AD-AO84 926 NAVAL POSTGRADUATE SCHOOL MONTEREY CA F/S 13/13FINITE ELEMENT PROGRAM SUITABLE FOR THE HEWLE1TPACKARD SYSTE--ETC(U)

MAR 80 R R MALLORY

UNCLASSIFIED N

1I 211 112.

IWO__ 112.01.1 1.8

1111.25 INi .HIl

MICROCOPY RESOLUTION TEST CI-$TNATIONAL BUREAU OF STANDARDS-1963-

LiE1NAVAL POSTGRADUATE SCHOOL

Monterey, California

00

THESISFAFINITE ELEMENT PROGRAM SUITABLE FOR THE

DESKTOP COMPUTER

by

Ray Ricardo Mallory

March 1980

Thesis Advisor: G. Caritin

Approved for public release; distribution unlimited.

LAIS ST Q~UALITY PACOTICABIOU__ CMY Tj* TOD DfC CcNTAINED

SIG'IIFlCANT NUJ1A3R DI ?kGLW

80 5 30 068

DISCLAIMER NOTICE

THIS DOCUMENT IS BEST QUALITYPRACTICABLE. THE COPY FURNISHEDTO DDC CONTAINED A SIGNIFICANTNUMBER OF PAGES WHICH DO NOT

REPRODUCE LEGIBLY.

Rtflf TFENAT PAGE 331033 CMPLCTING rO*g

1REPORT MUMEENW 12. *OVT ACCESION NeO.NCPIN' AALGMNF

S. TYPE OF REPORT At .EIoo COVERED

A Finite Element Program Suitable for the/ master's Thesis;Hewlett-Packard System 45 Desktop March 1980. MME

Computer itt ---- =Pa~e"POImaw 011. REPORITNU*

=AT______ 6. CONTRACT On GRANT NumbuEfl)

IA Ray Ricardo Mallory /

9. 0911FRMNwi ORGANIZATION NAME ANC ADDRESS 10- PROGRAM ELEEN1T. PROJECT. TASK

Naval Postgraduate School AR9A a WORK UNIT NUMBERS

Monterey, California 93940/ j.'

I I CONTROLLING OFFICE NAMIE AND ADDRESS 'I.NEP Kpa T~

Naval Postgraduate School ~QMonterey, California 93940 99

14. N.QNITOMING AGENCY MAMIE at AOD ESS(If diflfenat #Mo ConfIIII84 Offie..j a. SECURITY CLASS. (Of title t***vt)

(~ /l'~9 . i~.$? -~Unclassified

ASFCATION DOWNGRADING

1S. OaSTROGUTION STATEMENT (of thi. Naomi)

Approved for public release; distribution unlimited.

17. DISTRIOUTION STATEMENT (Of tie 064tr600 00#6Md In 8004k 34. if dUnifm tw ASPIel)

IS. SUPPLEMENTARY NOTES

19. MET WORDS (Confine an 1evor. side 01RneOOaRMY And IdefflifI for biaeh n~bt)

Stress Analysis, Finite Element, Computer Program, ComputerGraphics

20. AIISj 3 7 CT (Ce"110wo an pfeoo wdu of "600eeW and Idmedil Ai 61" biee* Noe)

The object of this work was to explore the possibility ofscaling a Finite Element program to fit in a small desktopcomputer. A program originally written in FORTRAN IV andcalled STAP was selected for the present project. This programis contained in Bathe and Wilson's Numerical Methods in FiniteElement Analysis and was conceived to solve static linear

DD 1473 goDITION OP I Nov a IsS OS1SOLETE NLSIFE(Page 1) SIM 0102.0t*s.S0I I UCLASSIINO TI AS(RmOeQ

.6U11T CLSIICTO OF T"16 -A4 (WIN D406 9-

NCT.R.Q TVT.neCuMvv T CLA66,ICATYON @U Tom Pe o floee t . n

(20. ABSTRACT Continued)

elastic analysis.The program was rewritten in an enhanced form of BASIC

language suitable for the Hewlett-Packard System 45 desktopcomputer. Presently, only the truss element is available;however, the program is structured so that other elementtypes may be easily implemented. A plot of the input meshis included in the program output.

Ateessiog ForAre s3 W"/

tUhim uncedJustrfltion____

A'iata alnd/orDJist special

DD For 1473 UNCLASSIFIEDS/4 0102-014-6601 2 *ego," es. ,¢BoaY or teo WheiI eM ae. an#~'

Approved for public release; distribution unlimited.

A Finite Element Program Suitable for theHewlett-Packard System 45

Desktop Computer

by

Ray Ricardo MalloryB.S., North Carolina State University, 1962

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN MECHANICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOLMarch 1980

Author

Approved by: __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Thesis Advisor

Chairmah', Wdprtment of Mechanical Engineering

Dean of Science and Engineering

3

ABSTRACT

The object of this work was to explore the possibility

of scaling a Finite Element program to fit in a small

desktop computer. A program originally written in FORTRAN

IV and called STAP was selected for the present project.

This program is contained in Bathe and Wilson's Numerical

Methods in Finite Element Analysis and was conceived to solve

static linear elastic analysis.

The program was rewritten in an enhanced form of BASIC

language suitable for the Hewlett-Packard System 45 desktop

computer. Presently, only the truss element is available;

however, the program is structured so that other element

types may be easily implemented. A plot of the input mesh

is included in the program output.

4

TABLE OF CONTENTS

I. INTRODUCTION--------------------------- 10II. THE HEWLETT-PACKARD SYSTEM 45 DESKTOP

COMPUTER--------------------------------------- 12

A. USER FAMILIARIZATION---------------------- 12

B. FEATURES----------------------------------- 12

III. GENERAL FORMULATION AND IMPLEMENTATION---------15

A. CALCULATION OF STRUCTURE MATRICES---------16

1. Geometric, Material and Loading Data- 16

2. Element Stiffness--------------------- 17

-3. Assemblage of Structure StiffnessMatrix--------------------------------- 18

B. SOLUTION OF EQUILIBRIUM EQUATIONS------20

C. EVALUATION OF ELEMENT STRESSES------------20

IV. PROGRAM ORGANIZATION-------------------------- 22

A. SUBROUTINES-------------------------------- 22

B. PROGRAM TAPE------------------------------- 23

C. DATA TAPE---------------------------------- 25

D. FLOW CHART ANTD STORAGE ALLOCATION---------- 25]

V. PROGRAM CAPABILITIES--------------------------- 30

A. 2-D PROBLEMS------------------------------- 30

B. 3-D PROBLEMS------------------------------- 30

C. MESH PLOTS--------------------------------- 31

VI. REMARKS, RECOMMENDATIONS AND CONCLUSIONS ---- 32

t A. RUN TIME----------------------------------- 32

B. ERROR MESSAGES----------------------------- 33

___ __ __ __ __ __ ___ __ __ __ __ __

C. RECOMMENDATIONS ------------ 34

D. CONCLUSIONS------------------------------ 35

APPENDIX A: USER'S MANUAL-------------------------- 37

APPENDIX B: EXAMPLE PROBLEMS----------------------- 48

SSAP-NPS PROGRAM LISTING----------------------------- 75

LIST OF REFERENCES----------------------------------- 98

INITIAL DISTRIBUTION LIST--------------------------- 99

6

LIST OF TABLES

I. Program "CREATE" Listing---------------------- 38

1I. Program "PURGE" Listing---------------------- 40

III. List of Variables ---------------------- 46

IV. Program "INPUT" Listing for 3-D TrussExample------------------------------------------ 52

V. Data Output for 3-D Truss Fxample------------54

VI. Program "INPUT" Listing for 2-D TrussExample------------------------------------------ 62

VII. Data Output for 2-D Truss Example------------64

____ 7 _

LIST OF FIGURES

1. The Hewlett-Packard System 45 DesktopComputer ------------------------------------- 13

2. Subroutine Organization and Segment Overlay - 24

3. SSAP-NPS Flow Chart ------------------------- 26

4. Array Storage Allocation and Tape DataFile, Input Phase ---------------------------- 27

5. Array Storage Allocation and Tape DataFile, Element Input Phase -------------------- 28

6. Array Storage Allocation and Tape DataFile, Assemblage and Solution Phase --------- 29

7. 3-D Truss Example --------------------------- 51

8. 2-D Truss Example ---------------------------- 61

8

MLr - r.- -.-

ACKNOWLEDGEMENTS

The author wishes to express his appreciation to

Professor Gilles Cantin of the Naval Postgraduate School

for the original idea of this thesis project, for super-

vising the work, and for his help and encouragement.

He also wishes to thank Lt. Dan Burns for his assistance

and suggestions in the utilization of the H-P System 45

desktop computer.

9

I. INTRODUCTION

There is no doubt that the finite element method

represents a rare breakthrough in engineering analysis.

This technique, in the last twenty years, has been sharpened

and refined and is today commonly utilized in stress analysis

and solid mechanics. Many computer programs have been

written for finite element analysis of static and dynamic

problems, with most of the general purpose codes written in

FORTRAN for large institutional computers. The last few

years have seen the advent of the personal computer, and

their memory, though limited at first, have been rapidly

increasing in capacity. With today's desktop computers it

now seems feasible to write a limited purpose computer

program which can be implemented on a desktop computer that

has enough memory capacity to handle the large memory

requirements that finite element analysis requires.

This project's purpose is to write a computer program

that, although simplified in various areas, shows all the

important features of more general codes. The

Static Structural Analysis Program (SSAP-NPS) is a simple

computer program for linear elastic finite element analysis.

It is written in an enhanced form of BASIC language and

has been tailored to be utilized on a Hewlett-Packard

System 45 desktop computer. The main objective of the

program is to show that a finite element similar in

10

procedures and overall flow to more general codes

[References 1,2] can be practical for use on a desktop

computer. Only a truss element has been made available in

SSAP-NPS; however, the code can be used for two- and

three-dimensional truss analysis and additional elements

can be added with general ease. The program also has a

mesh plotting capability for verifying the input mesh.

In writing the SSAP code, the organizational structure

and procedures of a FORTRAN code called STAP (Static

Analysis Program), contained in Reference 3, was closely

followed.

11

II. THE HEWLETT-PACKARD SYSTEM 45DESKTOP COMPUTER

A. USER FAMILIARIZATION

A photograph of the H-P System 45 desktop computer is

shown on Figure 1. Users of SSAP-NPS must have famil-

iarity with its features, operation, and some knowledge

of its programming language, BASIC (Beginners All-Purpose

Symbolic Instruction Code). References 4, 5 and 6 are

manuals which were instrumental in the creation of the

SSAP-NPS program. Reference 4 contains all the informa-

tion a user would need to utilize SSAP-NPS. In general, a

user must be able to do the following:

i. Load and run a program stored on tape.

2. Create and size data files on tape.

3. Store and purge files on tape.

4. Create and store a program on tape.

B. FEATURES

The H-P System 45 computer for which SSAP-NPS was

created has the following optional features:

1. Thermal Printer

2. Option 203 - 64K memory

3. Optional tape drive

4. ROM graphics package.

All of these features are required in using SSAP-NPS.

12

4I

E4

:1 13

The requirement for 64K memory may not be absolutely

necessary but since the program itself requires about 20K

bytes of memory and the only other memory option larger

than 20K is 32K, it would leave only 12K of memory

available for execution of the program. This would limit

the program to relatively small problems.

14

---- --- --- ---. .......... .-j°..... ....-------................----.........--....----.....----- ------------------------------------

III. GENERAL FORMULATION AND IMPLEMENTATION

The basic idea of the finite element method is to

divide a complex problem into a series of simple inter-

related problems that are easier to solve. The whole is

modeled as an assemblage of discrete part, or finite

elements.

In the case of a truss structure, the model is an

assembly of pin-connected rods having force and displacement

characteristics known from strength of materials. If each

interconnecting member is identified as an element, then

equations can be written expressing the element properties.

The unknown forces and deflections can then be found for

the overall system by combining these individual equations

according to the laws of equilibrium, and then solving the

resulting system of equations.

Using matrix notation, the model for a linear static

system is expressed as

(K] {U} = {R}

where (K] is the square symmetric stiffness matrix of the

assembled structure; {U} is the column matrix of unknown

nodal force displacements; and {R} is the column matrix of

forces applied at the nodal points.

15

... . ............ ......-._________________...... ...__.. . ...__"__

This relationship gives the linear correlation between

nodal forces and nodal displacement in the form of a set of

simultaneous linear equations which is common to problems

in static analysis.

After a structure has been idealized as by above

relationships, the stress analysis can proceed in the

following three phases:

1. Calculation of structure matrices [K] and {R)

2. Solution of equilibrium equations

3. Evaluation of element stresses

The procedure for accomplishing the above analysis phases

is described only in general terms in the following discus-

sions. For more complete discussions with illustrative

examples, Reference 3 (Chapters 3, 6, 7) and Reference 7

(Chapter 24) should be consulted.

A. CALCULATION OF STRUCTURE MATRICES

The calculation of the structure matrices [K] and {R}

is performed as follows:

1. Geometric, material, and loading data are read and/or

generated.

2. The element stiffness matrices are calculated.

3. The structure matrices are assembled.

1. Geometric, Material and Loading Data

Corresponding to each nodal point are a number of

degrees of freedom, i.e., for the truss element there are

three translational degrees of freedom at each node. An

16

Low I a .i-

array named the ID array identifies which of these degrees

of freedom are actually used in the analysis. The ID array

is of dimension number of degrees of freedom per node x NUMNP,

the total number of nodal points in the system. If ID(I,J) =0

the degree of freedom corresponding to the Ith degree of

freedom at nodal point J is active; and if ID(I,J) = 1,

the degree of freedom is nonactive. After all active

degrees of freedom have been identified by zeros in the ID

array, they are replaced by equation numbers by columns,

starting from 1. The total number of equations is deter-

mined by the maximum value in the ID array and is called

NEQ. Nodal location with respect to a global X,Y,Z set of

axes and the loading at each node are also input at this

stage. In the program the non-zero values are stored in

an array R. Other data read or generated are the connectivity

data and material property sets. Connectivity data require

the element node numbers that correspond to the nodal point

numbers of the complete assemblage. Material property sets

identify different material properties that can be

referenced by each element.

2. Element Stiffness

This phase consists of calling the appropriate

element subroutines for each elemeht. For SSAP-NPS

only the truss element subroutine is available, but other

element subroutines could be added. Chapters 3 and

4 of Reference 3 detail the general procedure to calculate

17

element matrices. The nodal coordinates for the elements,

material properties and loading information which have been

read and stored are required here for element matrix

calculations. An element matrix is stored on tape until

it is needed later when assemblage to the structure matrices

is performed.

For a truss element, the element stiffness matrix

is the well-known matrix

[K. -]

where A and E are the element section area and E is Young's

modulus obtained from the material property data input.

L is the element length which is calculable from the nodal

coordinates and connectivity data. The element matrix is

expressed in terms of local axis. The local element matrix

is transformed using direction cosines to obtain the

required global element stiffness matrix for use in the

assemblage of the structure stiffness matrix.

3. Assemblage of Structure Stiffness Matrix

The structure stiffness matrix [K] is calculated

by direct addition of the element stiffness matrices, i.e.,

(K] = I[K.]

18

To perform the summation, each element matrix [K.] could be

written as a matrix of the same order as the stiffness

matrix [K], where all entries in K. are zero, except those1

which correspond to an element degree of freedom. In

practice, only the compacted element stiffness matrix,

which is of order equal to the number of element degrees

of freedom, together with a connectivity array LM are

required to be stored. The connectiviy array relates to

each element degree of freedom, the corresponding assemblage

degree of freedom.

The distinct form of the structural stiffness matrix

lends itself to two major reduction schemes that result in

saving computer storage space. One is the storage of only

the items within a non-zero band; and, by virtue of the

matrix being symmetric, the items in the half band width are

the only ones needed for the analysis. The second is to

use the "skyline" technique whereby the columns above the

main diagonal are stored in a one-dimensional array of

length NWK. The column height information is given by array

MHT. A pointer array called MAXA is used to store the

addresses of the diagonal elements of [K] in the A array.

Chapter 6 of Reference 3 and Chapter 24 of Reference 7 give

details of these two storage reduction schemes which are

used in SSAP-NPS.

19

B. SOLUTION OF EQUILIBRIUM EQUATIONS

Program SSAP solves the simultaneous linear equations

given as

[K] {U1 = {R}

by using a direct solution technique based on Gauss

elimination (See Reference 5). The basic procedure of the

Gauss elimination procedure is to reduce the coefficient

matrix (i.e., the items in the reduced stiffness matrix)

of the equation to an upper triangular matrix from which

the unknown displacements {U} can be calculated by a back

substitution. This process is called the Triangular

Decomposition of (K].

Subroutine COLSOL (see Section IV and Program Listing)

is used to obtain what is called the (LI[D][LT I factoriza-

tion (triangular decomposition) of a stiffness matrix and

also to reduce and back-substitute the load vector. The

complete process gives the solution of the element equili-

brium equations.

C. EVALUATION OF ELEMENT STRESSES

Once the nodal point displacements {U} have been

obtained, element stresses are calculated in the final

phase of the analysis. This is accomplished by calculating

the element compacted strain-displacement transformation

matrix and then extracting the element nodal point

20

-' ... . ......... 1_ ..._ ..._ _. ...__ _. ... .. "__ _... .'_..-_. . ... .._ _...__ _-_ ' ... " '- ... "

displacements from the total displacement vector using

the LM array of the element. By establishing the strain-

displacement transformations, and relating stress to strain

by Hooke's law, stresses can be calculated at any desired

location in the system.

21

03

IV. PROGRAM ORGANIZATION

A. SUBROUTINES

Program SSAP-NPS is written in an enhanced form of

BASIC language for use on an H-P System 45 desktop com-

puter. The program consists of a main body and 14 sub-

routines, each of which is designed to compute one or at

most a few basic steps in the solution process. The

following is a list of the subroutines and a brief

description of their functions.

1. INPUT - Reads, generates, and printsnodal point input data tocalculate equation numbers andstore them in the ID array.

2. LOADS - Reads nodal load data, calculatesload vector for each load caseand stores it on file "ILOAD".

3. ELCAL - Loops over all element groupsfor reading, generating, plottingand storing the element data.

4. ELEMNT - Calls the appropriate elementtype subroutine. Only the trusselement is presently available.

5. TRUSS - Sets up storage and calls thetruss element subroutine.

6. RUSS - Truss element subroutine to read,generate element information,assemble structure stiffness matrix,and performs stress calculations.

7. COLHT - Calculates column heights.

8. ADDRESS - Calculates addresses of diagonalelements in banded matrix whosecolumn heights are known.

22

9. ASSEM Calls element subroutines forassemblage of the structurestiffness matrix. Only the trusselement is presently available.

10. ADDBAN - Assembles upper triangular elementstiffness into compacted globalstiffness.

11. COLSOL - Solves finite element staticequilibrium equations in core,using compacted storage and columnreduction scheme.

12. STRESS - Calls the element subroutine for

the calculation of stresses.

13. WRITE - Prints displacements.

14. PMESH - Plots structure mesh and numberseach node.

B. PROGRAM TAPE

The SSAP-NPS program is stored on tape in two separate

files called "SSAP" and SSAPO". This tape, called the

program tape hereon, is inserted in the H-P System 45

standard tape drive. In using the program, the "SSAP" pro-

gram file is loaded via the keyboard into the computer

memory. When the program is run, it automatically gets

the "SSAPO" file when it is ready to use that portion of

the program. Figure 2 shows how the program segments are

organized. On the figure, the arrow indicates the part of

the program to which the "SSAPO" segment is overlayed.

All subroutines below the arrow on the "SSAP" segment are

discarded from the computer memory since they are no longer

needed.

23

"SSAP" Segment

MAIN

ELEMNT

TRUSS

RUSS "SSAPO" Segment

OverlayELCAL ADD)PES

COLHT ASSEM

INPUT ADDBAN

LOADS COLSOL

PMESH WRITE

STRE SS

Figure 2. Subroutine organization And Segment Overlay

24

C. DATA TAPE

Before the program can be run, a data tape must be

inserted into the H-P System 45 optional tape drive. The

data tape contains the data files onto which computed data

will be stored and retrieved by the program. These files

are named "Al", "A21",1 "A311 "A4" 1"A5" ", "ILOAD" , "IELMNT",

and "MESH". The "MESH" file is required by the plotting

subroutine (PMESH); the other files are discussed further

in the next section. The data tape also contains the files

which will store input data for the program. These files

are named "COOR", "LOADS", and "TRUSS". Appendix A

discusses the input data required for these files.

D. FLOW CHART AND STORAGE ALLOCATION

Figure 3 shows a flow chart of the program. Figures 4,

5, and 6 give the data files used and storage allocations

used during the various program phases. Definitions for

the variables are given in Appendix A.

25

Read nodal point data(coordinates, boundary

START conditions) and establishequation numbers in theTD array.

Calculate and store loadvectors for all load cases.

Read, generate, and storeelement data.

FPlot mesh.

Read element group data, andassemble global structurestiffness matrix.

Calculate L*D*LP.T factorizationof global stiffness matrix.

FOR EACH LOAD CASE jRead load vector and calculatenodal point displacements.

SRead element group data and N

.calculate element stresses.EN

Figure 3. SSAP-NPS Flow Chart

26

Data File Storage Array

3;NUMN7PID

Q Z>-X Coordinates

Y Coordinates

Z Coordinatesj

diD......... (EQ+2)R for each load case

Figure 4. Array Storage Allocation and Tape DataLocation, Input Phase

27

Data File Storage Array

ID

X Coordinates

A NUM"P Y Coordinates

Z coordinates

MHT

IOAD "-*NLCASE R for each load case

TELUNTElement group data

MESH 6*NUME Connectivity array+ NUMEG from element group

data plus number ofelement groups

Figure 5. Array Storage Allocation and Tape DataLocation, Element Input Phase

28

Data File Storage Array

ID

MAXA

Reduced stiffness A

LALCASE R for each load case

IELMN 2*NMMATElement group datai +13*NUNE

MESR *NUMEConnectivity array+NUMEG from element group

data plus number ofelement groups

Figure 6. Array Storage Allocation and Tape DataLocation, Assemblage and Solution Phase

29

-, I -i iii * . .. . . . .

V. PROGRAM CAPABILITIES

A. 2-D TRUSS STRUCTURES

The SSAP-NPS program can solve for two-dimensional

truss structure displacements and stresses for a given

loading condition. The program output will first docu-

ment the input data, automatically generating missing

input if asked to do so by the input data. It will then

print out a plot of the structure in the XY plane, with

nodes properly labeled by node number. The nodal displace-

ments in the X, Y, Z directions are tabulated; and finally,

the axial stress for each element is tabulated. A positive

stress denotes a truss element in tension; a negative

stress, an element in compression. A 2-D truss example

is included in Appendix B. The displacement and stress

calculation phase is repeated if more than one loading

case is input.

B. 3-D TRUSS STRUCTURES

SSAP-NPS can solve three-dimensional truss structures

with the same capabilities as for the 2-D structures

discussed above. In addition, an isometric mesh plot of

the structure is outputted. An example in Appendix B

illustrates a program run for a 3-D truss structure.

30

''' h ' ' ' ' ' r~i* . I ... .-- .- -. .-. :. , .. ' . . , -' ,

C. MESH PLOTS

Describing the structure model or data input seems

trivial at first but can be the source of much frustration.

Mesh points are commonly missed or tabulated incorrectly.

The computer is unforgiving and one small error can make

subsequent work meaningless. Some form of data screening

is desirable, such as a graphical display or plot of the

structure models which can be generated from a minimum of

geometric input parameters. Mesh plots will verify that

the analyst and computer are, indeed, working on the same

problem. This feature is only usable on an H-P System 45

computer that has the graphics package option installed.

SSAP-NPS produces two types of mesh plot outputs: a

mesh plot on rectangular XY axes for two-dimensional

problems where all nodes are located on the XY plane; and

an isometric mesh plot for three-dimensional truss

problems. The positive XY and Z axes for the isometric

plot are 120 degrees apart with the positive Y axis in the

vertical direction. The positive X and positive Z axes

are, respectively, 120 degrees clockwise and 120 degrees

counterclockwise from vertical. In order to avoid having

two-dimensional problems plotted on the isometric axes,

their nodes should be located on the XY plane.

31

Lt. .__._._._._._._._._._.I.

VI. REMARKS, RECOMMENDATIONS AND CONCLUSIONS

A. RUN TIMES

In using the program, the size of the data files created

on the data tape has a significant effect on the amount of

time it takes to complete a run. This is because the pro-

gram stores and retrieves data from various locations on

the tape during a run and much time is spent winding and

rewinding the data tape.

As an illustrative case, take the 2-D truss structure

example in Appendix B. With all eleven data files

encompassing approximately 100 defined records on the

data tape, the problem took about 5 minutes from start to

finish; however, when the data files encompassed roughtly

500 defined records, the problem took more than 9 minutes

to finish. The program simply had further to go to access

the different data files during the execution of the program.

It is therefore important to size the data files such that

they are not much larger than required to hold the data

stored in them. This will require some knowledge of mass

storage techniques and definitions. One rough'guide in

sizing the amount of information a data file can store is

that a standard defined record can hold 256 bytes of infor-

mation and that a real number takes up 8 bytes while an

integer number takes up 4 bytes. Therefore creating a data

file with 5 defined records means that the data file can

32

hold approximately 5 x 256/8 real numbers or 5 x 256/4

integer numbers or any combination thereof. For more

detailed information on data storage and bytes per variable

type see References 4 and 5.

B. ERROR MESSAGES

From time to time when running the program, error

messages are encountered. When this happens, the program

pauses. In most instances, the error can be traceable to

a data input error, especially if error 17 (subscript out

of range) is encountered. In cases when error 88 (read

data error) or error 81 (mass storage device failure) is

encountered, it may mean that the data tape has been

stretched from continued use. In some cases it is curable

by pressing the CONT key; otherwise the data tape may need

to be re-initialized or replaced and the analysis must

start from the beginning. Error 59 (physical or logical

end-of-file) indicates that one or more of the data files

is too small to hold all of the data to be stored in it.

When this happens, the program line number at which the

error occurs is displayed by the computer and the statement

will reveal the data file for which the error occurred.

This problem can be alleviated by enlarging the data file

size, i.e., purging the existing data file and recreating

it with an increased number of defined records. Other

software-related error numbers and their causes are

summarized in References 4 and 5.

33

C. RECOMMENDATIONS

As stated before, run times are greatly affected by

the winding and rewinding of the data tape. One cure for

this is to store and retrieve data from disc mass storage.

Using disc mass storage also means that larger problems

may be handled because of the increase in storage capa-

city. Alterations to the program should be minimal to affect

this change in mass storage identifier for data files.

As presently written the SSAP-NPS program can only call

upon a truss element. However, the program is structured

such that other element types can be added in the form of

additional subroutines. The addition of other element types

would also require certain subroutine modifications to allow

for the increase in the nodal degrees of freedom; and a mesh

plot subroutine would have to be written for the added

element types. The addition of other element types would

be a major improvement and would make SSAP-NPS a truly

general purpose finite element analysis program. An increase

in computer memory size would, in this case, be imperative

if the program is to be expected to handle realistically

large problems. This means the addition of some type of

disc mass storage medium and foregoing the use of the data

tape altogether.

The isometric mesh plot output is a great visual aid

for an analyst. A plot of the deformed shape of the

structure after the loads are applied is another visual aid

34

I.i i i"..

that would be very helpful in quickly assessing the effect

of loads on a structure. In order to effect this for the

program as presently structured, it would involve creating

additional data files and rewriting the plotting subroutine.

Another type of plot that would be helpful in quick analysis

is the stress plot. This would indicate at each node loca-

tion the magnitude and direction of the principal stress at

that node so that the analyst can, at a glance, locate

critical loads for a structure. The effort required for

this improvement would be similar to that in effecting a

deformed shape plot.

D. CONCLUSIONS

The SSAP-NPS program is a useful tool for finite element

analysis of truss structure. It has a graphics capability

for plotting an input mesh in 3 orthogonal planes. It also

has automatic mesh generation capability.

Using a data tape for storing and retrieving data is an

effective means of reducing computer memory allocation

requirements; however, this comes at a price of increased

run times due to tape winding and rewinding during a run.

It is clear that in order to make the program more effi-

cient that data mass storage must be on a disc storage

medium.

The program is structured so as to make the addition of

other element types possible by appending new element

35

I'?

Isubroutines. This is a necessary step in making the

program a more general finite element program.

Additional graphics programming would also be desirable

in order to fully take advantage of the H-P System 45

graphics capabilities. This could take the form of

effecting a deformed structure mesh plot and a stress plot.

36

-- -~ . .

APPENDIX A

USER'S MANUAL

In this Appendix the data input required for the

input data files "COOR", "LOAD", and "TRUSS" on the

data tape are discussed. The units for any one problem

must be consistent. Error messages or erroneous results

in the running of the problems almost always indicate input

errors so careful preparation and review of input data is

an especially important task for the user. At the end

of the Appendix is an alphabetical list of variable names

and a brief definition of each.

A. DATA TAPE PREPARATION

Before the input data can be stored, the data tape must

have the required data files created on it, with each file's

number of defined records sufficiently large to handle the

analysis at hand. Table I. is a program listing which will

create and size the required data files. The size indicated

for each data file is strictly arbitrary and were sized to

handle the example problems in Appendix B. The user can

change the record size to suit a particular analysis. This

program must be run with the data tape in the T14 mass

storage medium. If the data tape already has the data

files on the data tape but file sizes need to be altered,

37

i'i

it 9.-L

z0L

E-a.ww

w

LJL

J,) 0. 1

CL WWWWWWWWWL

W )QL)QQL L L

I~d Cii 38

Vthe program listed in Table II will purge the existing

files from the tape and the file creation program can then

be run to recreate the data files with the new file sizes.

B. DATA FILE "COOR"

This data file contains the problem title, control

information and the nodal point information. The data

variables are recorded on the tape sequentially in the

-following order:

HED$, NUMNP, NUMEG, NLCASE, MODEX

N, ID(l,N), ID(2,N), ID(3,N), X(N), Y(N), Z(N), K(N)

The second line of data variables is defined here as the

nodal information set for node N.

1. Discussion of Data Variables

HED$ is the title information and is limited to 50

alphanumeric characters, including blanks. The title must

be enclosed in quotation marks which are not counted as

part of the 50 character limit.

NUMNP is the total number of nodes and controls the

number of times the program will look for the nodal point

information indicated on the second line of data.

NUMEG is the number of element groups. The total

number of elements are dealt with in element groups. An

element group consists of a convenient collection of

elements. Each element group is input as required by data

file "TRUSS" to be discussed later. There must be at

least one element group.

139

• -.. .. I-

000w

*I w

'.a.

w CL(p t

.

VC LO-0C:- JL

I.--L LXC X U.-X x

p.-D o - - I-

.40

NLCASE gives the number of loading cases to be

applied to the structure.

MODEX is the flag which indicates the solution

mode which gives the user the option of checking the data

without executing the analysis (MODEX = 0). If MODEX = 1,

the execution mode, the program goes on and solves the

problem. The program gives the user the option of over-

riding the MODEX setting by pausing execution and asking

the user if he would like to continue the solution. This

is a helpful feature if the user, after reviewing the input

data as listed decides that the program should go

ahead and solve the problem. The user is thus saved the

trouble of changing the value of MODEX and starting the

program from the beginning.

N is the node (joint) number and must have a value

1 < N < NUMNP. Nodal data must be defined for all NUMNP

nodes and may be input directly or automatically generated

(see discussion of variable KN). The last node that is

input must be NUMNP.

ID(l,N), ID(2,N), and ID(3,N) are, respectively,

the X, Y, and Z translation boundary condition codes. A

boundary condition code can only have a value of either 0

or 1. A value of 0 indicates unspecified (free) displace-

ment; a value of 1 specifies fixed displacement, i.e., the

node is not free to move in the specified direction.

Concentrated forces may be applied only in the directions

where ID has a value of zero.

41

One system equilibrium equation is required for

each unspecified (ID(M,N)=O) degree of freedom of the

structure. The total number of equilibrium equations is

defined as NEQ and is always less than or equal to 3 times

NUMNP. Specified (ID(M,N)=l) degrees of freedom are removed

from the final set.of equilibrium equations.

X(N), Y(N), Z(N) are the X, Y, and Z coordinates

of node N.

The nodal information sets need not be input in

node order sequence; eventually, however, all nodes from

N = 1 to N = NUMNP must have nodal information defined.

Nodal information sets for a series of nodes

[[NN 1+I*KN, N1+2*KN, ... , N2 1

may be automatically generated from information given on

two consecutive sets:

SET 1 - Ni , ID(I,N I ),..., X(NI), KN

SET 2 - N2, ID(,N 2),..., X(N2), KN

KN1 is the node generation parameter given for the

first set. The first generated node is NI+l*KNI; the

second generated node is N1 +2*KN 2, etc. Generation

continues until node number N2 -KN1 is established. Note

that the node difference N2 -N 1 must be evenly divisible

by KN

42

In the automatic generation of nodal information

sets, the boundary condition codes [ID(M,N)] and the

coordinates [X(N), Y(N), Z(N)] for the generated sets are

interpolated linearly.

C. DATA FILE "LOADS"

This data file contains the load data for the structure.

Each load case requires following data entered sequentially.

The total number of load cases is defined by the value of

NLCASE previously recorded on data file "COOR".

LL, NLOAD

NOD, IDIRN, FLOAD

The second line of data is defined here as a nodal load

data set.

LL is the load case number and must be put in ascending

sequence starting with one and ending with NLCASE.

NLOAD is the number of concentrated loads applied in

this load case and defines the number of nodal load data

sets that will be input.

NOD is the node number to which a particular load is

applied.

IDIRN is the degree of freedom number for the load.

If IDIRN = 1, the load acts in the X direction. If

IDIRN = 2, the load acts in the Y direction; IDIRN = 3,

in the Z direction.

FLOAD is the magnitude of the load and must be acting

in the global X, Y, or Z direction only.

43

D. DATA FILE "TRUSS"

Truss elements are two-node members allowed arbitrary

orientation in the X,Y,Z coordinate system. The truss

transmits axial force only, and, in general is a six degree

of freedom element (i.e., three global translation

components at each end). The following sequence of data

is recorded on the tape data file "TRUSS" for each element.

The total number of element groups (NUMEG) was defined

earlier on tape data file "COOR".

NPAR(l), NPAR(2), NPAR(3)

N, E(N), AREA(N)

M, II, JJ, MTYP, KG

The second line is defined here as a material/section

property data set. There are NUMMAT sets of material/

section property data sets. The data sequence on the third

line is defined here as an element data set.

NPAR(l) is equal to one, the truss element type number.

NPAR(2) is the number of elements in this element group.

NPAR(2) is equal to the variable NUME in the program.

NPAR(3) is equal to the number of material/section property

data sets to be input. NPAR(3) is equal to the variable

NUMMAT in the program, and must have a value equal to or

greater than one. If a zero is entered, it is defaulted to

one.

44

. ... . . . .. j j

For the material/section property data set, N is the

number of the data set and must be in ascending order

beginning with one and ending with N = NUMMAT. E(N) is

Young's modulus, and AREA(N) is the section area. The

Young's modulus and section area of each truss element

defined by each element data set is identified by one of

the data sets input here.

For the element data set, M is the truss element number

and must have a value 1 < M < NPAR(2), NUME elements must

be input and/or generated (see KN discussion below) in

ascending sequence beginning with one.

II is the node number at one end of the element. JJ is

the node number at the other end of the element. II and JJ

must not be equal and must have values between one and

NUMNP.

MTYP is the value of material property data set

applicable for the element.

KG is the node generation increment used to compute

node numbers for missing element data sets. If KN = 0,

its value is defaulted to KN = 1. If there are "J" missing

element data sets between element M = K and M = L they are

generated using MTYP of element number K and by incrementing

the node numbers of successive elements with the value KG

for element K. Table III is a summary list of variable

names used in the program and a brief definition for each.

45

TABLE III - LIST OF VARIABLES

A Reduced stiffness matrix

AREA(N) Truss section area for material/section propertydata set N

E(N) Young's modulus for material/section property

data set N

FLOAD Load magnitude

HED$ Title of problem (<50 characters)

IDIRN Degree of freedom numberIf = 1, X directionIf = 2, Y directionIf = 3, Z direction

ID(P,N) Translation boundary code for direction P,node number N.

II Node number at one end of truss element

JJ Node number at other end of truss element

K Structure stiffness matrix

K Element stiffness matrix

KG Node generation increment for element data sets

KN Node generation increment for nodal informationsets

LL Load case number (1 < LL < NLCASE)

M Truss element number

MAXA Array to identify diagonal element addresses

MHT Column height array

MODEX Flag indicating solution modeIf = 0, data check onlyIf = 1, execution

MTYPE Material/section property set for element M

46

T - i

N Node number (I < N < NUMNP) or material/sectionproperty data set number (1 < N < NUMMAT)

NEQ Number of equations

NLCASE Total number of load cases

NLOAD Total number of loads for load case LL

NOD Node number to~which load is applied

NPAR(1) Truss element case number (=1)

NPAR(2) Number of elements in the element group;NPAR(2) = NUME > 1

NPAR(3) Total number of different material/sectionproperty data set, NPAR(3) = NUMMAT > 1

NUME Number of elements in the element group;

NUME > 1

NUMEG Total number of element groups; NUMEG > 1

NUMNP Total number of nodal points; NUMNP > 1

R Load vector array

U Displa cements array

X(N), Coordinates of node N in global system axesY (N) ,Z (N) ,

47

APPENDIX B

EXAMPLE PROBLEMS

In this Appendix the procedure in a typical run for the

SSAP-NPS program is discussed. Then input and output

for two example problems are presented.

A. TYPICAL PROGRAM RUN

In order to complete a run for the SSAP-NPS program, the

user must have two tapes: a data tape, and a program tape.

The data tape may or may not have the required data files

already created on it. The program tape must have the

following programs on it: "SSAP", "SSAPO", "CREATE" "PURGE"

and "INPUT". "SSAP" is the first segment of two segments

comprising the finite element program. "SSAPO" is the

other segment of the program. "CREATE" is a data files

creation program and is listed on Table I of Appendix A.

The defined record size for each data file may be changed

to suit a particular problem. "PURGE" is a tape data files

purging program and is listed on Table II of Appendix A.

This program clears the data tape of existing input data

files if file sizes need changing. "INPUT" is a program

which writes the problem input data on the three input data

files as discussed in Appendix A. This program varies

with each problem. Sample listings are given in the coming

examples.

48

The following sequence of steps is what is involved

in completing a program run. Steps 2 and 3 are omitted if

the data tape already contains the data file and the user

does not want to change any of the file sizes.

1. Insert the data tape in T14 mass storage, and

the program tape in T15 mass storage. Turn on

computer power. Execute PRINTER IS 0 statement if

Thermal printer output is desired.

2. Load and run "PURGE" program. Skip this step if

the data tape does not have data files already

created on it.

3. Load and run "CREATE" program.

4. Load and run "INPUT" program.

5. Load and run "SSAP" program.

The program takes over from this point. If the value of

MODEX = 0, the data check only mode, the program will pause

and give the user the option of either stopping the program

or continuing. If the user continues the run, the program

sets MODEX = 1 and completes the run. This pause in the

program serves the purpose of giving the user time to review

input as interpreted by the program and choosing either to

go on with the calculations, or stop to make corrections

and/or changes.

49

B. 3-D TRUSS EXAMPLE

Figure 7 shows a three-dimensional truss example with

loads in the X, Y and Z directions. A tabulation of the

load magnitudes acting on locations (nodes) 1, 2, 3 and 4

are included in the figure. Table IV lists the input

program "INPUT" which would be required to accomplish step 4

above. Preceding comments are included here only as a

means to indicate which variables are being defined by the

lines that follow. Table V is the resulting output for

this example.

C. 2-D TRUSS EXAMPLE

The truss in this example has two load cases as indi-

cated in Figure 8. Table VI is the "INPUT" program

required for step 4 as discussed earlier. This input

program illustrates how automatic mesh generation can be

utilized to minimize input data. Preceding comment state-

ments are included to indicate which variables are being

defined by the lines that follow. Table VII is the result-

ing output for this example problem.

Node locations for 2-D trusses should be located on the

XY plane in order to avoid getting an isometric mesh plot

in the output.

50

. ., .. .

40

40 X

30

Fi -10 F ' =-10 F =10 F 4X 10

= 10 F2Y ' -10Y -100100 F = -120

F = -105 P2Z =-105 FU3 -165 F4 Z=-165

Figure 7. 3-D Truss Example

11 51

I--

0 0

. 2=

Q*= 00Q.)- 0 p

1-4

I. . .

aU a

04 xX -r : 0a -

M 0 -Cm -M 0 'M rSi O-3,0 ( 0s s U') , r :0V krO -- - -- - e.1si

0 - 52

I

CL-

M0

Ii

w

w

-J

LL.

I-

I-

wti

-

= '

X- I= =

x ~ ~ ~ I IxCi v k() Z m i -x I LML 0 xmI

I-- CL CL ,- LL CL C. T ,-- . ,- LL C-. -. C 1-, 0 -. a -. U.mO D-o 0 - m AI'4 0 M km O.J 0N IJ-'44 -4 . aMM-MO -M

,M I N Cg - LO ID . w C 0 N to" - y r. O N Q - ,J "- 10 W N CO Ch a, ,.w0 M 'n I . f v v )- v " 'x ) a a, U') aaU' U') 4' ) a .I

~53

'i'C .OC)' 'W l--Nl-C J O W -a

W C). 0 a

zLI

o 000 0

w C-n (DI 00

100X

0 --- kM

0(00000

0 00 0MI 0 0 0D inC" zm 0000imIn000

omMI a-

0n .l It

04 cc L)0)

U W N !D-D S

.4 Q)o T o 7 Z %A.0 ( =Cw m A WV1 W

0- F -IWJ t. 0~N 00rr

0L U ') UI-a, 0 C~. Z O

I-a~ ~~~~ W.Z.J aa 0 I aZ0 I- &71

'A ...J C O ~ CL.- 0"0Z

- S '~z.-WW 54

Lew

0 00 0000 0000000000000000000000000

0 0000000I- ))0 0 0 00 0 0zw 00000000- ~ 00000000

Z000 00 0

Z L)

0000 0

0 0 0 S00

~~~~~. Nl C.J CJ d 0

I I

LU N C' ID In 'A. 0 (S 0 0

LU

i~A: N D O ---- 0 S -'

-CL

*Z Z0UCd fl- C

CI- 55

it 11

.~ ~~i am 'm 00 0

0- -NMw

IZ I I I A

III; I- -

zA I-L

I- L k . j C ) ?i*j f.A (j 'J ", ("I CI w I. wC4 w

OLw w-LU W

'Q I-' w Jx z -1

4fLO LawI l )

fl~ 0 L..WMJ

*~1.

56

rw

ww

u w

w Q w

LL.

- ~ Y M, Z+k

0 DwI 7 0 0 ID

I-~ i.-j 'i iZ J Z w S

- 'I Coo

- 57

//

I

N /A /

/

I N/ I

// \, /

I /I I:

/ /

\\ / NV/

'Ms

Nj-.-.--- \~

C

N'.

I-o-J3-

zccl'hi

58

. -~ .. - -. w - - -*---- - --- -

zw

00&00000 Q C

a. wg N Go 0 a 0OOO%.t0 00

-. 'D .36) 0 'J O0 0 01 t ' OD N t 0 0 0 0

lflCN W C00 00-

WO.V n O 0U000000O 00

aommwo-. 0 0

O 00 0 0 0 0 G

W.I~C 0 0 0 0If 'C 0 0 0 0I

CL 7,N rc\0 0

OJCTCd kI4 10 C 0 0

IT- CL ItW4 ~ ~ 0 0j

CCC CIL C'D 0 0 w SLli WZ: N 0 -r r. -

A C Li

zw

La5)9 )I' f ~. -

CL

ma

XJ

.i . .. ..

U. V. ' 0.7 0 ( 0 0 N N 0 0

.- .~ r. '.g . . .a. . . .C .C

C' T..- - - M~ TNk- Ci 0' . .S I' N I 'I C5*=Mc~jN-

J.)

'SJ

fr wj a:

60

(a a

442 - Co

I E-4

N 'N

C,4 -,.

44hi -~4I

r4N

-A N -N.C

01

61

X r$4 I

C. C-

m =-

'm a

lm CO

> = 0 N(T

.4 -.

* z -'I

-4 -* c - wIL O '

0 0 ,C 0 ' DG(

On a - - -a - - - - I-.k4..k

o i- z:: .D- )0ID .62

LL~)* ~.'t( jLmz.1lal)aa (

ujI

isCA

NO -

n X rj CN A I I cI i"

U ~ 6a

t C"I cl LU r- I x 0 -:-Z - 2 -- : IU-) 1, m10 a

Co 00NC

s a w za

X L V xI y f)w Q , ' I xcxI xIC. a. a.I La La. CtL .a KEa .L

*DI 2-,000- M0-2i -N' M00G00 '.000I 'a CANW oIDr D(,- d V4 JDNX ' . r l )NX T

- I:) 63

I-j w

W 0 C1w000

-M 0 IMfm0D0C

0m4

0 OD* S)-M0

MM 0S IM C CD

-M k- 00DM

0~~~I 0l -'mi N -

1- CL. LL. 0 A-')Sl llnC

*= -.1 10

- I LL& ON. W -

= 0 , r- CWW

> Win Wj -N0=. m - PaCU-

o3=4 5- -iQ- .

0e 0 2. lz

E- CL L ** W. W

00 00 0 0 00000

00 0 0 00 0 0 0

,O'D

,;: .q---- - - - - - ~

-M P0x 00 0 00 M0 0 =Il G M 0 0 -M 0 0

Z

,, ,i -, -, - -, Cd ,d ,'] ,'" r ,fl Cf fn

w

C4 ",A,- -- - -- ,-; ,,

M

ru0 t

1.5

'I a]

a:OS

,3: ,-

I - --,1 ... .;6

0

w

LI

2 - --- ~C-1 01 CA CA

cc

= w

LUI

66

In =1

Ir4

0t

an i

Ir w It u

wl LL x x LL.

H - Hj -D

44 Co n

.... .. ... ..

liii w

LLJg

CW

LUii

LL Lii

LL Q

w TQ

z Cl ,)

~ Lii I-Wi

-

W Li Lu

w w C4LU W -j

68

-mo

LX

-j-E

wL =--- - ------ - -- - -- .- - - -- -- - -- -

'-- I-

z S

I-.I

I J Wo-a- = in ---.0""

w - - - - - - -('Il C'. -', % -

-

LUJ

ww

69

(0

A_______r1

-2

K 11Si

I-

-J

w

70

w

'I G I i InI In In in In In On OMC In IMM~0 000000000O()0

(DI00 0 0 0- In 0 0n 0 0M 0 0 0 In 0 0-00I 0 In0 000000iInIn0 0000n

In fl 0 IVI0 -0 n MI 0 In 00

17 D 0 00 0 0 IN tot hI0 000007 I0S lo I I I M IN V 0 N WO I

- 0 'C 7 ' CI) rri -n . tj) .7M 0

CA ~~~ ~ ~ O -Mi i0 Nt 7 CTN N wi 0 0 CC -&

4- li~Llt 1 11 1 1:

ww

= a t- .awiawwwwwwwwwwlwlw

CA w .' C."a, 0 nZ M ..7'1tC)Z )'toOl "C0 0

' I-1' C. La o0 MC nC)(' D0 oiI ~ Mt

p. Izwi x 4o(.C.I

CII u:- r*X

a WJ a -j '

L. LZmC4 a. wn.- 00 = Q)

A -J WXn Y.. '.0 C C. 0 to N L0, ' ' k -)In' t- s- X. -' -S --

71

-

w

O .. . . . .. ..+++++++1+++++++ + + + 1I44IL r 'A 0O00 00O0000 0000000 oIn... O.-.wna

LU I

0 ~ ~ ~ 1 I 0 IM I n000000G00- ID IM II Q I~ 7 N0 -= - V ns00-nn00n0 OC %NT . j*P- fvNu 7 0U 0C0NWu,0k '-C

00 7400000;0000oo U;oooo 4o.4

0cc0 0 .fO O o O oo o ~ o o

I) t

ix 000 00 000 000 00 00. .oo -. ..72.

w

i i ; II iii i iiii-JWWLWw wwwwwwwwwwww

CDlOO 00 00 00 00

0 0000 00000000,2000

0 000 00000 0 0 O 0T 00 I0I0000000000III

I-J w w w w w w w

*L0

w -:D 01 IN ~ r-. f '1 r'N M C f t t N N-

M It 00 C -CO~') b t r-k 0D

I I

w-rfw bjW tetlr-rlptww nt4Im .jm l _ fl 00(n-M='S _nfl "nr-f'-00 T

,_ 1 )". ')c M~uwa WW W WW01wWWWM

Xd .0 ki N 0 , -%

La I

ILJ Lat ,L- CO N 0ia0 rL 1

CC73

(-

0

Cr

w

0 0M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0M 0 0M -0 IM 00 0~'D 0 0, & 0-0

,J) 00000000 oo000 00000000WO'nC' N 0 Oe)P CX 0 0 0 00 0 0 00 00 m00-M0-M 000 ' N 00--.i

I- 0i00000000 M0 0 M0 00 0-'.OcMOcotD('0MO -M -i -M0 f00'i000 0G'%000 In N.)000000flN-M

0 -Mi 0 0 ,A 0 S 0 0M 0 00 ' 0 0 0 0m S 00 0- 0l 0D 00C) om

UP 4- u, 4 U U U UO U U1 In UM U U 0 _n U U CU M !U LO U UA U U 01 Ur U0

..- C-J V. Li* Ui' *!g CJ CAJ Vg t 'D Zn in ' - -7 (N J N --r7 'S C N -n LO T7 '0 l

CXC

--1 -- - ' UC.

74

z l

Q 0 W

z z

LL 0 0,

'i 0- - -

2T X

w AA W X.CWZ 'V C* 0 0 =ZW_

3 T S m*1 x ~LL. L.

I-~~ I--OZZ

a. W S U) (A 0 z =

aA Q. - T. a:I - =tL mu ==IL z 11 a i-c

X 3l a 00.0 M-

-0 =V 'T if) '4 - 1

CO 0 CL -- 0. ZC)I-0f 0 X 0 1-1 U4WT

(J) IxU=rX -1 - Zi) Z II 0)-.4 I) D wIociMJC

I- a. X =X(4.T - MM L 0 40CLC

N C ZN- 7 CA N . a0ZWI4 'JN N Nk

0.. C - c

'2-tx (j T- o

.- -L= j () t

x ax

0

WI II ix*Da:0 *) 0

0 Ix ri - *:m W 9

rj E * --.1

0 z Mr" cc a,

a 0 111'S-

W*J Il o . dg % C I- - x Na1 1.- 2- W M7 Z Z 11W O

go go pq CO' TO + Li0 --. -= - ' m-1 CT C: * r 0. W 0 ..- = "S CC. a. 1

X .U. 07 20 W ..- I i- lo

z -z - iiz WOO- II a W 1 O

'n - a1 >7 u c Iw ccmx - l 0- z -Ix flc N SI. .1 I'Uwow fw o- m w = Vx

O a a 0- 0. I W 'XS a ) 0 = 0 - Ua

-. t~- oi ax * ca a 5 ~)76

ILU

of'

j.I

W

OMU; Q

:A m

CL

... Y W

"s' Z IEII .l -'z Z : BE "7 L) 4 '. - 1

ly LC

"3 ,. v m 0 i

a.C

._j s~ a.

Xn

U') IT. z Oi

IxI

Qx Lu

'sOsX4 0- CwzY r 'n Q c U

>- U.c= l'- a- ixW (M

w >0 aC (CL.QUr # - I-- xCm W I i 0 0w (j)C

IXCt: 4 0 x a Ix +O x+ c W U Wc =W If ?- X-)

O=El ' i-a r~ a 'rw O> n 'A = -

C , -(7u( CO"-X 7310 I f a a Q" if

rz ft3w4 - I- i- 0 l'- x -x L N 0 .=U .. aI- XS = =- LX I.- z j1I

It) If 11 If 11 It L I- ~ M0 0 *9L )( -

ix ac: * n 'iIxl ix . 0 kA T -C- 0 G ix 4 Wm6CL ( a- LL a'~ -UoL i= )CL Z Q - Ig(- -) -L

Lu 0000000

-a N N N N N N N N N N 0C 0) 0 0 C 0 w) 0 0 ) 0. 0 0% 0% 0% 0 0 T. ON 0% - - -- - -

77

-A

X 4S zz E tzz rI- .

a Q f) fm

z 4.

w i z I- -i

'0) CA (.

IX C-1 w 5

3L W w

0i f f 1 x 5 00 U.CIL a1 . I- a

.1 -W IV

Lu 3 i * X- a-Lu) aj I> - a

0 'A W L CD. U- 1- 0" r '4z

o W - - (0 3 w a. r (D(5) T-~ I X CO) a JL - j e

-j ot oi fmcr( -7 0 w as moS- C..- A- IX = 0 0 01 =i

S 4iL Q( - zW _j i x - SLu -4 0) m CL O. 1.JI Cj f oQ L . 1, *X 0a Lu a. C OW 9. or -

-A Lr ) 'U Wu I. *-- U)a ') 2_4 'viu (5) to I.0 33 W2 S ... 04aac m' T s T- W .- z M L sQ - a.n. L

cC' - IX a f4 X pL o L 0 -1 1 .. .-- 1 W0 - --In NC) 0- in- 2a D0( DO M C . oe vC CD -0 co 0.m0Q 0000

M- N -I % ON 4* 5- N*4v00NW 0 -N0 Vo Za M *kU LQ 0 a C'r

'- 'r. .- I- S X1- ) )- . )78-

oI"OD N

a 0

r3%LL

z .a -

to

a i i40

aM 0 S

-C 0 V

*4 - UIn m .-

0 T.I vI

aM = cZall .1 . L =u ;;

a l . U (1 4wW.v T (

- 1 .- 0)> . 0 N L a - .It If 3D=t ) - - wfe

q ) 4p A - at Q) 0-L .'M

+ ~ ~~ Z %A 0 -flr QpU(EYa ~ ~ ix=3= P -Q.C)'- 01=s, = ~

-- 0 0 Go0 _jW6DW D I =%11ItOf( C ()zz-JQ - ~ xx 3 x'- ao a t

44 Mr Mn w X 0 CCxttt CIXI

-r ~ ~ v4 v) -'Y to 0 n00000k Zlt.Dk Dko -N NNr -O 3O

O a 1 '-"'U- a9

ap'Vr

. : I.

I

- s ) '- ix W

I (r-=

II x z 0)

- W.lix (

'W -Z) m IEL

(00)

= L t w 0) - = ) X. -L U' -Ill 0 - -

ILA. W WC' LU)

- Se

r wenS'w

W-fW-W 'Sfr-L LL:CLV l-C arj IV D a1 -l. e (z in Z: 0.

Ill o Z:. .- I =C -Tkix SI 'Sx C L At- - eac CL N-c

.iC flw' Zc = =f a W m

0W IllC' ^- ES M~ 0ixa'D- ci ': =- '-' u x =L N Cd mixw

W0W0-I-0 UM 0:.-~ WL) -j :DQ

WwwZ a j at It 11. IL w XX..Jl 'S 0)fl .a' Ix Z~e 1-- 1, -- -; PlbW ;N

W1Wz L - WI- x 5 I- w z 4 .iZ(

m ( xixI :: Z::x eaE Z L.w I ix *at f -wIIt 1N D

eaaaaccy OD.2 0 0 0aNU 0 ZW W N -Zr -N0 W NZ0 Y-'

a~~0 C .. e ' 0..13 00) Wy ON '% aa WM Nnr 0000 ----WCCSC -- SJ-------: 2 OL ~ Z

SWWW S M '-'ixi'-. 101 X0

II

0)

L.

la.

L Z

I-

a)" lie

• Q. (M-

13 ' :r 40 1.- L w- .

• -C; z CD S . X 1 C M--+ - " - O 2 . -- . --

-Q, D k-,,) . • . - - r . N -i M=acZ" v -1 ai 0 UP U a--" '2" X: w--= I-- O

I-- ai 11 -If _j -. f a0

11_ 1 . cc u2 t x . zccw

If..;,,x I I -I ko Q ( LL I-- CL (L. - ,. "

Ic V) il -I

I-- CA. -- 6 - =N N N (Z X: : .S X M- - I- -w I- (E Cc I--

X. xi -. ** =\ U-ibiQC L7- ZC EC j - CCeML

M)~~~ It. 1a If .M C-)VU)kDN j jl V 4 In toNm3%C 4 .7 oNO

, el C i (1-i q,I ¢ ': :.i4 (4¢,j ( ,.j c) " (,) enO 'n M3 k. (,) en r v v r 't v, vj v, 'r" 't Un) I to3 in 4n to U") to 01¢4 j C4 ol C-1 l N "-, I ',J C, 4 N J J N "%H J C-J NJ N, N.% Of4 N N% N N', N N% -1 N." N N" N' N.M N k % N. N kN

I: :ti -a a. -81

-J

w

cL) I

m o

W- 0)

\t - o Z V_

N a) a.)x 0 z12 :

N4 + 141_j V a - 9bQI.

It x~ -,_ j L. . i I.-

Q. It I- I.- (A W- . ) Z -I- jI-" Q wC

TM NMV D -0 0 N0 -r l -COi 0 Nr V 'r aN( C-N"V 01.DW 4 W a WU 'D~ 'D W4 ,NNPr DO O(D 0O 0Z DM O %(%O

(1-j~ t-j es No AN( 1(7 AO w ANN0 ,jO 4N( d0 ~

Nc' ' a 0 I-82

v 4 rVr. C a

(11

L;

a,"

z xZ >'

C, S

C

LL fvLa -

l'- z

CN' 9,;4 • III C"

-- % -- 1;% = 7 tW,, a : ,,. - ~

a: It w 0 cn

OQ~~~9 3 L -. -

.. j W _j W _j 0 (W) kD w o..b .

:3+ v1 -. 1I + DC LU Y,) - uJ 0 U do QJ 0U W.:-

1-- X, -. v -J I CO -_j 1 -- _j p" It 4- =" *- = E- X. MJ =J - = CL W5 0

lb CD -1 11 , _j ^ + 0 _ - 0 .. 0 .. L X M W 0 u z I.-

83

4k4'

Q..

z -IF

a, =V I.-X

= a W f-) = -

-i~~ z -I

4.' - l a -W ICc x I--X (

= ' -1 -"= lbX\

Q6 E -f- Zn a , aW G 3 G %0. (J) NI In X 0 ) I n, va

3; zl -- a 0 =wS(OD c. *io W -0 11 1 6 - D

0.Za4 1a aVa4 Vi a, L 4.

i-. - ' zj Mwz CW xii- '-1 L TC a x a 4 IM N r.WOIdOW W W.-'"..M..C - 4 n . 0 -w

M i--r M3 W L LC ) a W4 (A 4 )I - .

wnO )l *.z'a a.lr-z sr nroo no no -00ww wla w

4 ZZ~ ZO% a = '-' 4-84

Id CLL a mUS

* uvi

CL w

I-C P M C"-M 1-

-'4 a Z ;

3 S' CL

-M Q C Es

Ex -r XEQ -

I-C' w w M'T' k

U - * al*- I- m Cp I )-U~ k -1

a 0. X L) U) '0 aLxV X

FA ~ ~ W~j EQ. - jEw Q Q- nm :-=c -I c311C' E.L aC I- a= a;X ="w X

0 j - x WE w 0 0. =Z == -VZ1 sQMozC 1 1 Mca m ~ E.' X'-'a-0 a;"O - - N CL' CC C- C- U V's-Cx ,

Ir ~ .t0 WDin-. ZDa Jf) PIDN( - 0 - --%'nV1 D %- ME 0I 3 3N -i- - D D0 ) 0 D O 0O a 0' =N IC ON 373%O 0 DC 00

m . * mi cr oe 1 l r l v 1 r ~ 1 - TrvI vI

a ~l DE kM .'.85X

w &

m w - a.

PA 0 013c C

1 -X -d ig

0 ~ ~ ~ - 04 a '^

= ix a - - C aM.J = W3 aOWW w0xI x w w w

QldwY .- N w -fZ LLL LILL > N M E S-CX- X

ODN 0 -%,O N CV 0 0% 9-

I .- Vx.~I-. I L 5.s's.5'- -l-86 I

cc U

x C)Ww

x -r

cec

- a V

0 w I

Xo -W

La q

LN- z- t0 Z0 (

OW Vfl N 0% CC

a Q x 0 0 0,- a _0V C T XW: T N do

M acc: x 0 - -

aj 2C . X1 '1a

OD OD~ N ODi = U

ac- :- x OX, aN a

cc 04 I= : -a.* C

LU z 0 acar. a =0 a w") wwtasc 'U

ma +~a 0L M4 CL3

CD N UZ j - : < :rdO - CN S r )C)u wa aCV =a = 0a z ,EU)'

(D I. LU == - I-- WX ' a OWW W 4 MW M=-N 0 0 ,. -,A WC a ' 0 03Q0% 3 UCJ) a

a a(c m :Qa0 (Ya = C US

=l = C aj CL 0-aJ.- a'WX Zl 7- =S I--a~ Q W

H- Qm I. - Nm S CV: aa( ) = 04- Z nC~~~ ir. flLL M: ZL 0.X F'XC X i

W _j U0 - _j 0. U Z L CN -C-tILeW CL = S W eLC. i L iL 0 L C 0 - - - 0 U'mrr Z a9( DC 0 (D.t Mo on CC OM( v( D DC D0C V(

a " OU)I)Ls 01 0U f a0 QOO .. 'iJ)t k D% % Dk N3 N) aZ N U UU II V N il '-r-N -C D DO OC

.. i-..Opt..V L .~f: fl ~~.''-CC CC CC '87

II--- 4 Z H s rr r rr r ocw O

IX offibI

r

kit2. (r 4

xx. 4 a

ODDzC2 . !S

'a Q~ ru ;

020 0

;.l a. CD( -- M ) **aac mZCt ,C

a. 111 -Es; mQ(S

'a-. '0 toCa t

T3.fl - W am W 2 w c

NL L 's- 1' U- -1.. I'.S)O

4C 0 m 0 ZN 2a NL 02 aaW W q L NC Y

AN V a f1 kXI X im (0 L W U WL

QOO M- z 0- LU ( -

(CI-O C - X M1- 0 C a U 0 W *)m -aaU) ~ w wm LO - X 02 .4 0 L.Z0

*~~~~~~.. a......0L. . * 0. '".C f

N N* 4.

- .. -,. ItON C

N '- N v

-.1" 1(1

I I w.

X X'- X C-

CA. + X >- , c ;

X X c cI" I: I;

,S i I X .*

>c x

0 XW -r

Qa X S S

.. JI a "l- l--- .a, w a+N

ES C. I-- - Xw--. a a a.' > -- . 0 a.' -, - I ---

N a.- .- ' .- 1 ,- . M W Z ..- , I I I' I C >E c 0.+ X>C> = .e %W

0X X c * -> -- I X X-

4-l - I 1 t. a A vo . ,)-. X

N SE S >- 0 '.- .- *'-

0 CCX I4 a 0 W ',* ag W C) aaNM

0-1- s: XE X 0 Xli' --" +,,v,4.

M -~ Wi" -

L X X N -.1 I, M (% -* -

a &I O 1 4.J f- I > . a 'U> -Ue a -A -s -* 0 c c :saf111"O aOC 1 CoS - S X - I -II La 1 t a I' X Dl "

NEW Z. S,-I-I"I- Ic - -. a0 1-- X t_.-.,, - X. -A4 foxS 0 A0 C-~ il= f L LIL t

, -v LCC CX -L M .X" > W 's. .

c~ ' a- 44.-- * " X .I- . . ->IJ I--I- IIUOXIAXI1F)

i- 0 J0 5 X E- . . . . .- - _ X CE-. CC '-.'.

- ID N -0 -,

N M V W- N -* kl- N OD . 0--a aITXo-XSWXN-IaOC - 0N* *aOWC ~ ~ ~ ~ ~ ~ ~ 1 0. In-'- to~r Ir Si Wk Dj iC) SS 'za

89

killl

x IS 4

4:-:A

C N N

:5- + >- 14 +A LL

rC V- Lo vL0co- I- I 's ' a.01

> - I- - X W a . N-INW I0m 0

IC 1 C-'' a W.~I TU ^ k :, (d 4 ) CX. k() _j' U)Xa S

X+ W :4 N4 MN Z- a* X 01-

Ia5 -I- 6 * O W - - II -a 'sO.f 4 t . C

x C ( CWC a, 0CI C ~ a O NN C NW N )c Z 0) 0r(C) 1.

e WL O NX .- -Z .. a - lI- a' a> - X L.- M UOC1 -X j La - >-0 W-- C )0 ~f aad =1 Wt 51 11-30. oar a1 44 * I j - A Xu I-f .' - NO NstO'sX. .. I- . 1--0

4.-a -4CJ WZ zE4W U N DOC o WX IX)C0- ('~ £K C M0 'N-.SW L ILO' a.4 =.N 2 Z U.. *X *X7 LL - JI 4 Q -

- Ct)0U - (3) > 0. 4 5 u 04 W 0 0d X > X 00 0. (D (P 0 -3)9 0 ( ) X 0 0d 0' f0

-r) ?U) Wt-- NOD ON~ - - Cd C9) V U') Ito N OD 1% 0 - NC', V' U) Wf N ( Q. 0 - 'N ) V' U' W N CO 17 0

____ _______90

4' D

tA k

4 44U

z CL

'A OX L 0 C

C) 'Us W= (0 4ix 0 r w 1

1:c 'A W 'AW(P Q. 0) c

w as w I

0L L~ W'to M a af j- WI x X 't t

MI = T (- w z -13 a =~ a1 ~ -0 A+W a LI xt

- E L T -0U3) a. ms 1)0)E1. 1xx1 m X Yat M 1-zA (

W.NJ NN N 30) a Y

cro ~91

LL I,LL.'

0)V41'

ap

o 'Aw z

LL C

wi * m il

-. 1

U. +

S WC IW TO

I-- ~ ~ ~ ~ S I-- 3 . x0 m 4 I xx11. 1

ap ~ ' '. X=n(2If 43 (D 0 E O

z = =r 0 CD-;9 4 -~ _jI- 1 j-

a c M i Nca.' fr Ix i 3)k .

£ U) H aj

kD 4 W C0NMV0 , 0I U) r 0 N, LDr v0 3C9 UW ' i

S. *~ Q.Z2

(71

UU

L~ . L

LI w

0w V j -ix a '- -

M = v'-

- w .- T x-04I

I- 41 x1-1 .

(ra G - Ill)sy. 0 0x .

if0 0, TWk)f-I .

C- X LL ;cxC WVWI

- *lf =- -1 . l.x V zY a

-. 93

w

LL

w

- SM

-C w

o rLLL

Xm- z

=L W

'Z1 L LL r

+ WA'-,

WLW- M

:r W x C L kMd) z

f-~~ ~ ~ +idXx^=0- m , .CD M 1z + 4 4 w .^" CX X

16 1 oy . U'? 4 XfluL0 . + I-- SW'.E-,o m0-ea Q 11 Q - 4O ( Xxi ro v~ a Y. 4 V x f

0C 0 --O W 10 14 - I1 11 1 -- 13

-W.4f r,- OD) 13 iIZ' 0WN0 e)V0 DN0l-0 N T0W

~c Ja w-u94

.c ir ...... m-.s-o.*-'wr-~~~ZI TOM 0II..0-I '

AD-AO64 926 NAVAL. POSTGRADUATE SCHOOL MONTEREY CA F/S 13/13A FINITE ELEMENT PROGRAM SUITABLE FOR THE HEWLETT-PACKARD SYSTE--ETC(U)

MAR A0 R R MALLORY

UNCLASSIFIED ML

lill IIl IiIIIIL

111IL2 1 6

MICROCOPY RESOLUTION TEST CTNATIO04AL BUREAU OF VTADARD-1963-,

'Km

0-m

-J

. -

*X : -k3 i

-r 03> - -r' - ,

Ir 0.:. 4 . *.,.Z ,= ' '..

" I- It tZ .Q U)N X MW

~~ ~ MZ Lx ~ h X 3 m It v ,. X it") IxI ""n } ~

I- N:: C &'aa 64 +- -Ix Sl S1 ccI C l'l-x I - -QUU = M MMX

i W 0 0 a- .el s CMf 01.6u." -I - .- Z --. M M- X M Z M + n, - 0 .I ) " US C f C W IU.'K CO V *StoN3 * a NM 0 0 I 0 , N M V 0WNIN S.) S

95

'a 0 5 a e.Q 's s.5 t x IX I) 5

NN-A I

- a.

z w

Q 4-

aa.

M 2 CL

LX s

(V) I% x C

w0 m x r CL M 0+ 0 sr w Ix 0 W (A (a -

0 + WU) L. '00 CC 0 0 m'

04 I .4 604 X Y N N, L (C

ix ~~~~ ~ ~ ~ NN'-. Y z jama y = wa. x -j 33 .. Ex 1 (Lm tjU a

W~~~0 m .W1- -04 &X11a x XM X -I MI- EE L A A C -->

MMXO 3U.W aA-w wwm mQ. " mm ww40~~~~~~~~ ~ ~ ~ a L d - J=MX0E -- UQ- 0 LC KXIG .( c - O D D D D 3 ( i C D o mM V a0 CZ

kiD~~~~~ ~ ~ ~ ~ ~ P.0 (P Mt)A nI NO %# t . 0 %O v t0WN1D0 3

96,

WLAO

'A,In

'-L

rMC

vi.

xx

g)OD

L) -D j Z1 A n c

o3L= - a a

w CK0 op I I -() -"0

a A -U (C aZ O -0 U) a ZW Z kX E 0 00 1 Da

(C Li W2. a.-

9 ~ ~ ~ ~ ~ o DY U).M 0 )U XI 1C ixi

MO3~- W W = i (0) a) (0 a b (X L WA Zw 4 CL Z LiL M U t Z C( XLL W

a a aU- 97

REFERENCES

1. Wilson, E.L., "SAP - A Structural Analysis Program,"Report UC SESM 70-20, Dept. of Civil Engineering,University of California, Berkeley, 1970.

2. Control Data Corporation, EAC/EASE2 - User InformationManual/Theoretical, 1973.

3. Bathe, K.J., and E.L. Wilson, Numerical Methods inFinite Element Analysis, Prentice-Hall, Inc., New York,1976.

4. Hewlett-Packard Co., Hewlett-Packard System 45 DesktopComputer Operating and Programming, 1977.

5. Hewlett-Packard Co., Hewlett-Packard System 45 DesktopComputer Mass Storage Techniques, 1977.

6. Hewlett-Packard Co., Hewlett-Packard System 45 DesktopComputer Graphics ProgrammIng Techniques, 1978.

7. Zienkiewicz, O.C., The Finite Element Method, McGraw-Hill Book Co. (UK) Limited, London, 1977.

8. Ralston, A., A First Course in Numerical Analysis,McGraw-Hill, 1965.

I

98

- - -........... ~- .

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22314

2. Library, Code 0142 2Naval Postgraduate SchoolMonterey, California 93940

3. Chairman, Code 69Mx 1Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940

4. Professor Gilles Cantin, Code 69Ci 5Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey, California 93940

5. Ray Mallory 2Code 753Naval Coastal Systems CenterPanama City, Florida 32407

99

S'Am