 163 to
600 CALL SYMBOL (XDIST, YDIST,. 14, AB, 0.,12)
i. Replace statement 750 in CONT 196 by
750 CI(Ltl) = XMIN
CI(Lt2) = DELP
CD(Ltl) = YMIN
CD(Lt2) = DELP
CALL LINE (CI, CD, L, I, 0, 0)
XDIST = XAXLEN + 3.
j. Change statement 800 in CONT 197 to
800 CALL PLOT (XDIST, 0., -3)
6. DRAW Changes
a. Change CALL PLT(3, ••• ) in DRAW 35 to
CALL SYMBOL (.5,.6,.07, BCD(l), 0.,12)
G-67
b. Change IX= and IY= in DRAW 115 and 116 to
XPAGE = (CI(IFL)-XS)/XI-. 05
YPAGE = (CD(IFL)-YS)/YI-. 45
c. Replace CALL PLT(3... ) in DRAW 118 by
CALL SYMBOL (XPAGE, YPAGE,. 07. BCD(l), 0.,12)
d. Replace DRAW 119 and 120 by
340 CI(3 ) = XS
CI(4) = XI
CD(3) = YS
CD(4) = YI
CALL LINE (CI,CD,2,l,O,O)
7. BDF Change
See the subroutine listing on the following pages.
G-68
SUBROUTINE PLTMIITYPEI PLTM 1c* ••••••••• * •••••••• * •• * • * ••••• * * ••• *PlTM 2C STA~DARD CALCoMP 'BASIC SOFTWARE' PLOTTI~G INSTKUCTICNS ARE USED PlTM 3C TO PLOT FINITE ELEMENT MESH PLTM 4'* ••••.••....•.....••.•.....•..•...•.PlTM 5
CoMMCN/BASIC/NUMNP,NUMEl,NUMPC,NUMSC,ACELZ,ANuVEl,TREf,VGL,IFRE~ PLTM 6CoM~CN/NPDATA/RIIOOOI,CooEIIOOOI,XRllOOOI,ZllOOOI,XlllOOOl,TIIOOOIPlTM 7CoM~CN/EloATA/IXIIOGO,5I,EPKIIOOOI,AlPHAIICCC' PlTM 8CoM~ON/PTT/IPLoT,TITlEI14I,RMIN,Z~I~,DElP,TILT,FACT,ICEF,XAXLEN PlTM 9DIME~SION XI71,Yl71 PlTM 10
c* •••••••••••••••••••••••••••••••••••PlY" 11C lABEL BEGINNING CF GAIA CASE wITH TITLE PlTM 12c* •••••••••••••••••••••••••••••••••• • PlT~ 13
IFIITYPE.EQ.ZIGo TO 50 PLTM 14CALL SYMBolI9.B,6.0,.28,BHSAAS 111,0.,81 PLTM 15CALL SYM8oLI7.7,5.0,.21,30hFINITE ELEME~T STR~SS ANAlYSIS,O.,3DI PLTM 16CAll SYMBLlI7.5,4.5,.21,32hof AXISYM~ETRIC ANC PlA~t SUlIDS,O.,321PlTM 17CALL SYM8oLI7.0,4.0,.21,37HbY JAMES G. CRCSE AND ROHERT M. JCNES, PlTM 18
10,371 PLTM 19CALL SYMBolI8.8,3.0,.21,19HPLLTS FOR DATA CASE,O.,IS' PlTM 20CAll SYMBUlI2.0,l.5,.21,TIILE,O.,80' PlTM 21CAll PLCTI24.,O.,-31 PLTM 22
c* •••••••••• * • •••• * ••••••••••••••••• *PLTM 23C LABel ElEMHT PlCT PUM 24c* •• • • • •••• • • • * • * ••• * •••••••••• * • * * • *PlTM 25
50 IFIITYPE.EQ.IICALL SY~8CLI0.,lO.5,.21,12hElEMtNTPlCT,O.,121 PlTM 26IFIITYPE.E~.21 CAll SYMbULIO.,10.5,.21,21HDEFOKMEO EIE~E~T PLeT, PlTM 27
10.,211 PUM 26C* ••••••••••• • •••••••••• * •• * ••••••• • *PLTM 29C OBTAIN MAXIMLM ANO MINIMU~ R ANC I PLTM 30c* •• • • •••• • ••• • • * • * ••••••• * * * •• * ••• ••PLTM 31
IfICElP.EQ.C.1 GC TO 100 PLTM 32I~AX:lMI~ PUM 33RMAX:R~I~ PLTM 34GO TC lle PUM 3~
100 llUNzllll PlTM 36lMUzlMI ~ PUM 37RMINzR III PUM 3BRMU-RMI h PC TM 39
C PlTM 40110 00 120 I:l,NUMNP PLTM 41
IFllIII.lT.ZMINI lMIN=ZIII PLTM 42IFllMAX.lT.lIIII lMAX=1111 PLTM 43IFIRIII.lT.RMINI RMIN=RIII PLTM 44IFIR~AX.LT.Rllll RMAX=RIII PLTM 45
120 CONTI NUE PUM 46C PUM 47
RA-IRMAX-RMINI/IIMAX-ZMINI PLTM 48IFIIPlDT.EQ.11 RA=l./RA PlTM 49IFIRA.lT.l.l GO TO 150 PLTM 50
co •• • • * * ••• * ••• * * * * * •• * •• * • * * * * * * • * * *PLTM 51C Z-AXIS VERTICAL AND R-AXIS HORIZONTAL PLTM 52c* •• • • • ••••• * • * • * * * •• * * * * * • * * * • * * •• • *PLTH 53
TlLT:l. PUM 54CAll SCAlEIZ,lO.,NUMNP,ll PLTM 55
C PUM 56C CALCULATE PARAMeTERS TG SCALE THE X-AXIS THE 5AME AS THE Y-AXIS PLTM 57C PL TM 58
IFIDElP.NE.0 •• ANG.ITYPE.NE.21 G0 TO 125 PLTM 59lHIN=11 NUMNP+ll PUM 60DElP:IINUMNP+21 PLTM 61NDUM:RHIN/oElP PlTH 62DUM=NDUM PL TM 63RM IIl=DUMOoElP Pl TM 64
125 NDUH=RMAX/DELP PlTM 65DUM:NDUM PL TM 66RlASTV=IDUM+l.I*DclP PLTM 67XAXlEN=IRlASTV-RMINI/DElP PLTM 68
C PLTM 69C DRA~ AND LABEl AXES PL TM 70
G-69
C PlTM 71CAll AXlSI0.,O.,6HZ-AXIS,6,10.,90.,ZMI~,CELPI PlTM 72CALL AXISIC.,O.,6HR-AXIS,-6,XAXlEN,O,RMIN,OELPI PlTM 73
C PUM 74C DETERMINE ANC PLOT CCORDINATES OF EACH ELEMENT PLTM 75C PUM 76
XI61=RMI~ PUM 77XI7I=OELP PUM 78VI61=ZMI~ PUM 79VI7I=DElP PUM 80DU 140 N=I,NUMEL PLTM 81DC 130 M=l,5 PL TM 821=IXIN,MI PUM a3IFIM.E~.51 1=IXIN,l1 PUM 84XIMI=RIII PLTM 85
130 VIMI=1111 PUM 86140 CALL LlNEIX,V,5,l,O,CI PUM 87
GO TC 18C Pl TM 88c* •••• * •••••••••••••••••• * •••••••• * • *PLTM 89C R-AXIS VERTICAL AND Z-AXIS HORIZONTAL PlTM 90c* •••••••••••••••••••••••••••••••••• *PLTM ql
150 CALL SCALEIR,lO.,NUMNP,l1 PUM qzTIU=O. PL TM 93
C PUM <;4C CALC,LATE PARAMETERS TC SCALE THE X-AXIS THE SAME AS THE V-AXIS PLTM 95C PUM 96
IFICELP.NE.0 •• AND.ITVPE.NE.21 GO TC 155 PLTM 97RMI~=RINUMNP+ll PLTM 98CELP=RINUMNP+21 PlTM 99NOUM=ZMIN/DELP PlTM ICCCUM=NOUM PLTM 101ZMIN=OUM*OELP PlTM 102
155 ~OUM=ZMAX/DELP PLTM 103CUM=NCUM Pl TM 104ZLASTV=IDUM+l.I*DELP PlTM 105XAXLEN=llLASTV-ZMINIIOELP PUM 106
C PLTMl~C ORAl< AND lABeL AXES PUM 108C PLIM 109
CALL AXISI0.,O.,6HR-AXIS,6,10.,90.,RMIN,DELPI PLTM 110CALL AXISIO.,O.,6HI-AXIS,-6,XAXLEN,O,ZMIN,OELPI PlTM III
C PlTM112C DETERMINE ANC PLOT CCORDINATES OF EACH ELEMENT PlTM 113C Pl TM 11'0
XI61=ZMIN PUM 115XI7I=DELP PlTM 116Vl61 z RMH PlTM 117VI11=OELP PlTM 118DO 170 N=I,NUMEL PLTM 11900 160 M=I,5 PUM 120I=IXIN,MI PLTM 121IFIM.EO.51 1=IXIN,l1 PlTM 122Xl MI=HII PUM 123
160 VIMI=RIII PUM 124170 CAll LINElX,Y,5,1,O,GI PlTM 125
C PlTM 126C INITIALIZE PLOTTER PEN FUR NEXT PLCT PLTM 127C Pl TM 128
180 XOIST=XAXLEN+3. PLTM 129CALL PlOTIXDIST,O.,-31 PlTH 130RETUPN PLTH 131END PlIM 132
G-70
2. Change the dimension on F in COMMON / PTT / from 27 to 15
where it occurs in REST
3. Change the number 12 to 20 in CONT 163
4. Change 20A4 to SA10 in FORMA T statements 1000 and 2000 in MAIN
5. Change the dimension 5 of HEAD and Tl through T22 to 2, and
change the DATA declarations for T 1 through T22 accordingly
6. Use the CDC 6600 version of BDF listed on the following page.
G-73
SUBROUTINE BDFIFlT,BCDl ~DF2 IC•••••••••••••••••••••••••••••••••••••••••••••••••••••••••*•••••••••*•••~~F2 2C BDf IS A STANDARD AEROSPACE CORPORATION cec 6600 SUBPCuTINE BlF2 3C fOR CONTCUR PLOT ANNOTATION BlF2 4C•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••80F2 5
OI~E~SICN BOOI2I,RE4~AT(21 ~DF2 6C 301-2 7
ENCOOE(20,l,RE4~ATI fL T 8DF2 BI fOR~ATIE20.BI BDF2 ~
CECCOE(20,2,RE4~ATI BCDlII,BCDlll BUF2 Ie2 fOR~AT(2Al01 BDF2 11
C IlDF2 12RETURN BDF 2 13END BDF2 14
G-74
G.6 PLT360, IBM 1627 PLOTTING ROUTINE
The PLT360 subroutines are called from SAAS III for the purpose of
preparing element and contour plots. The following writeup of this plotting
feature has been excerpted from The Aerospace Corporation Mathematics
and Computation Center Programmer's Handbook. It is presented here as
an aid to potential program users in establishing their own plotting capability
and for use in understanding the calls to PLT.
G-75
S/360 LIBRARYPLT360Page 1 of 32Rev. 3 April 1967
lentification
LT360, IBM 1627 Plotting RoutineS/360 - Assemblerl. Clarke, February 6, 1967erospace Corporation, San Bernardino Operations
ontents of Write-Up
Gene ral De sc ription
Tape or On-Line Plotter Output
Storage Requirements (bytes)
Calling Sequence Usage
A. Setup EntranceB. Data EntranceC. Annotation EntranceD. Aerospace EntranceE. Cleanup EntranceF. Terminate Entrance
Extended Precision Plotting
EBCDIC Codes for Symbols and Special Annotation
Overlay Requirements
Errors
Notes on Title/Annotation Input
Multiple Report Plotting
A. Calling Sequences for Multiple Report PlottingB. Example for Multiple Report Plotting
Appendix A - Assembly Language Usage
Appendix B - Parameter and Work Storage Tables
Appendix C - Non-Standard Usage
Appendix D - Output Record Size Changes
Page 2
3
3
4
51011141516
17
18
19
19
19
20
2022
24
26
30
32
cknowledgment
PLT360 is the OS/360 Assembler language version of the 7040-7094 DCS
mtines PLT (AMOlB), PLTI (AMlOA) and PLTW (AMllA), which were major
,visions of RW CCP and RW CCP2, written by K. G. Tomikawa and J. R. Black
.er, respectively, in August of 1962, Space Technology Laboratories, Redondo
each, California.
G-76
S/360 LIBRARYPLT360Page 2 of 32Rev. 3 April 1967
General Description
This subroutine is called from FORTRAN programs to generate plots. User's
information is assembled and control is passed to PLTl, which generates plotter
commands. One plotter command is generated for each 1/1 Oath inch pen move
ment in vertical, horizontal or diagonal direction. One command is also gener
ated for each pen up or down movement. The plot records are output by PLTW /
360, onto a file named PLOTPLOT. All plot output generated in one computer
run is one file, and will be output to the same tape Or on-line plotter.
A maximum of seven dependent variables are permitted as functions of one
independent variable. All data must be in floating hexadecimal form in either
single precision (4 bytes) or extended precision (8 bytes). Data for any variable
must be stored consecutively in core. User's information controls scaling of
data and printing of scales and titles. Special annotation and the Aerospace
symbol may be written anywhere on the plot to the right of the plot origin at the
left hand side of the plot.
Floating point data are scaled to pen deflections as determined by parameters
specified in the CALL statements.
A Y data word in which bits 1-4 are all ones (approximately 1656
) is treated
as a missing point, and a small M is printed at the vertical level of the previous
point. Thi feature might be used to indicate telemetry dropout, etc.
c\ Y data word which falls off the scale (either above or below) is unconnected
to any other data point. If the point is more than a small fraction of an inch off
scale, a small W (for wild point) is printed at the margin in line with the wild
point. A W is only printed for the first of a series of off-scale points.
The independent variable must be in tabular form in core, stored in consecu
tive cells. The dependent variables must be stored in a like manner; however,
a block of the independent variables and blocks of the dependent variables need not
be adjacent storage locations.
G-77
S{360 LIBRARYPLT360Page 3 of 32Rev 15 September 1967
ape or On-Line Plotter Output
The number 1 entry to PLT will cause the file to be opened and the tape to
rewound. (OS{360 rewinds the tape when an OPEN command is given. )
The DDNAME of the plot file is PLOTPLOT • The user should refer to cur
mt operating procedures for specifying on-line or off-line plotting.
The plot output tape file is written in EBCDIC in 864 character records.
arger or smaller tape records may be written by changing two cards in PLT
~ck and reassembling PLT. See Appendix D.
:orage Requirements (bytes) .'.','
PLT
PLTl
PLTW
Total
E2416
142416
114 16
232C 16
362010
5156 10
276 10
(including 1728 bytes for buffers)
G-78
S/360 LIBRARYPLT360Page 4 of 32Rev 3 April 1967
Calling Sequence Usage
All fixed point numbers used by the plot routine must be four-byte integers.
Six calling sequences are available in PLT. They are:
1. Setup Entrance (required for every plot) which initializes tables within
PLT, sets up general information about scales, titles and symbols, and
outputs the title and left hand dependent variable scales if required;
2. Data Entrance (optional, but normally used) which scales user's data to
his specifications and outputs plot of date;
3. Annotation Entrance (optional) which is called to add special annotation
to the plot anywhere to the right of the dependent variable coordinate
scales;
4. Aerospace Symbol Entrance (optional) which will add the Aerospace
symbol anywhere on the plot to the right of the dependent variable coor
dinate scales;
5. Cleanup Entrance (required for every plot) which :is called at the end of
every individual plot to write the independent variable scales, additional
dependent variable scales to the right of the plot, and position the plotter
pen at the origin point for the next plot; and
6. Terminate Entrance (required) which is called at the end of all plots
output in one computer run, to write a message to the operator on the
plot file and write an end-of-file.
G-79
S/360 LIBRARYPLT360Page 5 of 32Rev 15 September 1967
lling Sequence Usage (cont)
Setup Entrance
CALL PLT( 1, N, XO, DX, X, DDX, NF, YOI, DYI, lSI, ISFI, YI, •.... ,
NT, TITLE, NTX, XTITLE, NTY 1, Y 1TITLE, .•.. )
Where;
1
N
Indicates a setup entrance.
Controls the output of plotter information. When N < 0,
any partially filled buffers are output before PLT returns
control to the calling program. When N ~ 0, any partially
filled buffers are retained in PLT until the next entrance
to PLT and filled with subsequent plotter commands. At the
end of each plot (cleanup entrance) all information is output.
,'.','
XO Starting value of X scale in the same units as data (floating
point).
DX Delta X per inch of plot, given in the same units as data
(floating point).
X FOR TRAN name of first value of independent variable (all
data are floating point) when DDX = 0, each data entrance
will plot data beginning at this point of the array and continued
forward in the array for the number of points specified. When
DDX F 0, the independent variable values are generated. X
will be the first value of the independent variable for a data
entrance and the value will be incremented by DDX for each
subsequent point in this data entrance. The values plotted
will be (X, X+DDX, X+2':'DDX, X+3':'DDX.••.•. X+(NP-I)':'DDX
(where NP = Number of points specified in data entrance).
X must contain the correct value of the independent variable
for the first point of each data entrance at the time the data
entrance is given.
G-80
S/360 LIBRARYPLT360Page 6 of 32Rev. 3 April 1967
Calling Sequence Usage (cont. )
A. Setup Entrance (cont. )
DDX = 0 when all independent variable values are stored in an array;
= the increment between each value of the independent variable
when they are to be generated by the routine (floating point).
NF Number of dependent variables. Extended precision values
may be plotted by adding flags to NF. See "Extended Precision
Plotting" for exact specification of flags. When single preci
sion data is to be plotted, NF = only the number of dependent
variables (integer).
YO 1 Starting value of scale for first dependent variable (floating
point).
DY 1 Delta per inch of plot for first dependent variable (floating
point).
lSI Symbol code for first dependent variable (integer).
= 0, no symbol
= 1, triangle ~
= 2, inverted triangle V= 3, hour glass[
= 4, star ~:~
= 5, spool II
= 6, += 7, X
ISF 1 = symbol frequency for first dependent variable. Symbol is
drawn at first point and every ISF 1th point thereafter (integer).
A connecting line is drawn between points.
= 0; point plot. Data points are not connected and a symbol
is drawn at every point.
G-81
5/360 LIBRARYPLT360Page 7 of 32Rev. 3 April 1967
'!ling Sequence Usage (cant. )
Setup Entrance (cont. )
FOR TRAN name of first data point of first dependent variable
(floating point)
}Repeat :ast 5 parameters for each additional dependent
variable after first one.
NT
TITLE
NTX
Number of EBCDIC characters used for plot title (integer).
Plot title - may be specified by literal data with number of
characters equal to NT or specified as an array and input by
means of a READ statement or a DATA statement. See
"Notes on Title/Annotation Input."
Number of EBCDIC characters used for independent axis
title. If NTX = 0, printing of X axis scale and title is sup
pressed (integer).
XTITLE X-axis title - may be specified in the same manner as TITLE.
When NTX = 0, the contents of XTITLE are ignored.
NTY 1 Number (integer) of EBCDIC characters used for title of first
dependent variable. If NTYI = 0, printing of Yl axis scale
and title is suppressed.
Y 1TITLE Yl axis title - may be specified in the same manner as TITLE.
When NTY1=0, the contents of Y1TITLE are ignored.
}
Repeat last 2 parameters for each additional dependent
variable after first one.
G-82
S/360 LIBRARYPLT360Page 8 of 32Rev. 3 April 1967
Calling Sequence Usage (cant. )
A. Setup Entrance (cant. )
e. g. , CALL PLT( 1,0, XO, DX, X, DDX, 3, YOl, DY I, lSI, ISFl, Yl, Y02,
DY2, 152, ISF2, Y2, Y03, DY3, IS3, 15F3, Y3, 10, 'PLOT
TITLE', 24, XTITLE, 24, Y 1TITLE, 12, 'Y -AXIS TITLE',
0,0)
In the above example, 3 single precision dependent variables are speci
fied. The plot title is specified in the calling sequence and will read PLOT
TITLE. XTITLE was input by means of a READ statement from a card as
follows:
DIMENSION XTITLE(6)READ(5, 100)(XTITLE(I), 1=1, 6)
100 FORMAT(6A4)
Y 1TITLE was specified by means of a Data Statement as fOllows:
DIMENSION YlTITLE(6)DATA YITITLE/'FIRST DEPENDENT VARIABLE'/
The title of the second dependent variable will read: Y -AXIS TITLE.
The title and scale of the third dependent variable will not be printed.
The plot file at the return from this setup entrance will result in the
partial plot illustrated on page 9: (YOl=O., DYl=l., ISl=l, Y02=10.,
DY2=-1., IS2=2, Y03=2000., DY3=10., IS3=3).
G-83
S/360 LIBRARYPLT360Page 9 of 32Rev. 3 April 1967
!ling Sequence Usage (cont. )
Setup Entrance (cont. )
Example of plot appearance at end of setup entrance:
PLOT TITLE
10. o.9. 1.
<l)~ 8. 2..0ro.~ 7. 3.....ro> 6. 4.~
'" 5. 5.<l)
"0 4. <l) 6.'" ~
<l) ~.~
0.. 3. f-< 7.<l)
C1 <1J2. .~ 8 .
~ ~<1J
~.... 1. 9..~
,~ ~
O. 10.
"':::J ~
The left dependent variable axis line is not drawn if no dependent
variable scales are printed on plot.
G-84
SUBROUTINE BeFIA,BI BOF3 1c* •••••••••••••••••••••••••••••••••• *eOF3 2( THE FURPCSE OF THIS SUeROUTINE IS Te CeNVERT THE NUMBER A INTO A BOF3 3( HOllERITH ARRAY THAT (AN BE USED AS A PlCT ANNOTATION INSTRUCTION BOF3 4C* •••••••••• * • • • * • • • • • • • • • • • • • • ••••• *eOF3 5
DIMENSION BIll BOF3 bCAll ENCOOtlBI BOF3 1WRITEI3l,11 A BOF3 8
1 FORMATIEl2.5) BOF3 9RETURN BOF 3 10END BOF3 11
G-71
G.5 CDC 6600 FOR TRAN IV COMPUTER PROGRAM LISTING
In order to run the SAAS III program on the CDC 6600, certain
program modifications must be made. The modifications are related to the
elimination of all double precision arithmetic and changes in array lengths
to accommodate plotting. The changes affect the Subroutines MAIN, REST,
POINTS, TEMl, TEMP, STIFF, QUAD, TRIS, INTER, MODIFY, SOLVE,
STRESS, SYMINV, CONTR, FLDIN, and MPROP.
In order to eliminate double precision arithmetic, delete the following
cards:
MAIN 12, 15
POIN 6
FLDN 9
TEMP 7
TEMI 6
STIFF 11,12,15
QUAD 9
PROP la, 15
TRIS la, II
INTE 6,7
MODI 7
SYMI 5
SOLV 7
STRE 10,11,12,13
CONT II, 14
In addition, change the dimension on E in REST 7 from 447 to 303.
In order to plot correctly, a number of changes must be made to
accommodate the CDC 6600 la-bit word length as opposed to the IBM 360
4-bit word length. These changes consist of:
I. Change the dimension on TITLE from 20 to 8 in COMMON/PTT/
where it occurs in MAIN, PLTM, and CONTR
G-72
5/360 LIBRARYPLT360Page 10 of 32Rev. 3 April 1967
Calling Sequence Usage (cont. )
B. Data Entrance
CALL PLT(2, NP, K)
Where:
2 Indicates a data entrance.
NP = Number (integer) of points to plot at this time. NP points will be
plotted for each dependent variable versus one dependent variable.
At each data entrance the first point is assumed to be at X, Yl,
Y2.•• Ym as specified in setup entrance. Points 2 through NP are
plotted from locations directly following the first.
K = 0 if curve of this data is to be connected to data previously plotted
on this plot.
= 1 if curve of data from this entrance is not to be connected to curve
of data from a previous data entrance.
G-85
S/360 LIBRARYPLT360Page 11 of 32Rev. 3 April 1967
llling Sequence Usage (cant. )
Annotation Entrance
CALL PLT{3, IX, IY, S, NA, ANNOT)
Where:
3 Indicates an Annotation Entrance
IX Distance in 1/100 inches from left boundary of plotting area to
lower left corner of first character (integer).
IY Distance in 1/100 inches from bottom of paper (30/100 inch be
low plotting area) to lower left corner of first character (integer).
S Character height in inches. If size is negative, the print line
will be rotated counte r clockwise about the XY refe renee point
by 90 degrees (floating point).
NA Number of Hollerith characters used for annotation.
ANNOT Characters of annotation to be written on plot. May be speci
fied by literal data or specified as an array and input by means
of a READ statement or a DATA statement. See "Notes on
Title / Annotation Input. "
The EBCDIC information is printed on the plot as specified. Distance
between character centers is the same as character height. Several non
Hollerith characters can be printed with the hexadecimal codes shown below.
fl 4F Plot Symbols 5F Suitable for drawing
V 50an axis.
X 4C6D Used to extend tail of
arrow..', 6F'" 7B+---D 5A
7C--+ 6A
X 6eG-86
5/360 LIBRARYPLT360Page 12 of 32Rev. 3 April 1967
Calling Sequence Usage (cont. )
C. Annotation Entrance (cont. )
Example of Horizontal Annotation for
CALL PLT(3, 300, 150,.5,3, 'ABC')
PLOT TITLE
'--_ __.------'ABC"
10.
9.
8.
7.
6.
5.
4.
riI 3.....4 2.f-<H
f-< l. ,>< •
O.
\300/100 inches \150/100 inches
(l/2 inch below plotting area,one inch in plotting area)
G-87
:alling Sequence Usage (cant. )
Annotation Entrance (c onto )
Example of Vertical Annotation for
CALL PLT(3, 300, 0, -.5,3, 'ABC')
PLOT TITLE
10.
9.
8.
7.
6.
5.
4.
3.
2.
1
O.u
3~0/ 10~ inche~ 0/100 inches
G-88
5/360 LIBR,ARYPLT360Page 13 of 32Rev. 3 April 1967
S/360 LIBRARYPLT360Page 14 of 32Rev. 3 April 1967
Calling Sequence Usage (cont. )
D. Aerospace Symbol Entrance
CALL PLT{4,MX,MY,51)
Where:
4 Indicates Aerospace symbol entrance.
MX Distance in 1/100 inches from left boundary of plotter area to
lower left corner of Aerospace symbol (integer).
MY Distance in 1/100 inches from bottom of paper (SO/lOa inches
below plotting area) to lower left corner of Aerospace symbol
(integer).
51 Is size in inche s (floating point).
G-89
S/360 LIBRARYPLT360Page 15 of 32Rev. 3 April 1967
,dUng Sequence Usage (cant. )
Cleanup Entrance
CALL PLT(5, IRSC, PLOTIM)
Where:
5
IRSC
PLOTIM
Indicates a cleanup entrance.
Test value for right hand scale option (integer). IRSC is
compared to the actual number of inches of plot. When the
number of inches of plot is greater than IRSC right hand
scales are also drawn.
The routine reports the IBM 1627 plotting time in minutes
through this location. This parameter should be printed off
line as a guide to estimated plot time (floating point).
If the plot occupies more than IRSC inches of paper, the independent
scales and titles will be printed at the right-hand edge. The independent
variable scale and title are printed in the bottom margin. The pen is moved
to the home position of the next plot. The plot address is increased by one.
G-90
5/360 LffiRARYPLT360Page 16 of 32Rev. 3 April 1967
Calling Sequence Usage (cont. )
E. Cleanup Entrance (cont. )
Example of plot appearance at end of cleanup entrance.
Cleanup Entrance(if plot is largerthan IRSC inches)
Pen location atend of CleanupEntrance
•o.
DataEntrance
9.
8.
7.
6.5.
4.
3.
2.
1.
o.
~ PLOT TITLE( ( 1-0-.-,---=--==-=---=..::..::....:=:=.--------1-0-.-,-----
9.8.
7.
6.
5.
4.
ril 3...:1t-< 2.Ht-< 1.><
o. 1. 2. 3. 4. 5. . 7. 8. 9.XTITLE
Cleanup' Entrance
F. Terminate Entrance
CALL PLT(6)
Where:
6 Indicates a terminate entrance.
This entrance writes a message to the plotter operator on the plot file,
writes an end of file and closes the plot file. This must be the last entrance
to PLT for one computer run.
G-9l
S/360 LIBRAR YPLT360Page 17 of 32Rev. 3 April 1967
xtended Precision Plotting
Extended precision data may be plotted by adding flags to NF (setup entrance).
11 the variables may be in extended precision, Or Some may be extended and
)me single precision. The following values must be added to NF to specify
,tended precision.
Independent variable (X).
First dependent variable (Y 1).
Second dependent variable (Y2).
Add 2 3 = 8
Add 2 3+ 1 = 16
Add 2 3+2 = 32
Seventh dependent variable (Y7). Add 2 3+7 = 1024
Examples
Number ofDependentVariablesto Be Plotted
3
3
7
7
1
SinglePrecisionVariables
None
X,Yl,Y2
Yl, Y2, Y3, Y4, Y5
All
None
G-92
ExtendedPrecisionVariables
X, Yl, Y2, Y3
Y3
X, Y6, Y7
None
All
NF
3+ 2':":' 3+ 2':":'4+2 ':":' 5+2 ':":' 6
7+ 2':":' 3+ 2':":'9+ 2':"n 0
7
S/360 LIBRARYPLT360Page 18 of 32Rev. 3 April 1967
EBCDIC Codes for Symbols and Special Annotation
EBCDIC codes for all letters and numbers are normal EBCDIC codes;
i.e., A ClB C2
etc.
1 Fl2 F2
etc.
Special Symbols
blank
x(
+tJ.
Vn$
w
/
+
x
M
40 or CO (either is acceptable)
4B period ~:~
4C plot symbol ~
4D left parenthesis ~
4E plus sign
4F plot symbol =
50 plot symbol ®5A plot s ymb 01
5B dollar sign
5C asterisk
5D right parenthesis
5E wild point
5F axis line
60 minus sign
61 slash
6A plot symbol
6B comma
6C plot symbol
6D arrow tail extension
6E mis sing data flag
G-93
6F plot symbol
7B arrow
7C arrow
7D prime (apostrophe)
7E equal sign
7F Aerospace symbol
S/360 LIBRARYPLT360Page 19 of 32Rev. 3 April 1967
Jverlay Requirements
Overlay is possible between completed plots for PLT and PLTJ. PLTW
oust be included with the main link because of the method by which OS/360 defines
>DNAMES and OPEN files.
:rrors
The following errors are recognized by PLT:
1. Illegal call codes (any number other than 1, 2, 3, 4, 5, 6);
2. Error in sequence of calls (setup entrance 1 must be given before any
other entrance); and
3. Number of functions less than 1 or greater than 7.
ate s on Title / Annotation Input
Titles in setup entrance and all annotation may be input in several ways.
lley may be included in the calling sequence as;
e. ,
.... , NA, 'XX .• XXX'
•... , la, 'ANNOTATION' , whe re the title is ANNOTA TION .'.','
Titles may be input by means of READ statements or DATA statements.
mr EBCDIC characters will fill one full word .
. • . . , NA, TITLE...•
e., DIMENSION TITLEl(3) TITLE2(3)
DATA TITLE2/'ANNOTATION2' /
READ(5,I)TITLEI
1 FORMAT(3A4)
CALL PLT (----, la, TITLE 1, ---)
CALL PLT (----, 12, TITLE2, ---)
G-94
S/360 LIBRARYPLT360Page 20 of 32Rev. 3 April 1967
Multiple Report Plotting
Multiple report plotting enables a user to generate several plots simultane
0usly. The resultant plots will be identical to normal sequential plots. One
useful application of these routines would be where large amounts of data are
generated (either read from tape or computed) and several separate plots are
needed of data generated simultaneously. In this case a setup entrance must
be given for each of the desired plots, then, as small amounts of data are
gene rated, data entrances may be given for each separate plot. When all data
and special annotations are complete, a cleanup entrance is given for each plot.
One terminate entrance should be given at the completion of all plots. Any
number of plots may be written concurrently. However, each multiple report
code causes a complete pass by the output processor of the operating system
over all plots written by one program. Therefore, plotting time may increase
when a large numbe r of multiple report codes are used. Codes may be re -used
for subsequent plots after the cleanup entrance is given. Any combination of
plot entrances may be given. However, a multiple report code must be reserved
for one plot until the cleanup entrance for that plot is finished.
The first parameter in the calling sequences to PLT must be negative for
multiple reports, and two additional parameters are needed.
A. Calling Sequences for Multiple Report Plotting
1. Setup Entrance
CALL PLT{ -1, A, B, n, XO, DX, DXl, DDX, nf, YOl, DYl, IS 1, ISFl,
Yl, .••• YOm, DYm, ISm, ISFm, YM, NT, TITLE, NTX,
XTITLE, NTYl, Yl TITLE, •••• )
G-95
S/360 LIBRAR YPLT360Page 21 of 32Rev. 3 April 1967
lultiple Report Plotting (cont. )
Calling Sequence for Multiple Report Plotting (cont. )
-1 Indicates a multiple report setup.
A = location which contains multiple report characters. The first
character must be a comma. The second and third characters
designate the plot and distinguish it from other multiple report plots.
A fourth character is not used.
B = name of an array which is used by PLT to save plot information.
This information must not be changed by the user between setup
and cleanup entrances. B must be dimensioned as:
15+8';'NF words (4 bytes per word)
[For one dependent variable B is dimensioned B(23); for two, B(31);
for seven, B(71); etc.]
The remaining parameters remain the same as in normal usage.
2. Data Entrance
CALL PLT(-2,A,B,NP,K)
-2 Indicates a data entrance
A Same as setup entrance for multiple report plot.
B Same as setup entrance for multiple report plot.
~p) Remain the same as for non-multiple report plot.
3. Annotation Entrance
CALL PLT(-3,A,B,IX,IY,S,NA,ANNOT)
4. Aerospace Symbol Entrance
CALL PLT(-4,A,B,MX,MY,SI)
G-96
,'.','
S/360 LIBRARYPLT360Page 22 of 32Rev. 3 April 1967
Multiple Report Plotting (cant. )
A. Calling Sequence for Multiple Report Plotting (cant. )
5. Cleanup Entrance
CALL PLT{ -5, A, B, IRSC, PLOTIM)
6. Terminate Entrance
CALL PLT{ -6, A, B)
or
CALL PLT(6)
Multiple report designation is not necessary for the terminate entrance.
B. Example for Multiple Report Plotting
Example: Read 20000 data points from cards
Plot Y vs X, Z and Q vs X
DIMENSION B{ 23), D( 31), X( 100), Y{1 00), Z( 100), Q{1 00)
DATAA,C/',AA',',BB'/ .:'
CALL PLT{-l,A, B, 15, 0.,1., X, 0,1,0.,1.,1,1, Y, 5, 'TITLE', 6, .:''X-AXIS', 6, 'Y-AXIS') .:'
CALL PLT{-l, C, D, 15, 0.,1., X, 0, 2, 0.,1., I, I, Z, 0.,1.,2, I, Q, 5 .:''TITLE', 6, 'X-AXIS', 6, 'Z-AXIS', 6, 'Q-AXIS') .:'
J=O
2 DO 11=1, 100
1 READ{5, 100)X{I), Y(I), Z(I), Q{I)
100 FORMAT(4FlO. 5)
CALL PLT{-2,A, B, 100,0)
CALL PLT(-2,C,D,lOO,O)
J=J+lOO
IF (J • LT. 20000) GO TO 2
G-97
V1ultiple Report Plotting (cant. )
3. Example for Multiple Report Plotting (cant. )
CALL PLT(-5, A, B, 10, PLOTLVl)
CALL PLT(-5, C, D, 10, PLOTIM)
CALL PLT(6}
STOP
END
G-98
5/360 LIBRARYPLT360Page 23 of 32Rev. 3 April 1967
S/360 LIBRARYPLT360Page 24 of 32Rev. 3 April 1967
Appendix A - Assembly Language Usage
PLT may be called from Assembler language programs with CALL PLT( ••• )
as in FORTRAN. Alternatively, PLT may be bypassed and PLTW and PLTI
may be called directly from an Assembler language program. However, direct
calls to PLTI and PLTW require:
1. Setting up a parameter table, PLTTBL, as described in Appendix B ':'
before the setup entrance to PLTl;
2. Setting up work storage table as described in Appendix B; and
3. Calling in sequence -
At beginning of all plots
PLTWSET(SETUP IO)
For each plot
PLT l(SETUP)
Any combination of data, change table, and annotation entrances
to PLT 1
PLT l(CLEANUP)
At end of all plots
PLT l(TERM1NATE)
PLTWTERM(CLOSE IO)
4. 10 Call Requirements
PLTWSET
L 15, =V(PLTWSET)
BALR 14,15
PLTWTERM
L 15, =V(PLTWTERM)
BALR 14, 15
G-99
S/360 LIBRARYPLT360Page 25 of 32Rev. 23 May \967
_ppendix A - Assembly Language Usage (cont)
5. PLT 1 Call Requirements -
Register 13 contains the address of work storage table.
Register 1 contains the calling code;
1 = SETUP2 = DATA3 = ANNOTATION5 = CLEANUP6 = TERMINATE7 = CHANGE TABLE
Example - Setup Entrance
TABLE2
PLOTID
LLHLABALR
DSDCDCDSDCDCDCDCDSDCDCDCDCDCDCDCDSDS
DC
15, =V(PLTl)1, HIll13, =A(TABLE2)14, 15
18FA(PLTTBL)CL4'",1IFH ' 1'H'O'F'O'A(PLOTID)4FD'Q',D'Q',D'Q'Dla', DID', DIO'DIal, Dla', Dla'D'O'H'864'H'D'A(TABLE2+408)26DCLl728
CL20'IPLOT NUMBER'
G-100
+72+76+80+84+86+88+92+96+ 112+136+160+184+192 BUFF CT+194+196 BUFF LOC+200+408 BUFFERS
S/360 LIBRARYPLT%OPage 26 of 32Rev. 3 April 1967
Appendix B - Parameter and Work Storage Tables
Parameter Table - Setup on full word boundary:
Addre s s of plot title
Number of characters in plot title
=0 No grid= 1 1" grid drawn over plotting area
=X'OO' Floating point data=X' FO' All data is pre scaled to plotter
increments with origin at lowerleft corner of plotting area.
=1 Means 4 bytes between addresses ofsuccessive independent variable datapoints
=2 Means 8 bytes between addresses=3 Means 12 bytes, etc.
Number of dependent variables
Location of XO (floating)
Location of DX (floating)
Location of DDX (floating)
Location of first data point in independentvariable array
Location of X title
Number of characters - X title
PLTTBL tot4
t6
t7
t8
tlO
t12
t16
t20
t24
t28
t32
t34
A
H
e
e
H
H
A
A
A
A
A
H
e =X'OO'=X'OF'
Do not print X scalePrint X scale
.'.."
t35
t60
t64
t68
t72
t74
eL2S
A
A
A
H
e
Set to zero before setup entrance to PLTI
Location of YO (floating)
Location of DY 1 (floating)
Location of first data point of first dependent variable
Symbol frequency (integer)
Symbol code - Hex code for symbol offirst dependent variable
G-lOl
S/360 LIBRAR YPLT360Page 27 of 32Rev. 3 April 1967
\ppendix B - Parameter and Work Storage Tables (cant. )
+76 A
+80 H
+82 H
+84 CL8
+92to
+123
to PLTI
+75 C Off scale treatment and Yl
scale 'X X 'I 2First HEX character (XIL-0 Wild point= I Mirror image of off scale data is
plotted=2 Off scale data is plotted mod 10 inches
Second HEX character-0 Do not print scale=F Print Y 1 scale
Location of Y 1 TITLE
Number of characters - Y 1 TITLE
Delta Storage - Y 1
Set to zero before setup entrance
Repeat data in PLTTBL+60 toPLTTBL+91 for second dependentvariable
-',','
Repeat for last dependent variable+60+( 32':'(NF -1))to +60+( 32':'NF)-1
Maximum size - 284 bytes (71 full words)
G-I02
S/360 LIBRAR YPLT360Page 28 of 32Rev. 23 May 1967
Appendix B - Parameter and Work Storage Tables (cont)
Call parameters and Work Storage Table - 408 bytes + BUFFER
Double word boundary.
TABLE2 +0
+72
+76
+80
+84
+86
+88
l8F
A
CL4
H
CL2
F
Used by PLTl to store register contents
Location of PLTTBL
Multiple report characters-normally", bhowever, the second and third position m<lYbe another report designator (e. g., XXb)
Unused
Number of this plot - Used in PLOT IDrecord
Initially contain zero
Count of words output stored by routine Should be ze ro initially
PLT 1 Setup Parameter
+92 A Location of PLOT ID (20 characters)
PLT 1 Data Parameters
+92
+96
F
H
Base address of data (added to address ofall data locations)
Number of points this entrance.
PLT 1 Change Table Parameter
+100 F New table address (used to suppress curveconnection between data entrances)
PLT 1 Annotation Entrance
+92
+96
+100
+102
+104
A
E
H
H
H
Location of characters
Size (floating)
Number of characters
IX
IY
G-103
S/360 LIBRAR YPLT360Page 29 of 32Rev. 23 May 1967
.ppendix B - Parameter and Work Storage Tables (cont)
PLT 1 Cleanup Entrance
+92
+96
+112
+192
+196
+408
H
A
H
A
IRSC
Location of PLOTIM
General storage used by PLTI (TABLE+112to TABLE+204 must be set to zero beforethe setup entrance with the followingtwo exceptions. )
Size of single buffer
Location of first byte of double buffer(Table+408)
Buffer (ln8 characters)
G-I04
S/360 LIBRARYPLT360Page 30 of 32Rev. 3 April 1967
Appendix C - Non-Standard Usage
Any usage of PLT other than that specified in the calling sequence must be
implemented by changing the parameter table in PLT. (See Appendix B). An
entry is available in PLT for this purpose (PLTTBL). However, with the excep- ,;,
tion of "8." these changes must be made after the setup entrance by means of
an assembler language subroutine. Some of the non-standard usages which may
be desired are:
1.
2.
3.
Grid (A one inch grid is drawn on the plot when the cleanup entrance is made.
This grid covers the complete plotting area from the left-hand to the right
hand dependent variable scales. This may be added anytime before the clean
up entrance. See PLTTBL+6. )
Prescaled data (Data is given to PLT already scaled in 100th inches refer
enced to the lower left corner of the data field, which is 1/2 inch above the
bottom of the paper and to the right of the left-hand dependent variable
scales. This may be specified before any data entrance. See
PLTTBL+7. )
Storage of data (Data is stored in non-consecutive bytes, but at a constant
increment. See PLTTBL+8. ) .'-,'
4. Location of data arrays.
5.
6.
Symbol frequency (PLTTBL+72)
Symbol (The symbol code itself must be added, not the number of the sym
bol. PLTTBL+74.)
"",
7. Titles (However, left-hand dependent variable titles and plot title are out
put by the setup entrance, so changes will have no effect. )
G-105
8. XO, DX, YO, DY may be changed in mid-plot merely by changing the num
ber stored in these locations. Any subsequent data scaling and scale values
printed will reflect the new values. No assembler language subroutine is
required for these changes.
S/360 LIBRARYPLT360Page 31 of 32Rev. 3 April 1967
~ppendix C - Non-Standard Usage (cont. )
A change to storage of data, location of data arrays, symbols and symbol
requency must be made for each variable desired before the affected data en
ranee is made. Changes to grid and titles may be made at any time before the
leanup entrance.
The following example changes the locations of the independent variable and
le first dependent variable arrays.
FORTDIMENSION Xl{lOO), Yl( 100), X2{l00), Y2{l00)REWIND9READ(9)(Xl(I), Yl(I), X2(I), Y2(I), 1=1,100)READ(5, 100)XO,DX, YO,DYCALL PLT(I, O,XO,DX,Xl, 0.,1, YO,DY, 1, 1, Yl,5, 'TITLE', 1, 'X',
1, 'Y')CALL PLT(2, 100, 1)CALL CHANGE(X2, Y2)CALL PLT(2, 100, 1)CALL PLT(5, 10,PLOTIM)CALL PLT(6)
100 FORMAT(4FI0. 0)STOPEND
,',','
\SMENTRY CHANGEUSING ", 15
iANGE STM 2,3, TEMPS TOR Store contents of registers to used.L 2, =V(PLTTBL) Address of plot table. ,'.
','
L 3, O{l) Address of independent variablearray.
ST 3,24(2) Store PLTTBL+24 ,',','
L 3, 4{l) Address of dependent variablearray.
ST 3,68(2) Store PLTTBL+68 ,'.0'
LM 2, 3, TEMPSTOR Restore registers.BC 15, 14 Return
:MPSTOR DS 2FEND
G-I06
S/360 LIBRAR YPLT360Page 32 of 32Rev 15 September 1967
Appendix D - Output Record Size Changes
The size of records on the plot file may be changed by changing two cards
in PLT and re-assemblying PLT.
Normal record size is 864 characters. In most cases, smaller record sizes
will decrease, and larger sizes will increase efficiency. The minimum record
size is 30 characters/buffer. Maximum is 864 characters/buffer for on-line
plotting and 1140 characters /buffer for off-line plotting.
The cards in PLT which must be changed to alter record size are:
BUFCT
DC
EQU
CLln8
864
These cards must be changed to the following:
BUFCT
DS
EQU
2';'REC\9RD SIZE
REC\9RD SIZE
i. e., for 950 character records
BUFCT
DS
EQU
CL1900
950
These cards are sequenced, columns 73-80, D0440431 and D0440432 in the ,;,
source deck that is on file in the MCC Program Library.
G-l07
5/360 LIBRARYPLT360Page 32-1Rev 8 January 1968
_ppendix E - Format of Plot File
All information output by PLT360 for all plots during one computer run is
ontained in one physical file. The file contains two types of records; operator
lformation records and plotter command records. All records are 864 bytes
1 length. When the plot information doe s not fill 864 bytes, the hexadecimal
haracters 'FF' are inserted in the byte following the plot information to desig
ate the end of the logical record. Any information following the I FF' is to be
;nored. The initial six bytes of each physical record are control characters
f the form, XXXGY (EBCDIC code) whe re:
is present as the first EBCDIC character of each record.
xxx
G
are multiple report designator bytes. These are input to PLT360
by the calling program when multiple report plotting is used. Under
nonmultiple report usage these are the EBCDIC characters: " b or
, , 1. The END OF PLOT TAPE record always contains the hexa
decimal characters FFFEFI in this field.
is present as the fifth EBCDIC character of each record. '.'
Y may be either T or G. T designates that the remainder of this record
is operator information and is to be sent to the typewriter. G desig
nates that the remainder of this record contains plotter commands.
An operator information record is output at the beginning of each plot and
: the end of all plots. These are in EBCDIC character code.
,XXXGTIPLOT NUMBER NNN (NNN contains the number of this plot
beginning with 001 for the first plot of the computer run.)
,XXXGTEND OF PLOT TAPE (This record is followed by an end of
file mark. )
Plotte r command records begin with the six byte s: ,XXXGG.
he remainder of these records contains EBCDIC plotter commands.
G-I08
8/360 LIBRARYPLT360Page 32-216 November 1967
Appendix E - Format of Plot File (cant)
When the last buffer is not full, the hexadecimal characters 'FF'
are inserted immediately following the last plotter command. The remainder
of the information is to be ignored.
The following codes are output as plotter commands.
HEX HEXFO PEN DOWN F5 -yFl +y F6 -X-YF2 +X+Y F7 -XF3 +X F8 -X+YF4 +X-Y F9 PEN UP
+Y is pen movement toward the top of the paper.
+X is drum movement in the direction of the take -up spool.
G-I09
G.7 MODIFICATION OF PROGRAM CAPACITY
Because of the inevitability of modification of this program by other
users, the following description of program size is in order. The SAAS III
program is written to fit in a 216K (54, 000 words) core storage of an IBM
360 computer (4 modules in an MVT system), in a 50K (50, 000 words) core
storage of a UNIVAC 1108 computer, or in a 142s K (49,168 words) core
storage of a CDC 6600 computer. Auxiliary FOR TRAN units are used and are
described in Appendix G, Section G. 1. Since the program is highly compact
due to extensive overlaying in COMMON, it is unlikely that the program will
fit on a computer with less core storage than noted above (e. g., the IBM 7094)
without sacrificing the program capacity. The capacity of the program can
be reduced or increased from the current 1000 nodal points Or elements, 6
materials, 12 material property temperatures, semibandwidth of 50, or
number of temperature cards by modification of certain dimension statements.
These are:
1. Number of Nodal Points - NUMNP - The dimension of 1000
in COMMON arrays R, CODE, XR, Z, XZ, T, and PST
must be changed to the new maximum value. The dimension
of R, Z, T in Subroutine FLDIN and Subroutine TEMI must
also be changed.
2. Number of Elements - NUMEL - The dimension of 1000 in
COMMON arrays IX, EPR, and ALPHA must be changed to
the new maximum value.
3. Number of Different Materials NUMMAT - The maximum
number of different materials is defined by the subscript 6
in the following COMMON statement:
COMMON/MATP/RO(6), AOFTS(6), E(12, 16, 6), EE(21),
POROTY(6)
G-ll0
4. Number of Material Temperature Cards - NT - The maximum
number of material temperature cards is given by the subscript
12 in COMMON /MATP /. In addition, the maximum number
appears in the following FORTRAN statements:
DO
DO
200
100
I
MM
= NT, 12
;:: 2, 12
MATL 60
PROP 27
5. Bandwidth - MBAND - The nodal point connectivity is presently
25, which can be changed to "b" by making the following changes:
COMMON/SOLVE/B(4b), A(4b,2b), NUMTC, MBAND
COMMON/SOLVE/X(c), Y(c), TEM(d), NUMTC, MBAND
where c =~ [4b(2b + 1)] truncated to nearest integer value
d = 4b(2b + 1) -2c
NB = b
NN = 2b
STIF 23
SOLV 11
NCODE(b-2,i)
MESH 18
MESH 19
Also, the mesh generation procedure ensures nodal point
connectivity by limiting the size of the I variable of the
I-J grid. (See Appendix A, Section A. 1 for a description
of mesh generation.) Note that b must be greater than or
equal to the maximum value of I plus 2 in order that the
bandwidth be sufficient to accommodate that mesh width.
Note also that the maximum mesh width for most mesh
generation problems must be less than or equal to 23 as
the program is now written. Therefore, to alter the
allowable bandwidth for mesh generation, the following
changes must be made:
COMMON/TD/IMINCi), IMAX(t), JMIN(b-2), JMAX(b-2),
MAXI, MAXJ, NMTL, NBC
DIMENSION AR(b-2, il. AZ(b-2,i),
DO 110 J = l,t
DO 100 I = 1, b-2
G-ll1
In Subroutine CONTR, the dimension of ANS is 2>:'NUMNP,
the dimension of X, Y, W is the same as AR and AZ in
Subroutine MESH, and the dimension of DUM is the length
of X + Y. Y is equivalenced to the midpoint of DUM plus 1.
Also, the number 25 is changed to the maximum value of I
in CONT 156 and 169. The dimensioning must be such that
l~'(b - 2) is less than three times the maximum number of
nodal points.
6. Number of Temperature Cards - NUMTC - The maximum
number of temperature cards is given by the subscript 1700
in the arrays X, Y, and TEM in COMMON !SOLVE. For
input of temperature fields on a tape, the number 1201 in
the EQUIVALENCE statement in Subroutine TEMI must
be changed so that it equals 1(2':'NUMTC - NUMNP) rounded
up to the nearest odd integer value.
The above changes must be reflected in the implied
length of COMMON in Subroutine REST.
In summary, the changes noted above affect the following cards:
MAIN 8, 9, 10, 11, 16, 18
REST 4, 5, 6, 8, 10
MESH 3, 4, 5, 18, 19
MNIM 3
NODE 3
POIN 4, 5, 7, 8, 10
PNTN 8, 9
FLDN 8, 10, 11
TEMP 8
TEMI 7, 8, 9
PBND 7, 8
G-1l2
APPENDIX H
EXAMPLE PROBLEMS
The following numerical examples with known solutions are presented
to illustrate the use of the various capabilities of the program and to provide
test cases for use when SA AS III is run at other computer installations. All
numerical examples were run on the IBM 360 MVT Computer System at
The Aerospace Corporation, San Bernardino Operations.
H.l HOLLOW CYLINDJ':R WITH UNIFORM INTERNAL AND EXTERNALPRESSURE (LAME CYLINDER)
The well-known Lam~ cylinder solution (Ref. 27) for an elastic
isotropic material is used to check the answers obtained by use of the
SAAS III program. The cylinder is idealized by four elements as shown
in Figure H-l.
The pertinent parameters of the cylinder are:
a. = 5000 psi r. = 1 in.1 1
ao = la, 000 psi r = 2 in.0
E = 3 x 106
psi IJ = O. 3
The computer output is displayed in Figure H-2, and the computer
results are given in Table H-l along with the exact results obtained by the
use of Ref. 27. As can be noted from the table, the computer results are
very close to the exact results. Even better results, however, can be
obtained by the use of more elements in the radial direction. Convergence
to the exact results is discussed in Appendix D.
H-l
z
27 8 9
(]) @4
5 6
CD ®1
O'j 2 3 0'0
No
~ 00..---....----2----.. r
Figure H -1. Four -Element Idealization of Hollow Cylinder
Table H-l
EXACT AND COMPUTER STRESSES FOR HOLLOWCYLINDER OF FIGURE H-l
Element Stresses at Element Centera a a a
rEXACT r SAAS gEXACT gSAAS
1 -7400 -7329 -15,933 -16,032
2 -9490 -9473 -13,844 -13,840
H-2
10
20
30
40
50
60
70
80
123456789012345678~012345b7890123456789012345678901234567H9012345678901234567090
LAM
EC
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ND
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STC
ASt
11
14
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33
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END
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H-2
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I(C
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)
H.2 HOLLOW CYLINDER WITH NONCYLINDRICAL ORTHOTROPYUNIFORM PRESSURE
This example was contributed by Dr. David Rodriguez. The computer
input is simple as is the output, yet the essential features of the transforma
tion of material properties are illustrated. The axisymmetric solid in the
example is a hollow cylinder with noncylindrical orthotropy under uniform
pressure and temperature. For the case in which the material is transversely
isotropic with the plane of isotropy inclined at 45 degrees to the body axes,
the resulting stresses are isotropic for the particular material properties
chosen. This result can be seen to be a consequence of the form of the
elasticity equations.
The following is a derivation of the material properties necessary to
give an isotropic state of stress in a body. An isotropic state of stress is
defined in body coordinates as
Urr= U\j\j = uzz = U
u rz = 0
and in local coordinates as
U = u\j\j = il = ilmm nn
u mn= 0
(H-l)
(H-2)
(H-3)
(H-4)
The strain transformation equations for a rotation of the plane of isotropy
45 degrees from the body axes are
1( €tn.m + Emu) (H-5)(; = (;
rr 2 nn
1( Erom + + iron) (H-6)(; = 2" (;
zz nn
(; = (; (; (H-7)rz mm nn
H-9
However, for the isotropic stress distribution, the shear strains are zero:
< = < < = 0rz mm nn
< = 0mn
but Our
< =~rr
(I-l-8)
(I-l-9)
(I-l-10)
Hence, upon substitution of the stress-strain relations in local coordinates and
when the Poissonts ratios are zero,
< = (J IE + a Tmm mm m m
< = (J IE + a T (I-I-II)nn nn n n
Egg = (Jgg/Em + am T
and using the strain transformation relation in Eq. (I-l-5), Eq. (I-l-IO) b"conws
(H-12)
Upon integration,
u =r
but
<gg =
::r dr = r [I (E~ t E~ )+ iu
rr
(I-l-13)
(H-14)
so, from Eqs. (H-3), (H-ll), (H-13), and (H-14),
~+aT=E I
Q.2 (E~ + E~)
H-IO
Tt 2" (H-15)
Solution of Eq. (H-1S) for T yields
T= a (E~ - E~ )/ (am - an) (H-16)
Equation (H-l6) represents a relation between imposed stress and tempera
ture and material properties which must be satisfied in order for an isotropic
state of stress to exist.
For the numerical values,
E = 104
psim
E = 0.5 x 104
psin
Ct = 10-4/ oF
m
Ct n= 2 x 10-4 / of
a = -100 psi
Equation (H-16) yields T = 100 degrees. For these values, it is easily
verified that ur is zero; hence, lOgg is zero. That result plus verification
of the isotropic stress state of 100 psi is shown in the computer output. The
shear modulus is immaterial since it is never utilized.
The geometry of the hollow cylinder is shown in Figure H-3.
a 1 2 3
ITIJJa
6
aITITI7 8 9
@ @5
CD ®4
z
2
No
~ O~---+----'*2----""r
Figure H-3. Hollow Cylinder - Uniform Pressure
H-ll
The computer output is listed in Figure H-4. Note that the loading
IS input by the use of pressure cards as opposed to the method used in the
first example, and that the angle of inclination of the local coordinates was
specified for each clement.
H-12
10
203
04
05
06
07
08
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
0RO
DRI
GU
EZTE
STCA
SEN
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NIF
ORM
PRES
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33
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H-4
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for
Ex
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2(C
on
tin
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)
H.3 HOLLOW CYLINDER WITH NONCYLINDRICAL ORTHOTROPY AXIAL LOAD
This example, as was that given in Section H. 2, is due to
Dr. Rodriguez. In a manner similar to that of the previous example. it can
be shown that. with the properties
E " 0.1 x 105 psi II " II " 0m m n
E " 0.33333 x 104psi am " C1 " 0
n n4
G " 0.25 x 10 psimn
and an angle of 45 degrees between the local and general coordinates. the
body in Example 2 given in Section H. 2. when subjected to an axial pressure
of 200 psi as shown in Figure H-5. has normal stresses of 100 psi in the
local coordinate s.
The computer output is given in Figure H-6.
z
(JITIJ]7 8 9
2
4® @)
5CD ®
6
Noi8~;:;
oo
1 2 3
ITIIJ(J
2
Figure H-5. Hollow Cylinder - Axial Load
H-19
10
20
30
40
50
60
708
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
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01
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H.4 THICK SPHERICAL SHELL OF A BILINEAR ISOTROPICMATERIAL UNDER UNIFORM INTERNAL PRESSURE
In order to illustrate the iteration and convergence ieaturcs for
bilinear material characterizations, SAAS III results are compared with an
exact solution for a thick spherical shell of a bilinear isotropic material
subjected to a uniform internal pressure (Ref. 28). The thick spherical
s hell has an inte rnal radius of 5 inche s, an exte rnal rad ius of 10 inche s,
an elastic modulus of 10 x 10 6 psi, a Poisson's ratio of 0.3, a yield
stress oi 104
psi, and a plastic -elastic modulus ratio of 0.5. The internal
pressure is 16,469.2 psi, which results in yielding at a radius of 7.5 inches
and less. (The yield radius was specified and the pressure required to
force such a yield radius was derived.)
The finite element solution is obtained for a wedge -shaped ring of
the spherical shell so that full advantage is taken oi the spherical symmetry
of the problem within the limitation of SAAS III to axisymmetric solids.
A schematic diagram of the wedge-shaped ring is shown in Figure H-7. A
short computer program was written to generate the geometry of the
wedge-shaped ring, subject to the constraint that H equal W in
Figure H-7 (i. e., an aspect ratio of 1 was prescribed) as closely as
possible for a fixed angle, C1. Thus, the angle C1 is varied to obtain
different numbers of elements in the radial direction.
The computer output is given in Figure H-8.
shaped ring is treated by use of the skew boundaries
in Appendix A, Section A-2.
Note that the wedge
capability described
Extensive numerical and theoretical results for the thick spherical
shell problem are given in Table H-2. (The numerical results were obtained
by use of The Aerospace Corporation EI Segundo Operations CDC 6600
computer.) The four-element solution is within 6 percent of the exact
solution whereas the 35-element solution is within O. I percent of the exact
solution. Note that the radial stress converges more rapidly than do the
H-24
axial and circumferential stresses (which must be equal for this spherically
symmetric problem). Note also that the four-element case presented in
Figure H-8 produced somewhat different results from those shown in
Table H-2. This is due to the different method of calculating stresses (see
Appendix A, Section A. 4). Since both methods apparently converge to the
same answers, as demonstrated in Appendix D, the difference is not
considered important.
z
w__H
+
Figure H-7. Schematic Diagram of Wedge-Shaped Ring
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4
H.5 HOLLOW CYLINDER COMPOSED OF TWO MA TERLALS
This problem was selected to demonstrate and check out the
porous media feature of SAAS III. The cylinder is idealized by ten
elements as shown in Figure H -9.
z
MATERIAL 1 MATERIAL 2. .I \t \
1 2 3 4 5 6 7 8 9 10No
'"~~ o
2.0 3.0r
Figure H-9. Ten-Element Idealization of Hollow CylinderComposed of Two Materials
The pertinent parameters of the problem are:
Pi = 0 for the inner material
5 -5for the outer materialPf = r - r
F = 0 (net axial force)z
Pout = 2.3
Pin = 0
E = I for both materials
V = 0 for the inner material
V = 0.2 for the outer material
H-34
The computer output is shown in Figure H-IO. The strain output
was omitted because of the obviously high values that result from use of
the above parameters. (These values exceed the field length of the
variable.) This is a demonstration only and is not meant to represent a
practical problem, whereas the allotted field lengths were designed to
apply to most problems.
That the following equations are the correct solutions may be
easily verified by consulting Ref. 24 where (f is the stress at the interfaceo
and is equal to 3.47. Generalized plane strain is assumed and (0 is the
axial strain.
Material 1
<1 = (-11.I2+2.78r 2)r
Material 2
<1r = <10
(-2.27 + 2°~i54 )_26i' 6 + 0.89347r5
r
- 44.2
= (II. 12 + 2. 78r 2 )
€o
<1 0-2r
UQ = - U (2.27 + 2:'2
454) - 44. 12
0
269.6 5+ 2+ O.36r
r
- 0.9082
Uz = U - 17.65 + O. 250r + (0 0
The computer results are plotted in Figures H-ll through H-13 along
with the exact results obtained by using the above equations. As can be
noted, the finite element results are excellent. However, better accuracy
can be obtained by USing more elements in the radial direction.
H-35
1 1 1 1
0.0
1.0
c.c
C.C
1.1
14
1
0.1
C.C
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10
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33
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53
3.0
35
59
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35
9.6
43
90
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69
0.3
16
12
5.1
11
25
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1.0
.0 .1 .0 .1 .1 .0 .0 .1 .1 .0 .0 .1 .1 .0 .0 .1 .1 .0 .0 .11
1020
304G
506
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801
23
45
67
89
01
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2.3
52
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2.4
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2.5
52
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2.6
52
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2.7
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.85
2.8
52
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CEN
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Fig
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5(C
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-"EXACT" SOLUTION (Ref 24)150 t:. SAAS III FINITE ELEMENT SOLUTION
125
.- 100'"...-b
75
50
25
o~,,",,"",~~~;;:::~~-~_~_-.--l2.0 2.2 2.4 2.6 2.8 3.0 3.2
RADIUS· in.
Figure H-ll. Radial Stress in Hollow Cylinder
H-43
30
20
--"EXACT" SOLUTION (Ref 24)
6. SAAS III FINITE ELEMENTSOLUTION
Nb
10
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RADIUS - in.
Figure H-12. Axial Stress in Hollow Cylinder
H-44
...
60
--"EXACT" SOLUTION (Ref 24)50 ~ SAAS III FINITE ELEMENT
SOLUTION
40
30
20 ~
10
o~__~__~__~__~__~__"'"2.0 202 204 206 2.8 3.0 3.2
RADIUS - in.
Figure H -13. Circunlierentia1 Stress in Hollow Cylinder
H-45
H.6 SOLID POROUS CYLINDER
This problem was selected to demonstrate and check out the porous
media feature of SAAS III. The cylinder is idealized by ten elements as
shown in Figure H-14.
z
0) 0 0) 0 CD 0 0 C9 CD ®0.1
0.5 1.0
Figure H-14. Ten-Element Idealization of Solid Cylinder
The pertinent parameters of the problem are:
-4Ct T = 1.5xlO
Pi = 100 for 0 ~ r ~ O. 1
. -100 (rZ- r - 1) for O. 1 ~ r ~ 1. 0Pi =
F = 1.0 (net axial force)z
Pout = 100
E = 106
IJ = 0.5
H-46
In the program, V = 0.5 IS not allowed. Therefore, V = 0.49 was used as
a reasonable approximation.
The computer output is displayed in Figure H-15.
That the following equations are the correct solutions may be easily
verified by consulting Ref. 24.
a = IOOr (l - r)r
a = a = IOOr (1 - r)Q r
10 Or (1 - r) - 17=a z =ar -17
The computer results are plotted in Figures H-16 and H-17 along
with the exact re sults obtained by using the above equations. As can be
noted, the finite element results are excellent. However, better accuracy
can be obtained by using more elements in the radial direction.
H-47
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--"EXACT" SOLUTION (Ref 24)30 6 SAAS III FINITE ELEMENT SOLUTION
01.Clo-----------......--......_-__....._---'o 0.2 0.4 0.6 0.8 1. 0 1. 2
RADIUS· in.
Figure H-l6. Radial and Circumferential Stress in SolidCylinder
H-56
N
t::l
10
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-10
--"EXACT" SOLUTION (Ref 24)6 SAAS III FINITE ELEMENT SOLUTION
-20~_~~__~__~~_~~__~_~o O. 2 0.4 O. 6 O. 8 1. 0 1. 2
RADIUS - in.
Figure H-l7. Axial Stress in Solid Cylinder
H-57
H.7 THICK SPHERICAL SHELL OF A MULTIMODULUS ISOTROPICMATERIAL UNDER INTERNAL AND EXTERNAL PRESSURE
In order to illustrate the iteration and convergence features of
SAAS III for multimodulus materials, SAAS III results are compared with
an exact solution due to Ambartsumyan and Khachatryan (Ref. 18) for a
thick spherical shell of an isotropic material with different moduli in
v , is O. 1. Note thatc
satisfy Ambartsumyan' s reciprocal relation
tension and com pre s sion subj ected to internal and exte rnal pre s sure. The
thick spherical shell has an internal radius of 5 inches and an external
radius of 10 inches. The tensile modulus, Et
, is 6 x 10 6 psi, the
compressive modulus, E, is 3 x 106
psi, the tensile Poisson's ratio,c
vt' is 0.2, and the compressive Poisson's ratio,
the moduli and Poisson's ratios
=
The internal pressure is 100 psi, and the external pressure is -100 psi
(tension on the surface).
As was shown in Example 4, the finite element solution is obtained
for a wedge -shaped ring of the spherical shell so that full advantage is taken
of the spherical symmetry of the problem within the limitation of SAAS III
to axisymmetric solids. A schematic diagram of the wedge-shaped ring is
presented again in Figure H-18. A short computer program was written
to generate the geometry of the wedge-shaped ring, subject to the constraint
that H equal W in Figure H-18 (i. e., an aspect ratio of I was prescribed)
as closely as possible for a fixed angle, a. Thus, the angle a is varied
to obtain different numbers of elements in the radial direction.
The computer output is given in Figure H-19. Note that the wedge
shaped ring is treated by use of the skew boundaries capability described
in Appendix A.
H-58
10
w--.
z
NC>.."'a 01
Ht
Figure H-l8. Schematic Diagram of Wedge -Shaped Ring
Extensive numerical and theoretical results for the thick spherical
shell problem are given in Table H-3. The numerical results were
obtained by use of the IBM 360/65 computer. The four-element solution
is within 6 percent of the exact solution. Further accuracy improvement
is achieved with increased elements, but the results are masked by the
truncation of stresses in the SAAS III program.
H-59
10
20
30
40
50
601
08
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24
9.
H.8 PLANE STRESS SOLUTION TO THE BENDING OF A CANTILEVERBEAM
In order to illustrate the plane stress option, SAAS III results are
compared to a two-dimensional theoretical result (Ref. 27) for a cantilever
beam loaded at the free end by a parabolical shear distribution.
The finite element model of a cantilever beam is illustrated in
Figure H-20. Two fixed end boundary conditions were investigated in
Ref. 27. The first sets the slope of the mid-surface at the end to zero.
The second sets the rotation of the normal to the mid-surface equal to zero
at the mid-surface. Both boundary conditions allow warping of the end
cross section and enable One to find simple closed-form solutions to the
problem. With SAAS III, the fixed end condition was obtained by specifying
zero longitudinal (u ) displacement of all nodal points at the end plusr
zero transverse (u ) displacement of the nodal point at the mid-surface.z
Since this boundary condition does not exactly duplicate those employed in
Ref. 27, absolute convergence of results will not be demonstrated.
However, the conditions are sufficiently similar so that the theoretical
solution serves as a reasonable check on SAAS III results.
The total shearing force applied on the ends of the cantilever is
1000 pounds. Young's modulus was 30 x 106 psi and Poisson's ratio
was 1/3. Part of the computer printout is shown in Figure H-21.
Figure H-22 is a computer-generated plot showing the deformed
shape of the cantilever with exaggerated displacements (u x 50). This
illustrates the capability of plotting deformed grids for rapid screening
of computer results. Figure H-23 is a computer -generated plot of con
tours of longitudinal stress, (J.r
As can be noted from the printout, the end point deflection of
0.01672 compares favorably with that given by Timoshenko (Ref. 27) of
u (r = 0, z = 0) = 0.016666 ..•.• and u (r = 0, z = 0) = 0.017333 .•.• forz zthe first and second boundary conditions respectively.
H-67
11.
9.
5.
(J)....~ -1.I~
-1.Z-AXIS
1.
Figure H-20. Element Plot
H-68
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30
40
50
60
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H-84Q
REFERENCES
1. R. W. Clough, "The Finite EleInent Method in Plane Stress Analysis,"Proceedin s, 2nd ASCE Coni. on Electronic COInputation, Pittsburgh,Pennsylvania eptember 9
2. M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp, "Stiffness and Deflection Analysis of Complex Structures, " Journal of theAeronautical Sciences, VoL 23, No.9, pp. 805-823 (September 1956).
3. R. J. Melosh, "A Stiffness Matrix for the Analysis of Thin Plates inBending, " Journal of the Aeronautical Sciences, VoL 28, pp. 34-42(1961).
4. R. W. Clough and Y. Rashid, "Finite Element Analysis of AxisymmetricSolids," Journal of the Engineering Mechanics Division, ASCE, pp. 7185 (February 1965).
5. Y. R. Rashid, "Analysis of Axisymmetric Composite Structures by theFinite Element Method, " Nuclear Engineering and Design, VoL 3, pp.163-182 (1966).
6. E. L. Wilson, "Structural Analysis of Axisymmetric Solids, " AIAAJournal, VoL 3, pp. 2269-2274 (1965).
7. P. E. Grafton and D. R. Strome, "Analysis of Axisymmetric Shellsby the Direct Stiffness Method," AIAA Journal, VoL I, No. 10, pp.2342-2347 (October 1963).
8. J. H. Percy, T. H. H. Pian, S. Klein, and D. R. Narvaratna, "Application of the Matrix Displacement Method to Linear Elastic Analysisof Shells of Revolution," AIAA Journal, VoL 3, No. 11, pp. 2138-2145 (NoveInber 1965).
9. R. J. Melosh, "Structural Analysis (If Solids," Journal of the StructuralDivision ASCE. PP' 205 -224 (August 1963).
10. J. H. Argyris, "Continua and Discontinua," Proceedings of the Conf.on Matrix Methods in Structural Mechanics, Dayton, Ohio, October 1965,AFFDL-TR-66-80, pp. 11-189 (November 1966).
R-l
11. L. R. Herrmann, "Elastic Torsional Analysis of Irregular Shapes,"Journal of the Engineering Mechanics Division, ASCE, Vol. 6, pp.11-19 (December 1965).
12. O. C. Zienkiewicz and Y. K. Cheung, "Finite Elements in the Solutionof Field Problems," The Engineer, Vol. 220 (1965).
13. E. L. Wilson and R. E. Nickell, "Application of the Finite ElementMethod to Heat Conduction Analysis," Nuclear Engineering and Design,Vol. 4, pp. 276-286 (1966).
14. E. L. Wilson and R. M. Jones, Finite Element Stress Analysis ofAxis mmetric Solids with Orthotropic, Tern erature -DependentMaterial Properties, TR-O 58(S38 -22)-, The Aerospace Corporation,San Bernardino, California (September 1967). (Available only fromthe Defense Documentation Center.)
15. R. M. Jones and J. G. Crose, SAAS II, Finite Element Stress Analysisof Axisymmetric Solids with Orthotropic, Temperature -DependentMaterial Properties, TR-0200(S4980)-I, The Aerospace Corporation,San Bernardino, California (September 1968). (Available from theDefense Documentation Center. )
16. R. D. Cook, "Strain Resultants in Certain Finite Elements," AIAAJournal, Vol. 7, No.3, p. 535 (March 1969).
17. E. M. Lenoe, D. W. Oplinger, and J. C. Serpico, "ExperimentalStudies of the Elastic Stability of Three -Dimensionally ReinforcedComposite Shells," AIAA Paper No. 69-122, New York, January 1969.
18. S. A. Ambartsumyan, "Basic Equations and Relations in the Theoryof Elasticity of Anisotropic Bodies with Differing Moduli in Tensionand Compression, " Inzhenernyi zhurnal, Mekhanika tverdo 0 tela,No.3, pp. 51-61 (19 9). Translation available as LRG-70-T-The Aerospace Corporation, El Segundo, California.)
19. S. W. Tsai, A Test Method for the Determination of Shear Modulusand Shear Strength, AFML-TR-66-372, Air Force Materials Laboratory,Dayton, Ohio (January 1967).
20. J. G. Crose, Finite Element Stress Analysis of Porous Media,TR-0200(S4816 -76)- 1, The Aerospace Corporation, San Bernardino,California (23 May 1969).
21. M. A. Biot, "Theory of Elasticity and Consolidation for a PorousAnisotropic Solid," Journal of Applied Physics, Vol. 26, No.2,pp. 182-185 (February 1955).
R-2
22. M. A. Biot, "Theory of Deformation of a Porous Viscoelastic Anisotropic Solid," Journal of Applied Physics, Vol. 27, No.5, pp. 459467 (May 1956).
23. M. A. Biot. "Theory of Propagation of Elastic Waves in a FluidSaturated Porous Solid, " Journal of the Acoustical Society of America,Vol. 28, Nos. 1 and 2, pp. 179-191 (March 1956).
24.
25.
26.
27.
28.
P. Tong and T. H. H. Pian, "The Convergence of Finite ElementMethod in Solving Linear Elastic Problems, " International Journalof Solids and Structures, Vol. 3, No.5, pp. 865-879 (September 1967).
Calcom Software Reference Manual, Section 1. Basic Software,alifornia Computer Pro ucts, Anaheim, California (February 1968).
S. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill,Book Company, Inc., New York, New York, pp. 58-60 (195l).
A. Philips, Introduction to Plasticitf' The Ronald Press Company,New York, New York, pp. 168-179 1956).
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