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NASA Contractor Report 198443 



Probabilistic Fiber Composite Micromechanics 



Thomas A. Stock 
Cleveland State University 
Cleveland, Ohio 



January 1996 



Prepared for 

Lewis Research Center 

Under Grant NAG3-550 




National Aeronautics and 
Space Administration 



PROBABILISTIC FIBER COMPOSITE MICROMECHANICS 



THOMAS A. STOCK 
Bachelor of Science in Civil Engineering 
Northwestern University 
June, 1983 



Submitted in partial fulfillnent of requirements for the degree 

MASTER OF SCIENCE IN CIVIL ENGINEERING 

at the 

CLEVELAND STATE UNIVERSITY 

March, 1987 



This thesis has been approved 
for the Department of Civil Engin ee ring 
and the College of Graduate Studies by 



Prof. Paul X./tellini, Thesis Conmittee Chairman 



C/UlkaymU /j?-Z4-&7 



Dr. C C. Chamis, Adjunct Associate Professor 



Dr. John H. Hemann, Department Chairman 



11 



ftCKNOU-EDGETEHIS 

This thesis is the result of work performs*! at NflSA- Lewis Research 
Center under grant MftG 3-550. 

I wish to express my gratitude to Dr. Christos Chamis, from the 
Structures Division at Lewis, for his insight, enthusiasm, patience, and 
wit, all of which contributed to my learning, of which this thesis is a 
small part. His example is an inspiration to ma, in school, work, and 
life. Dr. Bellini, my advisor, deserves thanks for patiently reading 
the many drafts, and supplying perspective and criticism to aid in 
clarifying my ideas. He has been a teacher and a friend. 



ill 



ABSTRACT 



Probabilistic composite micronEchanics nethods are developed that 
sinulate expected uncertainties in unidirectional fiber conposite 
properties. These netbods are in the form of computational procedures 
using Ifente Carlo simulation. The variables in vfcich uncertainties are 
accounted for include constituent and void volune ratios, constituent 
elastic properties and strengths, and fiber misalignment. A 
graphite/epoxy unidirectional conposite (ply) is studied to demonstrate 
fiber composite material property variations induced by random changes 
expected at the material micro level. Regression results are presented 
to show the relative correlation between predictor and response 
variables in the study. These computational procedures make possible a 
formal description of anticipated random processes at the intraply 
level, and the related effects of these on composite properties. 



xv 



PAGE 
1 
1 
6 



TABLE OF CONTENTS 

CHAPTER 

I. INTRODUCTION 

A. Background 

B. Purpose 

C. Formulation of the Model 

D. Method of Investigation 9 

1. Brief Description of ICAN 

2. Summary of Variables 

3. Ifcnte Carlo Methods 

E. Brief Sunnary of Results 

II. METHODS OF CALCULATION 

A. Overall Plan 

1. Input Structure for ICAN 

2. Constituent Property Variations 

3. Repeated Runs 

4. Data Collection 

B. Generation of Pseudo Random Numbers 23 

1. Uniform Distribution 23 

2. Normal Distribution 25 

3. Gamma Distribution 2S 

4. Weibull Distribution 28 

C. Property Distribution Assumptions 31 



9 

10 
13 
15 
17 
17 
17 
18 
20 
20 



D. Use of ICAN 47 

1. Composite Hicronechanics 47 

2. Laminate Theory 48 

3. Strength Theories 49 

E. Review of Applicable Statistical Concepts 51 

1. Sample Maans 51 

2. Sample Standard Deviation 51 

3. Confidence Interval Estimates 51 

4. Multiple Linear Regression 53 

III. RESULTS * 57 

A. Property Histograms and Distributions 57 

B. Fiber Strength Effect 73 

C. Matrix Strength Effect 76 

D. Fiber Orientation Effect 80 

E. Fiber Stiffness Effect 80 

F. Regression Msdels 1®1 

IV. DISCUSSION 13® 

A. Overview 13® 

B. Histograms and Distributions 131 

C. Confidence Curves 133 

D. Examination of Regression Models 134 

V. CONCLUSIONS 136 
REFERENCES 138 



VI 



APPENDIX A- SOURCE PROGRAM LISTINGS 14 ® 

APPENDIX B- ICflH PROGRAM DETAILS 154 

1. Conposite Micronechanics 155 

157 

2. Laminate Theory 

171 

3. Strength Theories 



VI 1 



LIST OF TABLES 

TABLE TITLE ?££ 

I INPUT DATA PCJR SAMPLING M 

II CASE 1 RESULTS 59 

III LONGITUDINAL MODULUS (ECU), SIMPLE 104 

IV TRANSVERSE MODULUS (EC22), SIMPLE 105 
V SHEAR MODULUS (EC12), SIMPLE 106 

VI POISSON'S RATIO, MBJOR (NUC12), SIMPLE * 107 

VII POISSON'S RATIO, MINOR (NUC21), SIMPLE 108 
VIII LONG. THERM. EXPANSION (CTE11), SIMPLE 109 

IX TRANS. THERM. EXPANSION (CTE22), SIMPLE HO 

X LONG. TENSILE STRENGTH (SCX5TT), SIMPLE HI 

XI LONG. COMPRESSIVE STRENGTH (SCXXC), SIMPLE 112 

XII TRANSVERSE TENSILE STRENGTH (SCYYT) , SIMPLE 113 
XIII TRANSVERSE COMPRESSIVE STRENGTH (SCYYC), SIMPLE 114 

XIV IN PLANE SHEAR STRENGTH (SCXYS), SIMPLE 115 

XV LONGITUDINAL MODULUS (ECU), INTERACTION 118 

XVI TRANSVERSE MODULUS (EC22), INTERACTION 119 

XVII SHEAR HDDULUS (EC12), INTERACTION 120 

XVIII LONGITUDINAL THERMflL EXPANSION 121 



vm 



XIX TRANSVERSE THERMAL EXPANSION 122 

XX POISSON RATIO, MAJOR (NUC12), INTERACTION 123 

XXI POISSON RATIO, MINOR (NUC21), INTERACTION 124 

XXII LONGITUDINAL TENSILE STRENGTH (SCXXT), INTERACTION 125 

XXIII LONG. COMPRESSIVE STRENGTH (SCXXC), INTERACTION 126 

XXIV TRANSVERSE TENSILE STRENGTH {SCTTT) , INTERACTION 127 
XXV TRANSVERSE COMPRESSIVE STRENGTH (SCYYC), INTERACTION 128 

XXVI IN PLANE SHEAR STRENGTH (SCXYS), INTERACTION 129 



IX 



LIST OF FIGURES 

FIGURE TITLE Pi^ 

1 Photomicrograph of Graphite/Epoxy cross 5 
section showing variation in fiber content. 

2 Conventional Ifedel 8 

3 Substructure Hbdel 8 

4 Coordinate Systems A1 

5 Order of ICflN input data cards H 

6 Command Input 19 

7 Constituent Variation Input 19 

8 Flow Chart of Probabilistic Integrated 22 
Composites Analyzer 

9 Uniform Distribution: general form 29 

10 Normal Distribution 29 

11 Gamma Distribution density functions 3© 

12 Weibull Distribution function 3® 

13 Normal Distribution Simulation 32 

14 Gamma Distribution Simulation 33 

15 Gamma Distribution Simulation 34 

16 Gamma Distribution Simulation 35 

17 Gamma Distribution Simulation 36 



18 Ueibull Distribution Sinulation 37 
Matrix Shear Strength 

19 Ueibull Distribution Sinulation 38 
Matrix Shear Strength 

20 Ueibull Distribution Simulation 39 
Matrix Tensile Strength 

21 Ueibull Distribution Simulation 4® 
Matrix Tensile Strength 

22 Ueibull Distribution Simulation 41 
Matrix Tensile Strength 

23 Ueibull Distribution Simulation 42 
Matrix Compressive Strength 

24 Ueibull Distribution Simulation * 43 
Matrix Compressive Strength 

25 Ueibull Distribution Sinulation 44 
Fiber Tensile and Compressive Strength 

26 Ueibull Distribution Simulation 45 
Fiber Tensile and Compressive Strength 

27 Ueibull Distribution Sinulation 46 
Fiber Tensile and Compressive Strength 

28 Typical Stress-Strain behavior of unidirectional 50 
fiber composites 

29 In-plane fracture nodes of unidirectional (ply) 5® 
fiber composites 

30 Sampling results for Longitudinal Elastic Modulus 60 

31 Sampling results for Transverse Elastic Modulus 61 

32 Sampling results for In plane Shear Modulus 62 

33 Sampling results for Poisson Ratio (major) 63 

34 Sampling results for Poisson Ratio (minor) 64 

35 Sampling results for Longitudinal Thermal Expansion 65 



XI 



48 Longitudinal Elastic Modulus, for various 
shape parameters of fiber orientation 

49 Transverse Elastic Modulus, for various 
shape paraneters of fiber orientation 

50 In-plane Shear Modulus, for various 
shape parameters of fiber orientation 

51 Longitudinal Tensile Strength, for various 
shape paraneters of fiber orientation 

52 Longitudinal Compressive Strength, for various 
shape parameters of fiber orientation 

53 Transverse Tensile Strength, for various 
shape parameters of fiber orientation 



68 
69 



36 Sampling results for Transverse Thermal Expansion 66 

37 Sampling results for Thermal Expansion Coupling 67 

38 Sampling results for Longitudinal Tensile Strength 

39 Sampling results for Longitudinal Compressive Strength 

40 Sampling results for Transverse Tensile Strength 70 

41 Sampling results for Transverse Compressive Strength 71 

42 Sampling results for In- plane Shear Strength 72 

74 



75 



43 Longitudinal Tensile Strength, for various 
shape parameters of fiber strength 

44 Longitudinal Compressive Strength, for various 
shape parameters of fiber strength 

45 Transverse Tensile Strength, for various 77 
shape parameters of matrix strength 

46 Transverse Compressive Strength, for various 78 
shape parameters of matrix strength 

47 In-plane Shear Strength, for various 79 
shape parameters of matrix strength 



81 



82 



83 



84 



85 



86 



xn 



87 



64 Longitudinal Compressive Strength; for various 
shape paraneters of fiber nodulus 

65 Transverse Tensile Strength; for various 
shape paraneters of fiber modulus 

66 Transverse Conpressive Strength; for various 
shape paraneters of fiber nodulus 

67 In Plane Shear Strength; for various 
shape paraneters of fiber nodulus 



88 



89 



54 Transverse Conpressive Strength, for various 
shape paraneters of fiber orientation 

55 In-plane Shear Strength, for various 
shape paraneters of fiber orientation 

56 Poisson's Ratio (major); for various 
shape paraneters of fiber orientation 

57 Poisson's Ratio (minor); for various 90 
shape paraneters of fiber orientation 

58 Longitudinal Elastic Ifcdulus; for various 91 
shape paraneters of fiber nodulus 

59 Transverse Elastic fbdulus; for various 92 
shape paraneters of fiber nodulus 

60 In Plane Shear ffodulus; for various - 93 
shape paraneters of fiber nodulus 

61 Poisson's Ratio (major); for various 94 
shape paraneters of fiber nodulus 

62 Poisson's Ratio (minor); for various 95 
shape paraneters of fiber nodulus 

63 Longitudinal Tensile Strength; for various 96 
shape paraneters of fiber nodulus 



97 



98 



99 



100 



Kill 



B.l Components of Stress acting on elenental unit cube 174 

B.2 Rotation of coordinates from 1-2 to x-y. 174 

B.3 Bending geometry in the x-z plane 175 

B.4 Shearing force deformations on straight cross section 175 

B.5 Stress and {foment resultants 176 

B.6 Laminate index notation contention 176 



xiv 



CHAPTER I 
IKTRODUCTION 

A. Background 

The diverse requirements of recent engineering applications have 
motivated designers to explore specialized structural and material 
systems. Ceramic materials, for example, have several attractive 
structural properties, such as their high stif fnessA*ight ratios, and 
low variation of stiffness and strength over wide ranges of 
environmental conditions. A significant disadvantage inherent to 
brittle structural materials is their vulnerability to failure due to 
cracks propagating from f lavs. The increased probability of a flaw 
occurring in a material as the volune increases leads to bulk strengths 
«hich are a fraction of the theoretical strength of the material. The 
size effect on material strength (Ref . 1) can be explained by the 
"weakest link" concept. Griffith ( Ref. 2) reasoned that very small 
solids, for example wires or fibers, might be expected to be stronger 
than large ones, due to the additional restriction on the size of the 
flaws. In the limit, a single line of nolecules mast possess the 
theoretical molecular tensile strength of a material. A consequence of 



the size effect on strength was the development of fiber composite 
materials which consist of thin, strong fibers bound together by a 
ductile matrix. The advantages of fine, strong fibers can explain the 
current trend toward increased use of fiber conposite materials in 
denanding aerospace applications. 

Properties of a composite laminate depend on the properties of the 
constituent materials, their distribution, and orientation. Laminates 
are composed of layers of unidirectional ly reinforced plies (laminae). 
The lamina is typically considered the basic unit of material in a 
composite structural analysis, which requires knowledge of the material 
properties of each individual lamina and its geometric orientation. The 
branch of conposite mechanics that predicts ply material properties 
based on the properties, concentration, and orientation of its 
constituents is knowi as conposite micronBchanics , and frequently 
incorporates the traditional ttschanics of Materials assumptions. The 
desired laminate is created by stacking of plies in specific directions. 
The integration of ply properties to yield laminate properties is called 
laminate theory. Laminate variables such as ply orientation and 
stacking sequence can be tailored to yield a laminate with the desired 
material properties. Thus, the laminated conposite is a suitable 
material for component design. 

Analysis of fiber composite structures is currently performed using 
a variety of computer codes. From the original codes based on classical 
micronechanics and laminate theory, recent codes (Bef . 3,4) have been 
developed which incorporate the current state of the art. Complete 



3 

nEchanical, thermal , and hygral properties are calculated, and can be 
used to compute response, Advanced failure criteria are used to 
calculate composite strengths. Environmental effects are also 
quantified. The usefulness of these codes has been deimnstrated by 
comparison with experimental and finite element results (Ref. 5,6). 

The analytical capability of many codes is limited by the 
deterministic nature of the computations. Specifically, fixed values 
for constituent material properties, fabrication process variables (i.e. 
constituent volume ratios) and internal geometry must be used as input. 
Ifevjever, random variations in these parameters are not -only expected, 
but easily observed experimentally. (See Fig. 1) 

The analysis of composite structures requires reliable predictive 
ntxlels for material properties and strengths. However, the prediction 
efforts have been complicated by inherent scatter in experimental data. 
Since uncertainties in the constituent properties, fabrication 
variables, and internal geometry would lead to uncertainties in the 
measured composite properties, the question arises: 

How much of the "statistical" scatter of experimentally observed 
composite properties can be explained by reasonable statistical 
distribution of input parameters in composite microraechanics and 
laminate theory predictive models? 

The increasing use of probabilistic methods in structural mechanics has 
been shown to provide a more realistic depiction of structural response 
due to load variations. (Ref. 7) The recognition that material 
parameters are characterised by a spectra of values ( that is, are 



statistical in nature ) rather than by a unique set of values, points to 
probabilistic nethods as a logical analysis approach. 




Fig. 1- Photonlcrograph of Graphite/Epoxy cross section 
showing variation in fiber content. (Ref. 19) 



B. Purpose 

The aim of this thesis i» to develop a oonputational oapability to 
simulate the probabilistic variations in the irechanical behavior of 
unidirectional fiber composites. The Ifonte Carlo ■ethod is used to 
simulate a variety of random processes, to quantify fiber coaposite 
material variations induced by random changes in composite fiber 
alignnent, constituent properties, and fabrication process variables. 
This random process description is an attempt to more accurately predict 
the behavior of manufactured materials, %fcich inherently include these 
random variations. The characterization of fiber reinforced composites 
through simulation of local nonuniformities provides an economical 
alternative to experimentation to measure material properties. 



C. Formulation of the Model 

Tte mxiel comronly used in characterizing fiber composites is based 
on the calculation of properties of the basic unit of an orthotropic 
ply. The layup geonetry is then used in laminate equations to calculate 
opposite prooerites (See Figs. 2a, 2b). In this «*, hoover , the 
basic unit is taken as the sub-ply. which consists of only one 
fiber-matrix level in the material, nicronechanics theory is used to 
calculate the properties of the assumed orthotropic sub-ply, each with 
randomly distributed fabrication variables and material properties. 
Distributed fiber directions, due to possible misalignment within the 
ply, are then used in the laminate equations to calculate ply 
properties. T*is substr^turing of the composite ply represents a novel 
attempt at characterization of fiber composite material properties based 
on probabilistically distributed constituent properties, individual 
fiber misalignment, and fabrication process variables (See Figs. 3a,3b). 

^is formulation is particularly well suited to the probabilistic 
description of fiber composite material properties. Since the 
^cromechanios and laminate equations can be used to calculate ply 
properties at any number of points in a ply, a tractable finite element 
structural analysis based only on simple distributional assumptions for 
physical parameter variations can be perform*. T*is nodel supplies a 
rational procedure for composite material property assessment, because 
it treats the material as the result of a series of random processes 
which occur at the intraply level. 



8 



^: V .... 


^^•tDpOtV' 


. . . -.._.. ___— 


Xm}ji 





ssaas 


XL -L\ 


$r 




(a) orthotropic ply 



(b) laminate 



Fig. 2- Conventional Model 




fiber 

misalignment 




(a) subply 



(b) ply 



Fig. 3- Substructure Model 



D. Method of Investigation 

1. Brief Description of ICAN 

The Integrated Composite Analyzer (ICflN) is a computer program for 
comprehensive linear analysis of multilevel fiber composite strictures. 
The program contains the essential features required to effectively 
design structural components made from fiber composites. It now 
represents the culmination of research conducted since the early 1970»s, 
at the National Aeronautics and Space Administration (NASA) Lewis 
Research Center (LeRC), to develop and code reliable composite mechanics 
theories. This user friendly, publicly available code incorporates 
theories for 

1. conventional laminate analysis 

2 intraply and interpiy hybrid composites 

3. hygral, thermal, mechanical properties and response 

4 ply stress-strain influence coefficients 

5*. microstresses and microstress influence coefficients 

6. stress concentration factors around a circular hole 

7. predictions of delamination locations around a circular hole 

8. Poisson's ratio mismatch details near a straight free edge 
g. free edge inter laminar stresses 

10. laminate failure stresses 

11. normal and transverse shear stresses 

12 explicit specification of matrix-rich interpiy layers 
\l\ f£ite element material cards for NASTOAN, MARC 

A detailed description of ICAN can be found in Reference (3). The 
ICAN code and documentation are available through COSMIC, the Computer 
Software Management and Information Center, Suite 112, Barrow Hall, 
Athens GA, 30602. 



10 



2. Sunirary of Variables 

The variables studied in this *ork csan be separated into two 
categories. The independent variables to be simulated using random 
sampling consist of the following (see Fig. 4a for fiber coordinate 
system) : 



Geometry: 

fiber orientation angle 

Fabrication variables: 
fiber volume ratio 
void volume ratio 

Fiber properties 

longitudinal elastic modulus 
transverse elastic modulus 
shear modulus, 1-2 plane 
shear modulus, 2-3 plane 
fiber tensile strength 
fiber compressive strength 

ffetrix properties 

elastic modulus 
matrix tensile strength 
matrix compressive strength 
matrix shear strength 



(THETA) 

(FVR) 
(WR) 

(EFP1) 

(EFP2) 

(GFP12) 

(GFP23) 

(SFPT) 

(SFPC) 



(OF) 
(SfPT) 
(SMPC) 
(SIPS) 



The dependent variables to be calculated using ICAN consist of the 
following ply properties, measured about the material axes (see Fig. 
4b): 



normal modulus in 1-1 direction 
normal modulus in 2-2 direction 
shear modulus in 1-2 plane 
Poisson's ratio for strains in 2 

by stresses in 1 direction 
Poisson's ratio for strains in 1 

by stresses in 2 direction 
Coefficients of thermal expansion 

in 1-1 direction 

in 2-2 direction 

coupling coefficient 



(sen) 

(EC22) 

(EC12) 
direction induced 

(NUC12) 
direction induced 

(N0C21) 



(CTE11) 
(CTE22) 
(CTE12) 



11 




(a) fiber 



i yV_ _£_ M .^flKJt 

SiCAJt 

(b) material 




Fig. 4- Coordinate Systems 




Fig. 5- Order of ICAN input data cards 



12 



Ply strengths in material directions 

longitudinal tensile JSCJOCT) 

longitudinal compressive (SCXXC) 
transverse tensile (SCYYT| 

transverse compressive (SCYYC) 

in-plane shear (SOWS) 



The descriptions above should be consulted periodically for the 
definitions of variables that henceforth will be referred to 
symbolically. 



13 



3. Monte Carlo Methods 

Complicated stochastic processes can be simulated by a variety of 
numerical methods generally referred to as Monte Carlo raathods (Ref . 8). 
The term refers to that branch of experimental mathematics concerned 
with experiments on random numbers. Since the advent of high speed 
computers, they have found extensive use in most fields of science and 
engineering, in analyzing many physical processes of a statistical 
nature, or where direct experimentation is not feasible. In general, 
they can be economically used to achieve a level of precision between 9® 

and 95 percent. 

ft Monte Carlo experiment refers to the procedure of randomly 
assigning a value to an independent random variable in a chosen itodel, 
and observing the dependent variable at the conclusion of the process 
being modeled. A Monte Carlo procedure is composed of n such 
independent experiments. When n is sufficiently large, the observations 
will yield, by virtue of the laws of large numbers, a statistically 
meaningful description of the physical problem. 

Tne form of Monte Carlo used in this study is as follows: 

1. Define the system model by assuming 

a. model regression function 

b. method of error incorporation 

c. probability distributions of all errors (for all independent 

variables) . 

d. any equations used to model the phenomena of interest 

2. Use the computer and random sampling techniques to select 
values of the independent variables. 

3. Calculate dependent (output) variables using the prescribed 



14 



equations. 

4. Estinate regression paraneters for the assoned radel. 

5. Replicate the experiment, each tiws with a new set of input 

values. 
6 Use appropriate statistical nethods to calculate properties of 

the distribution of paraneter estimates. 



15 



E. &-ief SuraiBry of Results 

ft ply made from the flS-Q-aphite /I1K epo«y composite system i. 
staled. The sonte carlo scheme is used to generate . nu*«r of 
response results, Wich are analyzed in graphical and nu^rical form, to 
5u pply a random process description cf composite ply elastic constants, 
therm,! expansion coefficients, and strengths. Histogram and 
distribution plots of results for assumed narrow and wide variations in 
input properties are compared with a deterministic base case for an 
aligned ply. The figures demonstrate the range of values that response 
tables assume for the example data under consideration. 

Confidence intervals are calculated for response variables in 
subsequent samples, which are normalized with respect to an appropriate 
independent variable, to yield plots of normalized response as a 
firo tion of fiber volun* ratio, for various values of distribution 
parameters for the related independent variable. These plots 
demonstrate the sensitivity of ply properties to randomly selected 
uncertainties in constituent and fabrication variables. 

Several multiple line-- regression models were calculated for 
response variables. The relative correlation of predictor (independent, 
tables with response is staled for all output properties considered. 
Varying levels of significance were achieved in the regression 
.guations, due to the differences in complexity of response variables. 
Elastic constants can be described adequately with simple r^n-essor 
fusions, a~l generally explain bet««. »^" P—" «* tte 
observed response variations about a «. The regression .-dels 



16 



studied far strength, although achieving better reliability with higher 
order regressor functions, demonstrate suoh low significance as to be 
practically useless for predictive purposes. This is not an unexpected 
result, because of the coup lex nature of strength behavior in cowposite 
neterials. 



CHAPTER II 
PETHODB OF CALCULATION 

A. Overall plan 

1. Input stricture for ICAN 

The input data for a typical execution of the available ICAN 

program consists of (see Fig 5) 

1. header card 

2. control cards 

3. ply data cards 

4. imterial system cards 

5. load cards 

For repeated use of the ICAN program, input data files must be 
created and used one at a tine. Each successive run of the master 
program (of ^ich ICAN is made a subroutine) vrites the input file from 
user-supplied parameters and calls ICAN. The ply data cards contain 
randomly generated fiber orientation angle values. The material system 
cards contain randomly generated values for fiber and void volune 
ratios. 



17 



18 



2. Constituent Property Variations 

Each siEoessive execution of ICfiH uses » distinct set of nmterial 
properties for fiber and matrix. The random number generation is 
performed with user-supplied parameters which are stored in a separate 
file. The options of using either generated properties or using the 
values contained in the resident data bank are available. Any subset of 
the parameters described may be generated or held constant with proper 
specification of the Boo leans which control the input to the ICAN 
program, (see Figs. 6,7) i 



19 



FIBER STRENGTH VARIES; CONSTANT FIBER VOLUTIE RATIO OF 0.30; TAPE 03131 
STDATA 13 1 15 T 
T 50 T F T T 
F 000.0 10.0 0.300 0.200 3.00 3 

F 
F 

p 1Y 70.00 70.00 .0 -000 

WATCRDAS-IOTHS AS_1I ? H ? -° '" 

PLOAD 10. 0.0 0.0 0.0 

PLOAD 0.0 . 0.0 0.0 

PLOAD 0.0 0.0 



OPTION 



Fig. 6- Command Input 



i. 3100E 08 0.3000E 07 

EFP1 I MoOOE 07 0.2000E 06 

EFP2 I S 2000E 07 0.2000E 06 



GFP12 5 B1000E 07 0.1000E 06 

GFP23 I S'loOOE 06 0.1000E 02 

SFPT I 4000E 06 0.1000E 02 

SFPC I 5000E 06 0.5000E 05 

ET1P I I i5ool 05 0.1000E 02 

S"PT I 03500E 05 0.1000E 02 

SriPC J •.}»« 5| dOOOE 02 



Fig. 7- Constituent Variation Input. Example for AS ; J graphite 
fiber and IMHS Epoxy matrix, with wide variations of 
stiffnesses and strengths. 



20 



3. Repeated runs 

The user* must specify the number of I CAN runs desired in a given 
sample. In this study, fifty (50) runs were used throughout, to take 
advantage of the simplification in statistics by using suitably large 
samples. From elementary statistics, it is known that any process that 
is the result of the combined interaction of several probabilities can 
be assumed to approximate a normal distribution. For phenomena that are 
assumed to approximate a normal distribution, the simplest forms ior 
calculating statistics apply to suitably large samples (usually greater 
than thirty). The sample size of fifty was chosen to supply a 
practicably large amount of data, within the restrictions imposed on 
computation time. 

The data generated by repeated execution of the ICAN routines is 
stored in a sequential access dataset, where the 50 output files are 
separated by end of file markers. This arrangenent allows a single 
Fortran unit to be used for output throughout. A simple flowchart of 
the data generation routines is shown in Fig. 3(a). 

4. Data collection 

The ICAN output files are searched to locate the specific material 
properties and strengths of interest in this study. The flowchart of 
data collection routines is shown in Fig. 8(b). After obtaining the 
sample of ICAN output, the investigator may choose to scrutinize 
parameters or calculate statistics aside from those chosen in this 
study. This is likely, in light of the large quantity of data available 
and the need for limiting the scope of this particular study to 



21 



representative properties. The user „ld have to supply additional 
code or adapt existing code to suit his purposes in this case. The 
coded mxlif ioations to ICAN used in this study are included in Append 



A. 



[ 



22 



C " m ) 



CKU 

sumoutihe 




CALL 
JUMOUTIIrt 
UMAT 



H 



6ENERATE. 

WtlTC 

MHtXM 

DATA 



/ Mil 7 

Air 



I MRS 



CALL 

SJMOUT1IC 

ICAMI 







CALL 
SUMOUTIRE 
ICAIM 



lEVINO 
DATA 
■ARK 



/MUTE 7 
dutwt ml 
UNIT « / 



TOFItf 
UNIT 

C 



@ 



(a) data generation program 





/ 28 / 










CALL 
STATISTICS 
SWaOUTIHCS 
















HOT 
MSTDSMS 
MB 

diotuwtjojb 




HOT 
CORFI DCIICC 
CURVES 




tOKSSlO* 
AMLTSIS 



(b) analysis procedures 

F1g. 8- Flow chart of Probabilistic Integrated 
Composites Analyzer 



23 



B. Generation o£ Pseudo Random Numbers 

An integral part of any nonte carlo simulation is the use of random 
numbers having a specified distribution *hich is assuned to characterize 
the process under study. Indeed, many statistics textbooks carry tables 
of random numbers as appendices. Situations using large samples 
require many repeated calculations, each with different "random- 
numbers. Since filling of a computer mznory with a large table of 
random numbers is wasteful, algorithms have been developed (Ref . 9) to 
generate streams of random numbers whenever needed in the process of 
calculations. The numbers used are usually obtained using sone form of 
a recursion relation, hence the sequence is terned pseudo-random. 

1. Uniform Distribution 

The starting point for many random number schemas is the uniform 
random number generator, which simulates a sample from the uniform 
distribution. A continuous random variable has a uniform distribution 
over an interval a to b ( b > a ) if it is equally likely to take on any 
value in this interval. The probability density function is thus 
constant over ( a,b ) and has the form 

= elsewhere 
The probability distribution function is, on integrating 

F(x) =0 x < a 

x - a 
_ a < x < b 

b - a 



24 



= 1 x > b 

The uniform distribution is shown in density and distribution form in 
Figs. 9a and 9b. 

Lehmer (Ref. 1©) proposed the congruent ial nethod of generating 
pseudo random numbers conforming to the uniform distribution. The 
recurrence relation takes the form: 

x. = (ax. . + b ) modulo m 
x v i-i 

where the notation signifies that x A is the remainder when (ax i _ 1 ♦ b) 
is divided by m. The multiplier a, increment b, and modulus m are 
integers. The starting value x_ must be assumed, and is known as the 
"seed" of the generator. Generators for which b = © are known as 
multiplicative. They are called mixed when b is nonzero. Because 
selection of the multiplier a and modulus m strongly influence the 
generator, most generators in use are of the multiplicative form. A 
discussion of the choice of parameters, maximum period, and degree of 
correlation of this generator is available (Ref. 11). 

For a given uniform random number u on the interval (0, 1) a random 
number x having a desired distribution F(x) is often obtained by solving 
the equation u = F(x) for x (Ref. 12). Since the process requires the 
determination of the inverse distribution function F (x), its use 
depends on the ease of deriving the expression or some approximation. 
The following sections describe the distributions used, and methods for 
generating random numbers on those distributions. 



25 



2. lfarmal (Gaussian) Distribution 

The most ooimon distribution is the familiar normal distribution, 
with the "bell shaped" density function, given by 



f(x;A*,a 2 ) = 



1 f (x-/0 2 1 

■ ■ exp : — 



-«• < x < oo, ft < », and o > 
with nean // and standard deviation a. The distribution function is 



written 



F(x) = 



JTaa 1 



exp 



(u-fi)' 
-2^~ 



du 



«hich cannot be expressed in closed form analytically but can be 
nunerically evaluated at any value of x. 

The Box-fcller or "Polar" method (Ref . 13) is most comronly used 
for generating random deviates from a nean to approximate the normal 
distribution. If x A and x 2 are independent uniform random variables, 

then 

0.5 



y t = o{-2 In x 1 ) W ' cos 2nx 2 + ft 



0.5 



y = o{-2 In x x r sin ani 2 + * 



are independent random variables with the standard normal distribution 
having nean fi and standard deviation a. 



26 



3. Gamma Distribution 

The gamrn distribution is a two-parameter distribution whieh is 
flexible in fitting * variety of random processes. It is a one sided 
distribution in that physical quantities that are limited to values in 
the positive range are frequently modeled by it. Its density function 
is given by 

A -Xx k-1 
f(x) = e x 

/•(k) 
Were x, X, k > «, and k is an integer. 

The parameters X and k nay be interpreted as scale and shape parameters, 

respectively. T(k) is the well known gamma function, 

r<» k-i -u... 
*"( k ) = Jo u e du ' 

which is widely tabulated. The gamra distribution function is given by 



X k 



f (*> = T(krj0 u 

r(k,xx) 



x 

k-1 -Xu 



du 



x > 



r(k f «l = I x" _ e"*dx 



= elsewhere 
where T(k,u) is the incomplete gamra function 

. T U k-1 
J® 
which is also widely tabulated. For integer values of k, 

T(k) = (k-1)! 
and the gaima distribution is known as the Erlangian distribution after 
A. K. Erlang, who introduced it in the theory of queues and !farkov 
processes. 



27 



Garni* variates are generated using the sapence 

satisfying the uniform distribution on the interval (€>,i). 

The recursion relation is 

1 

y L = - — ln u i« 



-hi-- — 1 "^ 



vtere x is a gamma var 



x 

i=l 
iate having parameters X and k (Ref. 14)- 



28 



4. Vfeibull Distribution 

The Ueibull distribution (Ref . 15) is mist popular **en modeling 
problems of reliability, material strength, and fatigue. The Ifeibull 
density function is given by 

f (x;a,fl) = o^~ 1 exp(-<iir) 
< x < co, a > 0, £ > 1 
where a and fi are the shape and scale parameters, respectively. The 
cumulative distribution function 

y = F(x) = 1 - exp[-(x/0) a ] 
leads iimediately to the inverse relationship 

p-^y) = x = - fi[ ln(l-y) ] 1/a 
as the desired ifeibull random generator i*en y is a uniform random 

variable. 

Figures 9-12 show the above distributions in analytical form. 



29 



fx<*> 



Fx<x) 



£T- 



■»■« 



■♦•* 



(a) density 



(b) distribution 



Fig. 9- Uniform Distribution: general form. 




-4 -3 -: 



2-101 2 

(a) density 




-a -a -i 

(b) distribution 

Fig. 10- Normal Distribution 



uo 



Hounj uou^Mlsta lin<H 3 « "21 ' 6 * J 



a -nam amptij 
T 1 T 



I 




SCO 



-\ oso «■ 



- sio 



-»oi 



SUOl* 3un J- 



/Citsuap uo^nq^siQ euiueg -U *6W 




w*t 



OC 



31 



C Distribution Assumptions 

n. tables chosen for — ««- « tto ~ «- * iCh "— »•• 

.ssv^tions can be mede to describe their distribution. The fiber 

» _i«, r.snect to ply axes is assuned to follow a 
geometric conf iguration with respect to p y 

mml distribution with mean - — (d*«) - — ™ 1 ' •*-« 
delation, to be specified. T** fiber .1- ratio is assumed to be 
„. lly distributed about some mean bet«en ... a* ».7. The -id 
„« ratio, *i=h is ideally -11. L — < » «»— •— 
distribution s„ewd toward ~~. («■ «« »» «- «" °^«~"°" 
..sed, a value of ze~ has a probability of zero. This model L ^sen 
hecause the state of —t present manufacturing technology precipes the 
fabrication of a fiber composite completely free of void.) 

The properties of individual fibers and matrix are varied. The 
„! and shear moduli are assumed to follow the normal distribution, 
and the straths are assumed to be tkitall distributed. 

Figs. 13-27 show the results of random, numoer generation in each 
distribution studied. The density (or histogram, and curative 
distribution plots are sho». Several -ibull and ga™e distribution 
simulations are — . to demonstrate the effects of assu»d parameter 
variations on the distribution sampling. 



32 



HISTOGRAM FOR 
NORMAL GENERATOR 



Standard 
Deviation 




1IOOO 

14000 

1*000 

12000 
w 

£10000 j- 

« 
-I 

x 0000 

s 

u 

4000 
♦000 
*000 



RANCE < t 

(a) histogram 

DISTRIBUTION OF 
NORMAL GENERATOR 

Mean « 0.0 

Standard 
Deviation 



1.0 




jo io in ifl ifl » ° 



RAN6E C E -01) 



o.jr^o- 

(b) cumulative distribution 

Fig 13- Normal Distribution Simulation with mean of 
3 n n and standard deviation of 1.0. 



33 





HISTOGRAM FOR 








GAMMA GENERATOR 






♦500 r 








4000 


X= 3.0 






»500 


k » 1 






3000 








«2500 

bi 

S 

5?000 

at 
it. 








1S00 






- 


1000 


- "-| 






soo 


i i*— i i ' ' i - 


-Jr- 


i I 





fc '5 lOiii A ft W 

•ANCE < C -01> 

(a) histogram 


J5 


♦ ~« 



DISTRIBUTION OF 
GAMMA GENERATOR 




■ a A A A A A 



■ AN8C < t -01> 

(b) cumulative distribution 
Fig. 14- Gamma Distribution Simulation 



34 



HISTOGRAM FOR 
GAMMA GENERATOR 




X - 5.0 
k « 3 



i A riT-A A A 

RAN6E < £ -01> 

(a) histogram 

DISTRIBUTION OF 
GAMMA GENERATOR 



"A — A A 




fr A A A A A A A - A 

CAN6C < t -01 > 

(b) cumulative distribution 
Fig. 15- Gamma Distribution Simulation 



35 



H1ST06RM* FOR 
GAMMA GENERATOR 




■ANSI « « -01> 

(a) histogram 

DISTRIBUTION OF 
6AHMA GENERATOR 




< t -on 



• AN6C * l "* 



uo^einuits uoiVKH-nsKl w««9 "LI * 6 U 
uoiVKUJ*sip aAH«inuna (<1) 
(to- a > 39Nva 

V* fl * V n . r V Q »* tf qt ' * ° 




9-1 

0*E - X 

dQiVa3N39 VWHV9 

jo NQiinaiaisia 

uieuBois.m (e) 
(10- 3 > 3SNV8 

V <VLV V V ^ V °'' ^ 




S * I 

dQ!Va3N33 VHWV3 
SQi HVdSQiSIH 



oooc 



- 000% 



- 0009 



- ooot * 



- ooooi- 



- 00021 



000%t 

ooott 

J OOOtl 



J\ 




001 

002 



oot 
a oo*: 



-I OOSo 



009 
004 

oot 

J 00* 



9C 



CUMULATIVE 




rteouENCv 




u 
-o 



38 



HISTOGRAM FOR 
HE1BULL GENERATOR 




*— ,, l'i l 1 * l'l A A 
RANtc < e oo) (ksi) 

(a) histogram 



nooo r 

14000 
1*000 
12000 

£10000 

« 

f tooo 

9 

u 

4000 



♦000 



*ooo 



DISTRIBUTION OF 
HE1BULL GENERATOR 



6 « 13 ksi 
a - 20 




A \\ A l'l A A 
MNte < e oo> (icsO 

(b) cumulative distribution 
Fi" 19- Weibull Distribution Simulation 
s * Matrix Shear Strength 



39 



HISTOGRAM FOR 
HE1BULL GENERATOR 



ISO 


6 


« 15 ksi. 




140 


a 


' 5 




1« 








120 


- 






S JOO 

Ml 
3 

S to 

m 

ib 








40 








♦0 








20 


J- 








1 Tt * * 


f • • 


j__ 


( 


L >, — 1% — fa — r* — Ti - 


11 


& 


RAxte < c oo> 

(a) histogram 


(Its 



DISTRIBUTION OF 
HEIBULL GENERATOR 




(b) cumulative distribution 

welbull Distribution Simulation 
" Matrix Tensile Strength 



Fin 20- WeiDU.ii 
J " Matrix 



4iJ 



*so r 



%00 




»so 




300 




2(0 


■» 


200 




... 


. 


100 


- 


to 




e »r- 



. HISTOGRAM FOR 
HEIBULL GENERATOR 

6-15 ksi. 
a - 10 




ti 



Y ^ro — rV — r* — rs ij *■ d 
KAHte < c oo> Usi; 

(a) histogram 



• V 



.DISTRIBUTION OF 
HEIBULL GENERATOR 

6 --15 ksi . 
a -10 




"J ■ 10 I* IT I* ••..«» «» 

■AMtC < C 00) (ksi) 

(b) cumulative distribution 
Fig. 21- Weibull Distribution Simulation 
Matrix Tensile Strength 



41 



itooo r 
1*000 

1*000 

12000 

~ 10000 

? tooo h 

9 
U 

4000 
♦000 J- 
2000 



HISTOGRAM FOR 
HEIBULL GENERATOR 




.. .- T* — rr 
rank < t oo> (ksD 
(a) histogram 

DISTRIBUTION OF 
HEIBULL GENERATOR 

B - 15 ksi. 
a ■ 15 



o V 



'i I'a^rt 




tV— rt — 1% A A 

... c c oo> ri^i) 

(b) cumulative distrlbuYTon 
Flo 22- Wei bull Distribution Simulation 
Fig- " £v"«„ T. n< n» Strenqth 



•Ante 



MeiDUII UlStriuw..— - 

Matrix Tensile Strength 



42 









. HISTOGRAM FOR 






HEIBULL GENERATOR 


%S0 


i» 






♦00 






B - 35 ksl . 


sso 






a - 10 


soo 


- 






£2S0 

w 

9 








Szoo 






is \ 


ISO 






r A 


100 


- 






so 


- 


~lj 




°ir* 


**¥ 


j — n — rt — » w *s *fl *» •< 

*ANSC < C 00) (kSl) 








(a) histogram 



itooo r 

1*000 

1*000 

12000 

£10000 

S tooo 

9 

u 

AOOO 

%000 
*000 



. 



.DISTRIBUTION OF 
HEIBULL GENERATOR 

B - 35 ksl. 
o - 10 



o» V i ' o l 2C A A rti A ri* rt i'o 

RANte < e oo> (ksl) 
(b) cumulative distribution 
Fig. 23- Welbull Distribution Simulation 
Matrix Compressive Strength 



^WM^IP •M^in«3 (q) 
( V S*) tOO 3 > Hy« w n, q u 
of y y *fr 




aoiva3N39 nnai3M 
jq NOiinaiaisto' 

uibj6oisim («?} 
(ts^) coo a > >•»*■ 

o * y ",* V ^ 



02 

tsi se 




aoiva3N39 nnai3M 
aoj uvasoisiH 



e* : 



44 



HISTOGRAM FOR 
NEIBULL GENERATOR 




rance < e on (ksi) 
(a) histogram 







DISTRIBUTION OF 






HEIBULL GENERATOR 




ltOOO 








IAOOO 




B - 400 ksi . 




1*000 




a « 10 




12000 


- 






Itl 
> 








— 10000 


^~ 




« 

ml 




/ 




W tooo 

9 

o 




I 




4000 




I 




♦000 




I 




>000 




■hi — ^U -\ — A — Jm — A — A — ztr 


-t>. 



~t» €9 *W •» ■"• '• " — w - 

■ansc < c oi> (ks1) 

(b) cumulative distribution 
FIq 25- Welbull Distribution Simulation 

Fiber Tensile and Compressive Strength 



45 





HISTOGRAM 


FOR 




HEIBULL 


6ENERAT0R 


too 








• 00 






6 « 400 ksl 


700 






a « 15 


400 








"500 

2»oo 

m 


f 







too 



200 - 



100 



*AN6C c I oi> (ksi) 
(a) histogram 



DISTRIBUTION OF 
HEIBULL GENERATOR 




9 *° " *° »*y«£™ (i ei>(Icsi} 

(b) cumulative dlstrtbut on 
c . , 6 uei bull Distribution Simulation 
F1g. 26- *™^ en$11 , and Compressive Strength 



46 



HISTOGRAM FOR 
HEIBULL GENERATOR 




A A A 



rancc c e on (ksi) 
(a) histogram 

DISTRIBUTION OF 
HEIBULL GENERATOR 

B . 400 ksi . 
o - 20 




A A A A A *'» 
■ance < c on (ksi) 
(b) cumulative distribution 
f<« 27- Welbull Distribution Simulation 
F1g. 27- y^eVVensile and Compressive Strength 



47 



D. Use of ICfiH 

TOis section describe, «. — «« theories - — *— 

asperated in «- K- »«— * ^ UC ~ titi0n 00n ~ nti0OS ' 
*r-l«ti— . - definitions are included in SpP^i" B. 

1. Composite Kicronechanics 
«. „ra~h of coeposite ,-chanics «*» relate, ply properties to 
^stituent properties is »— as «^site -erodes. * inputs 
^sist not oniy - constituent -terial P-P-ties (fiber an- -«-). 
« gec-tric elation and fabrication pr-ess <*** includes 

pi r byural, «— ». - — ^ PrcPertie " ^ a5SU " n0nS EOr 
equation development are: (Ref . 16) 

^ .fccbanics - ^-^"^SirSn^ieT" 005 ' 
^T^^tf^i^lotdsl^i Jto tne scbe-tic 

STSxf ^?it. ( cLtlt— t. behaue in , linear elastic »nner 

S^M e^at'the fiber^tri, interface. 
„. direction conventions an- ter-inolooy used in «. *-*«. 

are! .. Properties assured .Ion, fiber direction are call- 
longitudinal. , h-„,verse to fiber direction are called 

2 Properties neasured transverse » 



1. 

2. 

3. 

4. 
5. 
6. 



3. 
4. 



48 



2. Laminate Theory 

Classical laminate theory supplies a convenient procedure to 
predict the response of a laminate to external load. The theory uses 
anisotropic elasticity to obtain the stress-strain relationship for the 
basic lamina. The stress-strain relations of individual laminae are 
transform* to coincide with a global set of reference axes. The 
stress-stain law of the laminate in terms of the properties and 
distribution of individual laminae are calculated using a summation. 
Resultant forces and moments are defined by integrating the stresses 
through the thickness of the laminate. The plate constitutive equation 
is inverted, giving midplane strains and plate curvatures in terms of 
applied forces and moments. These strains and curvatures are 
substituted into the lamina stress-strain equation to obtain lamina 
stresses in the global system. The stresses obtained are then 
transformed into the principal material system of the lamina in question 
and compared with ultimate stresses obtained using failure criteria. 



49 



3. Strength Theories 

Trow *ke use of several assumptions. 
The strength theories in ICPM make use ot 

. that there are five characteristic values of 

First, it is assumed that there arc 

strength of a unidirectional composite: 
1 longitudinal tensile strength 
2. U^lt-i-1 c^pressi^ .tr-^h (3 separate criteria) 

a. rule of mixtures 

b. fiber microbuckling 

c. de lamination 

3. transverse tensile strength 

4. transverse compressive strength 

5 in-plane or intralaminar shear strength 

-«*.« usually associated vith these strengths are shov* 
The fracture nodes usually assw-* 

schematically in Fig. 29. 

0^ ply straths are calculate* (in «. ,1T — *— sy.te«), 
g eo»etric transforations are usee to calculate oolite failure — 
Tte process us«l i. briefly describe below. 

, Calculate 1— (A- — »« — > ~— " 1 ~ ^ """ 
to ply st«noths (in ply system) »- •** -*• 

2 Calculate .tain of *•«— «-* t0r e ~ h PlY - 

3 calculate -ni- - — U-. - •" P»". - " «" , ~ <J 
' as tbe failure street* of the co^site for a particular failure 

node. 



5© 



B iilT- W 



LONCnUOMAl 





\ \ N. MATRIX <1M 

\ \>raotn 
\-rurui 



MIXAUUMNA* 
S»CAR 



stram 



Fia 28- Typical Stress-Strain behavior of 
unidirectional fiber composites. 



I 



MUnfRuAMl 





fltOt COMf«SSI0H BCLMMNATION/ 

swttimc 





FIBER 
MCtWUCUINC 
« lantfeioWl eomr tsslon. 



f 



Id TnnswM tanstan 





MTnntvtrM 



W trmaalmr *!»•'. 



Fia 29- In-plane fracture modes of unidirectional 
s * folvl fiber composites. 



(ply) ttbe r compos 



31 



E . Review of applicable Statistical Concepts 

Cc^osit. property a~ calculate Tor 1^ sables — a 

^ ic set - -i.tr— - — — "- " ^ -— *" 
OTl l sa^lio, «-, does -t appl*. *— - — — " 
sufficiently large. 

1. Sample Ifeans 

Calculation of toe -an sample -I- P~~«s * «»— 



n 



nean = x = n 

vtere n = sanple size 

x .= sample data values 
1 , « mn the sanple nean is assumed to be the 

Tte population mean is unknot, so the sanp 

best estimator of the population mean. 

2. Sample Standard Deviation 

te e.ti«te of «. POPUl-ion — ~*«- - —— 
using the statistically efficient estimtor 

n ,1/2 

f n * («.-«)* 1 • niy > 

o - [ n - 1 i=l V * J 

3. Confidence Interval Estimates 

.. iB the area of statistical inference is the 
ftn important problem m the area A * ,_ 

.station Of Elation palters ,- » — — ■ *~* *~ 
_*. statistics. Peters i -.-«. — *— 



52 



nation of the sanpling distribution of a statistic S. 11* sailing 
distribution of S is assumed as approximately nornal (whioh is true for 
.any statistical distributions if n > »). Confidence interval 
estates are constructed for the statistic S. Thus, intervals are 
identified for wMch it can be asserted with a reasonable degree of 
certainty that they contain the paraneter considered. Obviously, the 
degree of certainty (or confidence level) will vary with the size of the 
interval chosen. Values of confidence coefficients, = c , are associated 
with confidence levels. For example, an actual sample, statistic S is 
expected to be found lying in the interval (x - V ) to (x ♦ zj) (where 
a is the unknown population standard deviation) sone percent of the 
tine. L* the z c value in this example be 1. Assuming a normal 
sanpling distribution, (with * c = 1) the normal distribution area 
function specifies that S falls between (x - a) and (x ♦ a) about 
68.277. of the tine. Similarly, the confidence of x lying in the 
interval (S - «) to (S ♦ a) is about 68. 2T/.. The endpoints of the 
intervals are known as confidence limits. Various confidence 
coefficients « , corresponding to frequently used confidence levels, 

have been tabulated. 

In this work, the confidence interval for neans is given in terms 

of the sample statistics by 



x + z 
- c 



itere ^ is the confidence coefficient, which takes on values of 
1.645, i!*», and 2.58* for the 90, 95, and 997. confidence levels, 



S3 



respectively. 

4. Regression 

The ter. -r^ssion- as used in the are. of statistic refer, to 
the process of formating a ..the^tical — .1 to explain ranoo-ly 
ohserved phe^ena. So.* rational f— for the »y each triable 
enters the -de! _t be asswed. Co^arison or the decree of fit of 

differ-nt assu«d — •»• »*-»»* leadS t0 * tetMr -* 1 ' "" to5i ° 
regression strategy used here consists of; 

X. «,su»e a -ItipU linear regression -*1. 11- nor«i equations 

for such a nodel are: 

{Y} = [X)ifi + (*) 

inhere 

(Y) = sector of dependent variable values 

[K] = mtrix of functions of independent variable 

{0} = regression "true" values 

{«?} = errors 

The nom*l equations can be solved as follow: 
[X ] T {Y} = [X] T [XK/3> + [X] T ( e > 

{b} = [xVWoo 

Were 

/b} = paraneter estiBBtes 

2 . „. , st«^rd statistical p«*ao, (™- 17) to esti-t. r^r^.ion 
paraneters . 



54 



3. Calculate properties of regression parameter distributions to 
assess nodel precision. 

In the event that [X T X] is singular, implying that some of the 
normal equations are linearly dependent, [X T X] _1 does not exist. The 
nodel should be expressed in terms of fe«sr parameters, or should 
include assumed restrictions on the parameters. 

The square of the multiple correlation coefficient, R , is usually 
calculated for each regression nodel, and supplies a convenient measure 
of the degree of fit between data values {Y} and values {Y> 
predicted by the regression equation. It is defined by 

Sum of Squares due to regression nodel 

2 . , - 

R = Total Sum of squares about mean Y 

T (Y. - Y)* 



Z (Y i - Y) : 



Frequently, it is necessary to determine if inclusion of particular 
terms in a regression nodel is worthwhile. To this end, the extra 
portion of the regression sum of squares which arises due to the terms 
under consideration is calculated. The mean square (defined as the sum 
of squares divided by the corresponding degrees of freedom) derived from 
this extra sum of squares can be compared with s*, the estimate of a*, 
to see if it appears significantly large. If it does, the terms under 



35 



consideration should be inclined. The statistic is frequently compared 
to the appropriate percentage point of the F- distribution, %faich is 

tabulated. 

Supopose the extra sum of squares due to a parameter, given that a 
number of other paraneters are already in the nodel, is calculated. 

Symbolically, 

SS(b.|b ,b 1 ,...,b._ 1 ,b. +1 ,...,b k ) i = l,2,...,k 

represents a one degree of freedom ( 1 df ) sum of squares which 

neasures the portion of the regression sum of squares due to the 

coefficient b.. This is a measure of the value of adding a ^ term to 

the nodel which previously did not include 0.. The corresponding nean 

square, equal to the SS (since it has one df ) can be compared by an 

F- test to s 2 . This is know* as a partial F- test for the single 

parameter 0., which is a special case of the F- test described earlier. 

The stepwise regression procedure (Ref . 18) is a structured way to 

insert variables in order of correlation until the regression equation 

is satisfactory. The partial correlation coefficient measures the 

relative importance of terms not yet in the wodel, to choose the next 

candidate for entry. The analagous statistic, F- to enter (or F- to 

renove) is usually evaluated for each predictor at every stage as though 

it were the last term to enter the model, to determine if terms retained 

at a previous step have become superfluous, because of soma linear 

dependence with terms now in the nodel. The largest F- statistic 

calculated at each step is compared with the appropriate percentage 

point of the F- distribution, and the predictor variable is entered (or 



56 



renewed) based on the significance of this F- test. Testing of the 
least useful predictor is performed at every step. The R* statistic is 
calculated, to provide a measure of the value of the regression at each 
step. This stepwise linear regression schene is used in this work 
because of its conputational economy, and because it allows the analyst 
to assess the relative influence (or correlation) betwaen individual 
predictor variables of a selected model and response for a particular 
data sample. Other sche.es are available (Ref. 18), such as backward 
elimination. The stepwise procedure is recoimended for its direct 
nature in testing the nodel with only significant predictor terms. 



CHAPTER III 
RESULTS 



A. Property Histograms and Distributions 

In this work, fiber and matrix properties are .IImnI to assune a 
range of val«s to assess the sensitivity of the composite ply 
properties to constant perturbations. Graphite fiber and epoxy 
matrix are used as the constants. Initially, t. separate samples of 
output data are generated and studied to demonstrate the effects of 
input parameter changes on composite material properties. These two 
cases are compared with a deterministic base case with no random input 
property generation. The data for all three cases is given in Table I. 
The results of cases 2 and 3 are shown in histogram and cumulative 
distribution form in Figs. 30 - 42. The results of the deterministic 
case 1 are summarized in Table II, and can be easily compared with the 
histograms and distributions. 



57 



TABLE I- INPUT DATA FDR SflfTLING 



58 



IHPOT 

THETA (degrees) 

A* 
a 
FVR 

P 

a 
WR 

X 

k 
EFPi(ksi) 

a 
EFP2(ksi) 

a 
CFP12(ksi) 

a 
GFP23(ksi) 

H 
a 
SFPT(ksi) 

fi 

a 
SFFC(ksi) 

fi 

a 
ETF(ksi) 

A* 
a 
SlfT(ksi) 

fi 

a 
SrPC(ksi) 

fi 
a 
SfFSfksi) 

fi 
a 



CASE 1 



0.0 



0.50 



0.01 



31000 



2000 



2000 



1000 



400 



CASE 2 



CA8E3 



400 



500 



15 



35 



13 







0.0 


0.0 


5.0 


10.0 


0.5 


0.5 


0.1 


0.2 


3.0 


3.0 


3 


5 


31000 


31000 


1500 


3000 


2000 


• 2000 


100 


200 


2000 


2000 


100 


200 


1000 


1000 


50 


100 


400 


400 


20 


10 


400 


400 


20 


10 


500 


500 


25 


50 


15 


15 


20 


10 


35 


35 


20 


10 


13 


13 


20 


10 



59 



toctf II- CASE 1 RESULTS 



PROPERTY 



CTE12 

saocr 
scxxc 



SCYYC 
SCXYS 



VALUE 



ECU 1575 ° ksi 

BC22 1065 ksi 

EC12 516 ksi 

HUC12 0.275 

NUC21 0.O18 

CTEli ®' 775 x 10 " 8 

-4 

CTE22 0.181 x 10 



O.OOO 
203 ksi 
165 ksi 



SCYYT ll - 74 ksi 



27.41 ksi 
10.01 ksi 



69 



HISTOGRAM FOR ECU 

LONGITUDINAL MODULUS 




HISTOGRAM FOR ECU 

L0N6ITUO1NAL MODULUS 



10 r 



o 

* 5 

a 

S *h 

or 

w 3 - 

2 - 

1 - 



tt» rrs . rs 770" 

RANGE < £ OS) 

(a) case 2 histogram 



I 



H 



ll 



III 



"h — ttt^ rh Hi 



".fl iT3 TT5 TI7 .T^ 
BAN6E < E OS) 

(b) case 3 histogram 



DISTRIBUTION OF ECU 
L0N6ITU0INAL MODULUS 



DISTRIBUTION OF EC1 2 
LONGITUDINAL MOOULUS 




■AN6E 

(c) case 2 distribution 



• AN6E 

(d) case 3 distribution 



Fig. 30- Sampling results for Longitudinal Elastic Modulus 



61 



HISTOGRAM FOR EC22 

TRANSVERSE MODULUS 



»o r 

■ 

7 

u 

Z 51- 

9 
ItJ T 

a 
*■ 3 

2 



a 






HISTOGRAM FOR EC22 

TRANSVERSE MODULUS 




RANGE < E 07) 

(a) case 2 histogram 



RANGE < E ° 7) 

(b) case 3 histogram 



DISTRIBUTION OF EC22 
TRANSVERSE KOOULUS 



SO 

*s 

♦0 
35 

«30 

ho 

9 

°»s 

10 



DISTRIBUTION OF EC22 

TRANSVERSE HOOULUS 



9oV? — n 




TtT— H- 



RANGE < E 07> 

(c) case 2 distribution 



RANGE < E 07> 

(d) case 3 distribution 



F1,. 31- Sanpims r-.lt. for Transvers* EU.t1c r*du,us 



62 



HISTOGRAM FOR EC12 
SHEAR HOOULUS 



< E 0*> 

(a) case 2 histogram 



HISTOGRAM FOR EC12 
SHEAR MODULUS 




10 r 



u 



' • , IT III i i 

O.oT .A* .08 ."2 Ttt 72 



BAN6E < E 07) 

(b) case 3 histogram 



USTR1BUT10N OF EC12 
SHEAR MODULUS 




DISTRIBUTION OF EC12 

SHEAR MOOULUS 



SO r 
45 
40 
35 

(to 

9 

»!S 

10 



A °.to — .d» .At — : 



RAN6E < E 04) 

(c) case 2 distribution 



7TT— Tic 



rfr- 

RANOC < E 07) 

(d) case 3 distribution 



F1g. 32- Sampling results for In- plane Shear Modulus 



63 



HISTOGRAM FOR NUC12 

POISSON RATIO- MAJOR 




.% •& •* 

RAN6E < E 00) 

(a) case 2 histogram 



HISTOGRAM FOR NUC12 

POISSON RATIO- MAJOR 



»o r 

■ 

7 



* 




RAN6E C E 00) 

(b) case 3 histogram 



HSTR1BUT10N OF NUC12 
POISSON RATIO- "AJOR 



DISTRIBUTION OF NUC12 
POISSON RATIO- «*JOR 




RAN6E < E ° 0> 

(c) case 2 distribution 



■o4* — vb — vb — rii 

RANGE C E 00) 
(d) case 3 distribution 



Fig. 33- Sampling results for Polsson Ratio (major) 



64 



HISTOGRAM FOR NUC21 

POISSON RATIO- MINOR 



HISTOGRAM FOR NUC21 

POISSON RATIO- MINOR 



10 r 



t 



nl 



If 



n 



If 



'?& '.sW .3V« 



,1*75 '.? ' » — Tt 

RAN6E < E -01) 

(a) case 2 histogram 




TTT2 6T* 0.* o.b 

RANGE ( E -01) 

(b) case 3 histogram 



Ho 



DISTRIBUTION OF NUC21 
POISSON RATIO- MINOR 




DISTRIBUTION OF NUC21 
POISSON RATIO- MINOR 



■hs — rsVs — rafc — rift 




RAN6E < c -on 
(c) case 2 distribution 



RAN6E < E -01> 

(d) case 3 distribution 



Fig. 34- Sampling results for Poisson Ratio (minor) 



65 



HISTOGRAM FOR CTE11 

IONS. THERMAL EXPANSION 



10 

• 
7 

». * 
u 

2 5 

at 
«" 3 

2 

1 





rr 



-?h 



P 



IU 



ft 



H 



rrrr 



• 



in. n 



HISTOGRAM FOR CTE11 

LONG. THERMAL EXPANSION 



io r 



TT55 — BTffr 

( E -06) 



TJT15 



RAN6E 

(a) case 2 histogram 



TO 




• ff.fc -B ' 

RANGE < E -06) 

(b) case 3 histogram 



TTT 



BTT" 



T.o 



1STR1BUTION OF CTE1 1 
LONG. THERMAL EXPANSION 



DISTRIBUTION OF CTE1 1 
LONG. THERMAL EXPANSION 




RANGE < E -04) 

(c) case 2 distribution 



-0.4 -D'.2 

RANGE ( E 



•04) 



(d) case 3 distribution 



F1g. 35- Sampling results for Longitudinal Thermal Expansion 



66 



HISTOGRAM FOR CTE22 

TRAN. THERMAL EXPANSION 



10 



N 



u . ; 



IT 

.17 



It 



■rh- 



TT1 

RANGE < E -04) 

(a) case 2 histogram 



HISTOGRAM FOR CTE22 

TRAM. THERMAL EXPANSION 



»o r 



J 



r 



ill 



1 



J 



4 



3 d .it — rr* rrs m ^iu rs; 

RANGE < E -0*> 

(b) case 3 histogram 



HSTR1BUTI0N OF CTE22 
TRAN. THERMAL EXPANSION 



DISTRIBUTION OF CTE22 
TRAN. THERMAL EXPANSION 




7*B zii 



RAN6E < C -Q*> 
(d) case 3 distribution 



F1g. 



RAN6E < E -0*> 

(c) case 2 distribution 
36- Sampling results for Transverse Thermal Expansion 



67 



HISTOGRAM FOR CTE12 
CROSS THERMAL EXPANSION 



6.0 
*.S 

*.o 

3.5 

**.0 

o 

*2.5 

9 

Sa.o 

a 
^1.6 

1.0 

0.6 

0.0. 



I 



HISTOGRAM FOR CTE12 

CROSS THERMAL EXPANSION 



JL 



10 r 



TITT 



72 DTD DV7 
RAN6E < E -05) 

(a) case 2 histogram 



— irlfc Got* 




j. 



JL 



RAN8E 
(b) case 3 histogram 



BT? BT8" 

i E -05) 



-V.i 



DISTRIBUTION OF CTE12 
CROSS THERMAL EXPANSION 



DISTRIBUTION OF CTE12 
CROSS THERMAL EXPANSION 




RAN6E < t "OS) »*"« « E - 05) . 

o 1. * -k ♦<„« (d) case 3 distribution 

(c) case 2 distribution v°; 

Fig. 37- Sanplinn results for Thermal Expansion Coupling 



68 



HISTOGRAM FOR SCXXT 

L0N6. TENSILE STREN6TN 



10 




■ 




7 


- 


- * 

u 

* 5 

3 

at 
■" 3 


- 


2 




t 


■ D, 1 

1 .1 J 



i 



im 



\ 



i- 



RANGE ( E 03) 

(a) case 2 histogram 



HISTOGRAM FOR SCXXT 
LDN6. TENSILE STRENGTH 



T2J 



s.o 




\ 


™ 




».S 








4.0 




m 


m 


3.5 








,.3.0 

u 

J2.5 

22.0 
ac 
"•I. 5 




i n 


J J 


1.0 




JU ■ 


u 




0.5 


- 


i i 




1— 


0.0 #| , 


.68 .1 


2 .6 .*u 




RANGE < E 03) 






(b) case 3 histogram 



DISTRIBUTION OF SCXXT 
L0N6. TENSILE STRENGTH 




DISTRIBUTION OF SCXXT 
L0N6. TENSILE STRENGTH 



ft TTT5 TT7 

RANGE < E 03) 



72. 




* 7T2 .1* 7JB T2* 

RANGE < t 03) 

(d) case 3 distribution 



(c) case 2 distribution 
Fig. 38- Sampling results for Longitudinal Tensile Strength 



69 



HISTOGRAM FOR SCXXC 

LONC. COMPRESS STRENGTH 




772 
RAN6E < E 03> 

(a) case 2 histogram 



HISTOGRAM FOR SCXXC 
LON6. COHPRESS STRENGTH 



S.O r 

♦ .5 |- 

4.0 

3.5 

„3.0 

u 

*2.S 

3 

S2.0 

at 

W 1.S 

1.0 

o.s 



tU — :h — ri*°-°.6T 



I 






—rU — tU — ris 



RANGE ( E 03> 

(b) case 3 histogram 



DISTRIBUTION OF SCXXC 
L0N6. COHPRESS STRENGTH 



DISTRIBUTION OF SCXXC 
LONG. COHPRESS STRENGTH 




F1g 



1 .12 •.»* 

RANGE < E 03) 

(c) case 2 distribution 

39- Sampling results for Longitudinal Compressive Strength 



RANGE < t 03) 

(d) case 3 distribution 



70 



HISTOGRAM FOR SCYYT 
TRAN. TENSILE STRENGTH 



HISTOGRAM FOR SCYYT 
TRAN. TENSILE STRENCTH 



to r 



at 



Ti ot 



B2 



nnl 



h 



-L 



vrur 

RAN6E 



UTTff 
< E 02> 



bt¥? — it'ib 




(a) case 2 histogram 



.'02 " 6.02 — DTK — OTTT 

RANGE C E 02> 

(b) case 3 histogram 



.1* 



DISTRIBUTION OF SCYYT 
TRAN. TENSILE STRENGTH 



DISTRIBUTION OF SCYYT 
TRAN. TENSILE STRENGTH 



iO 














f0 




-J 


♦5 


- 




/ 






»5 




I 


40 














40 


- 


J 


35 














35 




/ 


5! 30 












$30 


- 


f 


525 












525 

.j 




1 


§20 

9 

°IS 


™ 










§20 

9 

°I5 




J 


10 


- 










10 




| 


s 




J 1 


i 


• i 


ii ' 


5 






B.« 


-for 


■"8.TJ4 


0.10 0.14 


TTTli *. 


(U 


-.02 0.02 0.04 o.io I 




RANGE 


(.E 02) 






RANGE ( E 02) 






(c) 


case 


2 


distribution 








(d) case 3 distribution 



n* 



Fig. 40- Sampling results for Transverse Tensile Strength 



?i 



HISTOGRAM FOR SCYYC 

IRAN. COMPRESS STRENGTH 



20 
11 

1* 
1* 

o 

JlO 

S 

W ■ 

Or 
*■ 4 

* 

2 

0, 



« 




HISTOGRAM FOR SCYYC 
TRAN. COMPRESS STREMSTH 



io r 

7 

. * 



u 
* 5 



srre — DT2T 

RANGE < E 02> 

(a) case 2 histogram 



TJ^T5 — BTVs 



3 
2 

I 



1 



T7 



1 



1 



I 



■oir 



■BTT 



"6T1 

RANSE < E 02) 

(b) case 3 histogram 



ttU 



1STR1BUT10N OF SCYYC 
TRAM. COMPRESS STREMSTH 




DISTRIBUTION OF SCYYC 
TRAN. COMPRESS STRENGTH 



TJ^?T 



TJ7T5 0.25 0" 

RANGE < E 02> 




RANGE < E 02> 
(d) case 3 distribution 



Fig. 



(c) case 2 distribution 
41- Sampling results for Transverse Compressive Strength 



72 



HISTOGRAM FOR SCXYS 

IN-PLANE SHEAR STRENGTH 



RANGE 

(a) case 2 histogram 



HISTOGRAM FOR SCXYS 
IN-PLANE SHEAR STRENGTH 




-.00 



_L 



TJT55 oTTO" 

RANGE < E 02) 

(b) case 3 histogram 



bTT5 — oT20 



DISTRIBUTION OF SCXYS 
IN-PLANE SHEAR STRENGTH 



SO 
♦5 
♦0 
95 

Sfso 

ho 

9 

W IS 

10 

S 




6T 




DISTRIBUTION OF SCXYS 
IN-PLANE SHEAR STRENGTH 



tw — :*! — rfe — rfr ^o 

RANGE < C 02) 




85 :W B.B5 O.'lfl B.'lS U.'z o 
RAN6E < e 02) 



(c) case 2 distribution (d) case 3 distribution 

F1g. 42- Sampling results for In-plane Shear Strength 



73 



B. Fiber Strength Effect 

To show the effect of fiber strength changes on the longitudinal 
strengths of the conposite, several shape paraneters of the weibull 
Jistribution for fiber strength are assumed. The nonte carlo procedure 
is then conducted at several fiber volume ratio values. All properties 
are varied, except fiber volume ratio. The distribution paraneters of 
all properties except fiber strengths are held constant. The curves 
generated are showi in Figs. 43 and 44. In the figures the solid lines 
and symbols show the means of the 95V. confidence interval estimates for 
the sanple size of 5© chosen at each point. The points on both sides of 
each curve locate the upper and lowar bounds of the confidence 
intervals. The convention described is intended to provide a convenient 
indication of the dispersion of the sanple values at each point. 



74 



LONG. TENSILE STRENGTH 



65 r 



CM 




A a « 20 


?60 




D a « 15 


UJ 

^55 




V a = 10 


£50 






a. 






u. 






5*5 






h- 






>c 






£f0 


- 




<n 






^ 






=.35 






UJ 






»»J _ — 






-30 




*^F^r 


-j 






c 






525 






e . 






Z 20 ^ 

• 




, 


3 


.* 



• 6 



.7 



FIBER VOLUME RATIO 



Fig 43- Longitudinal Tensile Strength; for various 
shape parameters cf fiber strength. 



75 



LONG. COMPRESS. STRENGTH 



38 r 




FIBER VOLUHE RATIO 



.7 



Fig 44- Longitudinal Compressive Strength; for various 
9 shape parameters of fiber strength. 



76 



C. Matrix Strength Effect 

The effects of changes in matrix strength on oonposite strengths 
are studied by suitable variation of the shape paraneters governing the 
imtrix strength distributions. Analagous to the plots given for fiber 
strength effects, the matrix effects are showi in Figs. 45 - 47. 



77 



TRANS. TENSILE STRENGTH 



1 10 r 



.3 



£ a ' 20 
□ a - 15 
y a = 10 




FIBER VOLUME RATIO 



.7 



Fig. «- Transverse Tensile Strength; for various 
shape parameters of matrix strengths. 



78 



TRANS. COMPR. STRENGTH 



A 


a 


M 


20 


D 


a 


m 


15 


V 


a 


s 


10 




3 .+ .5 

FIBER VOLUME RATIO 



Fig. 46- Transverse Compressive Strength; for various 
shape parameters of matrix strengths. 



79 



IN-PLANE SHEAR STRENGTH 




.3 .* .5 

FIBER VOLUME RATIO 



.6 



.7 



Fi' 



47- In-plane Shear Strength; for various 
shape parameters of matrix strengths. 



90 



D. Fiber Orientation Effect 

Assuned valtes of the fiber orientation angle distribution 
paraneter are consecutively used in the «onte carlo procedure to assess 
the effects on several conposite properties. These plots are sbowi in 
Figs. 48 - 57. 

E. Fiber Stiffness Effect 

Assuned values of the fiber modulus distribution paraneter are used 
in the simulation to similarly assess the effects on the related 
conposite properties. The plots thus generated are show* in Figs. 
58-67. 



81 



LONG. ELASTIC MODULUS 



95.0 r 




.3 



. <r .5 .6 

FIBER VOLUHE RATIO 



.7 



Fig. 48- Longitudinal Elastic Modulus; for various 
3 shale oarameters of fiber orientation. 



82 



TRANS. ELASTIC MODULUS 



A o - io° 

D o « 5° 

v o « r 




.3 .<*• .5 

FIBER VOLUME RATIO 



.6 



.7 



Fig. 49- Transverse Elastic Modulus-, for various 
shape parameters of fiber orientation. 



83 



IN PLANE SHEAR MODULUS 



100 r 



eg 

o 90 h 



10 



A ° ■ 


10° 


D o ■ 


5° 


V o « 


1° 




.3 .* -5 

FIBER VOLUME RATIO 



• 6 



Fiq 50- In-plane Shear Modulus; for various 
9 ' shane oarameters of fiber orientation. 



.7 



shape parameu 



84 



LONG. TENSILE STRENGTH 



65 r 



? 60 

UJ 

£50 

o. 
u. 



05 - 






M-0 - 
35 - 



= 30 



25 
20 




.3 .* -5 

FIBER VOLUME RATIO 



• 6 



Fiq 51- Longitudinal Tensile Strength; for va 
shaoe Daraineters of fiber orientation 



.7 



nous 



85 



LONG. COMP. STRENGTH 



38 r 




.3 • * .5 .6 

FIBER VOLUHE RATIO 



.7 



Fia 52- Longitudinal Compressive Strength; for various 
M3 - «h,L narameters of fiber orientation. 



shape parameters 



86 



TRANS. TENSILE STRENGTH 



1 10 








c4 




A o « io° 




?100 




Do» 5° 




liJ 

Z 90 




V c = 1° 




£ 80 








Q. 








K 4 




• 




K 70 


■^^^^"^^^ 




^> 








>■ 




* ^^*^^^^&^^^^- • 




£ 60 




^^^^^^^^ 




cn 








^ 




^^^^^^^. * 




o B0 








ID 








= "tO 








_J 








« 








a 30 


- 






o 








* 




1 1 


i 


20 

• 


3 


.<f .5 .6 
FIBER VOLUME RfcTIO 


• 



F-Ig. 53- Transverse Tensile Strength; for various 
shape parameters of fiber orientation. 



87 



TRANS. COMP. STRENGTH 



1 10 r 



<u 

<?100 


UJ 


90 


*^ 






80 


a. . 




X 




to 

\ 


70 


o 




>■ 




>- 
o 


60 


«/> 




%^ 




o 


50 


IU 




rw 


*r0 


_J 




« 




X 

ec 


30 



20 



£» o = 10° 
V o - 1° 




.3 .* ' -5 

FIBER VOLUME RATIO 



• 6 



.7 



Fig 54- Transverse Compressive Strength; for various 
Mg - ch,np narameters of fiber orientation. 



shape parameters 



88 



IN PLANE SHEAR STRENGTH 




.3 .* .5 

FIBER VOLUME RATIO 



Fig. 55- In-plane Shear Strength; for various 
shape parameters of fiber orientation. 



89 



PDISSON'S RATIO (MAJOR) 




Fig. 



.3 .* 

FIBER VOLUME RATIO 

56- Polsson's Ratio (major); for various 
shape parameters of fiber orientation. 



9© 



POISSON'S RATIO (MINOR) 




.3 .* -5 

FIBER VOLUME RATIO 



.6 



.7 



F1g. 57- PoUson's Ratio (minor); for various 
shape parameters of fiber orientation. 



91 



LONGo ELASTIC MODULUS 



70 r ^ o = los 




FIBER VOLUME RATIO 



.7 



Fig 58- Longitudinal Elastic Modulus; for various 
9 shape parameters of fiber modulus. 



92 



TRANS. ELASTIC MODULUS 




FIBER VOLUME RATIO 



Fig. 



59- Transverse Elastic Modulus; for various 
shape parameters of fiber modulus. 



93 



IN PLANE SHEAR MODULUS 



70 


A 


a « 10% 






^ 










<\J 








1 


= 65 


- D 


o « 5* 






bJ 

.60 


- V 


o - 1« 




Sr'' 


£55 










~* 










O. 






r^ % 




£50 










\ 




• ^r^T 






CM 










oM-5 




s\S 






UJ 




^r ^^ • 






^ 










o*° 


- 








UJ 










= 35 


- 


^^^1 






« 










£30 If 








O 1 








z 

25 

• 




1 f 


• 


_J 


3 


- A .=> 


.6 


. 7 




FIBER VOLUME RATIO 







F1a 60- In Plane Shear Modulus; for various 
shape parameters of fiber modulus. 



94 



POISSON'S RATIO (MAJOR) 



68 




A 


o » 10« 






66 




D 


a « 5X 






6f 




V 


o « 1% 






^ 












S62 
i 










• 


"60 










; 


>■* 






• ^^ 


* ^^^^^ 




58 








* 




(M 












|56 


- 








. 


* i 






^ . 


• 


• 


5<r , 








• 




52 


- 




• 






^ ^% 






i 


• 


1 


50 

• 


3 




.* 


.5 


• 6 






FIBER VOLUME 


RATIO 





Fig. 61- Polsson's Ratio (major); for various 
shape parameters of fiber modulus. 



95 



POISSON'S RATIO (MINOR) 




. H- .5 

FIBER VOLUME RATIO 



Fig. 62- Poisson's Ratio (minor) for various 
shape parameters of fiber modulus. 



.7 



96 . 



LONG. TENSILE STRENGTH 



65 r 




.7 



FIBER VOLUME RATIO 
f<o 63- Longitudinal Tensile Strength; for various 
Fl9 * sXa?e parameters of fiber modulus. 



97 



LONG. COMP. STRENGTH 

38 r 



?36 

UJ 



£32 



in 

^30 h 

S28 



26 






22 



20 



A 


a 


s 


10% 


D 





s 


5* 


V 


a 


X 


IS 




.3 .* - 5 

FIBER VOLUME RATIO 



• 6 



.7 



Fig 64- Longitudinal Compressive Strength; for various 
shape parameters of fiber modulus. 



98 



TRANS. TENSILE STRENGTH 



1 dU 




A a « 10% 




+*> 








?1 10 




D a * 5X 




UJ 

J00 




V o ■ 1$ 




C 90 








a. 








s 








<: so 








H- 








>- 




^^^^^^ . 




£. 70 








tn 








^ 








» 60 








UJ 








- 50 








_i 








c 








« M-0 


- 






o 








ac 




i • 


• 


30 

• 


3 


FIBER VOLUME RATIO 


.6 



.7 



Fin. 65- Transverse Tensile Strength; for various 
shape parameters of fiber modulus. 



9» 



TRANS. COMPo STRENGTH 



120 




A 


o « 10X 




^ 










CM 

°110 




D 


o - 5% 




"100 




V 


o « IX 




G 90 


■ 








a. 










X 1 










*C 80 


_^^^ 








o 




^^^^^&% 






>- 










o 70 










<n 










^ 










c 6° 








^^^^^ 


UJ 










- 50 










—i 










« 










£ M-0 


- 








o 










* 




i 


i 


« 


30 

• 


3 


.*(- 


.5 


.6 



FIBER VOLUME RATIO 



.7 



Fig. 66- Transverse Compressive Strength; for various 
shape parameters of fiber modulus. 



100 



IN PLANE SHEAR STRENGTH 




FIBER VOLUME RATIO 



F1g. 67- In Plane Shear Strength; for various 
shape parameters of fiber modulus. 



.7 



101 



G. Regression Ifadels 

The output data of cases 2 thrown 11 are used as successive i«P«" 
to the regression schene. The goal of stepwise regression, as used 
here, is to neasure the degree of correlation between a dependent and a 
set or independent variables for a given -t of data. The outputs of 
the reoressions conduced show the independent variables accepted into 
the audel <bas«i on F-«es, criteria) in order of degree of correlation 
with the dependent liable of interest, along with the final B 
statistic. (The B 2 values represent t» scuare of the. nultiple 
correlation coefficient, a convenient measure of the fit hetween data 
values and values predicted by the regression equation.) 

The ordering of predictor variables by stepwise regression has 
several important uses. In this study, the sohe- facilitates easy 
investigation of the effects of -aterial changes on composite 
preperties. Si«* the nonte carlo sche- permits generation of large 
„ou»t, of data, the regression is easy, inexpensive, and can provide 
insigh t coaming the »nsitivity of dependent -riable, for ,s.»ed 
distribution, of predictor triable.. « variety of mteri.1 
.^figurations and constituent distributions are enamined, and a -del 
constructed for each dependent (or response, liable. It t be noted 
that the relative correlations of predictor tables with response 
cables -ill he functions of the assu«. distributions, the p~ticular 

■- _ .™i the functional manner in which the predictor 
data sanple considered, and the function. 

-iables are incorporated into the mel. 

. .Upl. regression -del was assu-d for each response triable. 



van 



102 



The first set of "simple" regression models uses as predictor functions 
only the independent variables as individual terms. To be more precise, 
the predictor variables used are not simply the independent variable 
values, for there are 15 of these for each layup. The arithmetic mean 
of independent variable values is thus used as the predictor variable in 
the first set of regression models. The only exception to this is the 
use of the sin* of the average of the fiber orientation angles as the 
angular dependence predictor, denoted by THETft in the tables to follow. 
The simpler response variables can be adequately described using the 
linear function forms in the regression models. The simple variables 
include the elastic constants, (ECU, BC22, EC12, NUC12, NUC21) and 
coefficients of thermal expansion (CTE11, CTE22). The results of the 
regressions performed in the "simple" manner are given in Tables III - 
XIV. In the tables the input labeled with HI through N5 and Wi through 
U5 represent narrow and wide distributions of all properties. Input 
labeled N6 through N10 and U6 through W10 describe the same 
distributions, except that the composite is assumed unidirectional, i.e. 
no angular variation. The distinction shows the reduction in predictive 
capability induced by deviations of the fibers from aligned orientation. 
The models assumed for the response (output) variables are of the 

form 

Y = B + B!X, + BjX, + B3X3 + . • • + B n X n 

vfcere 

Y = response variable (ECU, EE22, BC12, etc.) 

B = regression parameters to be obtained 
n 



103 



X = average of independent variable values through the 
thickness of the ply (TlErft, FVR, WR, etc.) 
Each nodel postulated contains all independent variables that 
appear in the equations for the related ply property (see Appendix B). 



104 





TUB 


SirPLE HDDH- 


uptrr 


FVR 


TEBIB ACCEPTED 


JuU * 

Ml 


9.3 


FVR.EFP1 


N2 


0.4 


FVR,EFP1,1HEIA 


K3 


0.5 


FVR.EFPl.THETA 


M4 


0.6 


FVR.EFPl.THETA 


N5 


0.7 


FVR.EFPl 


Ul 


0.3 


FVR.THETA.EFPl 


U2 


0.4 


FVR.EFPl.THETA 


U3 


0.5 


FVR.THETA.EFPl 


W4 


0.6 


FVR.THETA.EFPl 


U5 


0.7 


FVR.EFPl.THETA 


N6 


0.3 


FVR.EFPl.HF 


H7 


0.4 


FVR.EFPi 


N8 


0.5 


FVR.EFPI 


N9 


0.6 


FVR.EFPl.EM 1 


NiO 


0.7 


FVR.EFPI 


U6 


0.3 


FVR.EFP1.WR 


W7 


0.4 


FVR.EFPI 


UB 


0.5 


FVR.EFPi 


U9 


0.6 


FVR.EFPI 


wio 


0.7 


FVR.EFPI 



R* 



83.17 
92.63 
94.02 
94.59 
84.00 
64.49 
89.88 
72.85 
65.37 
57.83 

99.83 

99.81 

99.69 

99.74 

99.77 

99.13 

98.40 

98.90 

99.59 

99.34 



105 



tqct g TU- TRAtB VEBSE 1C0ULPS (EC22) 

SIMPLE IEDEL 
INPUT FVR TESTE ACCEPTED 



FVR,EFT>2 

FVR 

FVR,EFP2 

FVR.EFP2 

FVR,EFP2,THETA 

FVR,THETA,EFP2 

FVR f THETA,EFP2 

FVR,THETA,EFP2 

FVR,THETA,EFP2 

FVR > THETA J EFT2 

FVR,EFF2 
FVR f EFP2 
FWR.EFP2 
FVR.EFP2 
FVR.EFP2 
FVR.EFP2 
FVR,EFP2 
FVR.EFP2 
FVR.EFF2 
FVR.EFP2 



Nl 


0.3 


N2 


0.4 


K3 


0.5 


N4 


0.6 


N5 


0.7 


Ul 


0.3 


U2 


0.4 


U3 


0.5 


W4 


0.6 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


N10 


0.7 


U6 


0.3 


W7 


0.4 


UB 


0.5 


W9 


0.6 


W10 


0.7 



R 2 



83.50 
85.23 
91.83 
93.26 
93.06 
78.36 
90.73 
80.15 
86.05 
87.14 

87.13 
86.15 
90.97 
93.47 
92.05 
79.72 
70.71 
81.92 
88.62 
84.05 



106 



TABLE V- SHEAR MODULUS (EC12) 

SIMPLE MODEL 
INPUT FVR TEBM3 ACCfcyiM) 



HI 


0.3 


N2 


0.4 


M3 


0.5 


N4 


0.6 


N5 


0.7 


Ul 


0.3 


U2 


0.4 


U3 


0.5 


W4 


0.6 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


H9 


0.6 


H10 


0.7 


W6 


0.3 


W7 


0.4 


UB 


0.5 


W9 


0.6 


W10 


0.7 



THETA.FVR.GMP 97.01 

THETfl I FVR l GrP,Gn'23 98.85 

THETA,FVR f GMP,GFP12 97.50 

THETA.FVR.GMP 98.01 

THETA.FVR.GMP, GFF12 98.42 

THETA.FVR.GMP 91.79 

THETA.FVR * 94.27 

THETA,FVR,GFP23 93-71 

THETA.FVR 95.62 

THETA.FVR, GMP.GFP23 96.67 

FVR.GMP 97.66 

FVR,GMP,GFP12 98.02 

FVR,GMP,GFP23 96.65 

FVR,GMP,GFP12 97.11 

FVR,GMP,GFP12 98.55 

FVR.GMP.GFP12 96.93 

FVR.GMP.GFP12 92.45 

FVR,GMP,GFFi2 95.16 

FVR.GMP 97.18 

FVR,GMP,GFP12 96.90 



107 



Tpra f VI- POISSON'S RATIO. MAJOR fNUC12) 



SIMPLE MDDEL 
INPUT FVR TERTE ACUfaKm) 



THETA, EFP1 
THETA,FVR 
THETA, FVR 



Kl 


0.3 


H2 


0.4 


N3 


0.5 


N4 


0.6 


N5 


0.7 


Ui 


0.3 


U2 


0.4 


W3 


0.5 


U4 


0.6 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


Hie 


0.7 


U6 


0.3 


W7 


0.4 


UB 


0.5 


U9 


0.6 


uie 


0.7 



THETA, EFF1 
THETA, FVR 
THETA 
THETA, WR 
THETA, FVR 

FVR 

FVR 

FVR 

FVR 

FVR 

FVR 

FVR.WR 

FVR 

FVR,GFF12,EFP2 

FVR,EFP2 



96.39 
97.88 
96.60 



THETA, FVR,EFP1 98.32 

THETA,FVR,En , 2 96.62 



88.43 
84.62 
89.48 
84.05 
92.05 

97.83 
98.48 
97.77 
98.40 
99.17 
97.32 
96.45 
96.38 
98.34 
96.96 



108 



TABLE VII- POISSOH'S RATIO, MIHOR (HUC21) 
SIMT-E MODEL 



INPUT 


FUR 
0.3 


TERM3 ACCEPTED 


Nl 


THETA, FVR 


N2 


0.4 


THETA, FVR, EFP1 


N3 


0.5 


THETA, FVR 


N4 


0.6 


THETA, FVR, EFP1 ,EFP2 


N5 


0.7 


THETA,FVR > EFP1 


Wl 


0.3 


THETA, FVR 


U2 


0.4 


THETA, FVR, EFP2 


W3 


0.5 


THETA, FVR 


U4 


0.6 


THETA 


U5 


0.7 


THETA,FVR,EFF1 


N6 


0.3 


FVR,EFP1,EFP2 


H7 


0.4 


FVR,EFP1,EFP2 


N8 


0.5 


FVR,EFP1,EFP2 


N9 


0.6 


FVR,EFP1,EFP2 


N10 


0.7 


FVR,EFP1,EFP2 


U6 


0.3 


FVR,EFP1 I GFP12 


W7 


0.4 


FVR,EFP1,EFP2 


W8 


0.5 


FVR,EFP1,EFP2 


W9 


0.6 


FVR,EFP1,EFP2 


uie 


0.7 


EFP1,FVR,EFP2 



R» 

91.15 
94.78 
94.31 
97.18 
95.87 
90.87 
89.86 
91.93 
92.57 
94.78 

95.64 
94.90 
95.40 
93.12 
91.83 
87.73 
85.06 
84.29 
90.37 
91.42 



109 



TABLE VIII- LCHG. THERM. EXPANS ION (CTE11) 

S TITLE KXEL 
INPUT FVR TERTE ACCEPTED 



FVR, THETA, EFP1 
THETA, FVR.EFPI , WR 
FVR.THETA.EFPl ,WR 
FVR, THETA, EFP1.WR 
THETA, FVR 
THETA.FVR.EFFl 
THETA, FVR 
THETA, FVR 
THETA, FVR, WR 
THETA 

FVR,EFPi,WR 

FVR,EFPi,WR 

FVR,EFP1,WR 

FVR.EFPI 

FVR,EFPi 

FVR.EFPI 

FVR.EFP1 

FVR.EFPi 

FVR,EFPi,WR 

FVR.EFPi 



Nl 


0.3 


N2 


0.4 


K3 


0.5 


M4 


0.6 


H5 


0.7 


Ul 


0.3 


U2 


0.4 


W3 


0.5 


U4 


0.6 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


NIC 


0.7 


U6 


0.3 


W7 


0.4 


U8 


0.5 


W9 


0.6 


W10 


0.7 



R* 

90.29 
94.46 
95.72 
95.23 
87.63 
80.53 
78.91 
84.77 
74.37 
80.50 

97.21 

96.96 

96.53 

96.60 

96.24 

91.60 

90.88 

91.55 

96.03 

94.13 



lie 



TABLE IX- TRAMS. THERM. EKPAKSIOH (CTE22) 



SIICLE MXEL 
INPUT FVR TER»E ACCEPTED 



Nl 


0.3 


H2 


0.4 


H3 


0.5 


N4 


0.6 


M5 


0.7 


Wl 


0.3 


U2 


0.4 


V3 


0.5 


W4 


0.6 


U5 


0.7 


N6 


0.3 


H7 


0.4 


N8 


0.5 


N9 


0.6 


N10 


0.7 


US 


0.3 


M7 


0.4 


UB 


0.5 


W9 


0.6 


U10 


0.7 



FVR.THETA, WR 99.60 

FVR.THETA.WR 99-21 

FVR.THETA.WR 99.46 

FVR.THETA 99.69 

FVR.THETA 99.79 

FVR.THETA 95.04 

FVR.THETA, EFP1.WR 98.60 

FVR.THETA 95.19 

FVR.THETA 94.84 

FVR.THETA 97.98 

FVR.WR.EFP1 99.70 

FVR.WR 99.53 

FVR.WR 99.65 

FVR 99.67 

FVR 99.75 

FVR.EFP1 99.15 

FVR 98.81 

FVR 98.88 

FVR 99.47 

FVR 99.22 



Ill 



•ran F X- LOHG. TOBILE STRENGTH (SOgfll 



SIMPLE HDDEL 
INPUT FVR TERTE QCLkVlkl) 



FVR 

FVR.SFPT 
FVR 
FVR.SFPT.THETfl 

FVR 

FVR.SFPT 

SFPT.FVR 

EFP1.SFPT 

FVR.EfP 

FVR.SFPT 

FVR.SFPT 

FVR.SFPT.EFP1 

FVR.SFPT 

FVR.SFPT 

SFPT.FVR 

SFPT.FVR 

FVR 

FVR.SFPT 

FVR.SFPT 

SFPT.FVR 



Nl 


0.3 


N2 


0.4 


K3 


0.5 


N4 


0.6 


N5 


0.7 


yi 


0.3 


U2 


0.4 


U3 


0.5 


m 


o.e 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


N10 


0.7 


U6 


0.3 


U7 


0.4 


U8 


0.5 


W9 


0.6 


WIO 


0.7 



12.25 
43.72 
21.68 
43.68 
40.97 
33.37 
39.02 
26.13 
42.27 
33.55 

52.12 

68.43 

34.89 

49.00 

24.00 

46.61 

19.33 

33.13 

34.40 

37.65 



112 



Tfiaj: XI- LONG. COMPRESSIVE STREHCTH (SCXXC) 
SIMPLE MODEL 



input 


FVR 


TERTE flCUsKlkl) 


Nl 


0.3 


FVR 


N2 


0.4 


FVR 


N3 


0.5 


NONE 


N4 


0.6 


SFPC 


M5 


0.7 


FVR 


Ul 


0.3 


WR 


U2 


0.4 


THETA 


U3 


0.5 


GMP,SMPC 


W 


0.6 


THETA 


U5 


0.7 


NONE 


N6 


0.3 


SFPC 


N7 


0.4 


NONE 


N8 


0.5 


NONE 


N9 


0.6 


NONE 


N10 


0.7 


GFP12 


U6 


0.3 


FVR 


W7 


0.4 


WR 


U8 


0.5 


WR 


W9 


0.6 


GFP12 


W10 


0.7 


NONE 



12.25 
18.23 

8.52 

8.08 

8.02 

9.29 

20.59 

9.18 



11.30 



12.01 

9.40 

10.76 

9.85 

8.87 



113 



TABLE XII- TRANSVERSE TBSIIE S TRENGTH (SCYYT) 
SOTLE H3CEL 



INPUT 


FVR 


TER1E ACCEPTED 


Nl 


0.3 


FVR 


N2 


0.4 


FVR 


N3 


0.5 


SIFT 


N4 


0.6 


FVR 


HS 


o.7 


HONE 


Ul 


0.3 


FVR.WR.STfT 


U2 


0.4 


FVR 


U3 


0.5 


FVR 


W4 


0.6 


NONE 


U5 


0.7 


FVR.SMPT 


N6 


0.3 


FVR 


N7 


0.4 


FVR 


N8 


0.5 


FVR.EFP2 


N9 


0.6 


NONE 


N10 


0.7 


NONE 


W6 


0.3 


FVR 


W7 


0.4 


FVR 


US 


0.5 


SMT 


U9 


0.6 


FVR 


U10 


0.7 


FVR 



27.03 
32.91 
8.10 
41.92 

26.89 
41.43 
14.74 

31.05 

9.43 
8.19 
15.58 



33.87 

13.39 

8.62 

27.85 

32.77 



114 



TP»g XIII- TRflNSVESSE COWRESSIVE STRENGTH (SCYYC) 

SIPPLE IEOEL 
INPUT FVR TER1E flCCKFlED 



FVR.SWC 33 - 17 

pyjj 30.10 

NONE 

pyjj 38 . 93 

NONE 

FVR.WR 28.19 



Nl 


0.3 


N2 


0.4 


N3 


0.S 


N4 


0.6 


N5 


0.7 


Ul 


0.3 


U2 


0.4 


U3 


0.5 


U4 


0.6 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


N10 


0.7 


US 


0.3 


W7 


0.4 


W8 


0.5 


U9 


0.6 


Ul® 


0.7 



FVR 



43.26 



FVR.SWC 19 - 57 

NONE 

pyjj 15.85 



NONE 

NONE 

NONE 

NONE 

NONE 

pyjj 28 . 68 

FVR n ' S * 

NONE 

FVR 31.97 

pyjj 33.05 



115 



TPPi F XIV- IN PLANE SHEAR STRENGTH (SOWS) 

SIIFLE MODEL 
INPUT FVR TERTE ACCEPTED 



FVR,THETA,GFP12 28.51 



Nl 


0.3 


N2 


0.4 


H3 


0.5 


N4 


0.6 


NS 


0.7 


wi 


0.3 


V2 


0.4 


W3 


0.5 


U4 


0.6 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


N10 


0.7 


U6 


0.3 


W7 


0.4 


U8 


0.5 


U9 


0.6 


W10 


0.7 



FVR 
THETA 



FVR 
THETA 
THETA 
NONE 

NONE 
NONE 
NONE 

srps 

WR 



8.74 
14.96 



THETA,GFP12, FVR, SITS 31.84 

NONE 

THETA.WR.SIPS.FVR 48.16 



43.26 

8.40 

14.75 



8.25 
8.53 



FVR,S*FS,Gtt> 13.Q& 

NONE 

NONE 

GFP12.FVR 22.20 



SMPS 



17.73 



116 



Further regression node Is were studied, in an attempt to improve 
the predictive capability of the models, especially For the strengths. 
These models, incorporating higher order functions and combinations of 
predictor variables used in the simple node Is, show some improvement 
over the sinple models, proving the value of including the "interaction" 
effects of predictor variables in the regression wodels. In addition, 
the higher order interaction models can fit response functions over a 
wider range of fiber volume ratio, with associated improvements in the 
R 2 statistics. The data cases C0H1 and C0N2 contain selected points 
from the entire range of fiber volume ratios, to supply the samples for 
these runs. Furthermore, since higher order models are postulated, 
THETA is taken to be the cosine of the average of fiber orientation 
angles. The variable IWR is a "dummy" variable, that is a function of 
other variables in the model. It is defined as 

MVR = 1 - FVR - WR 
and is intended to represent an "average" matrix volume ratio over the 
thickness of the ply. The interaction models are shown in Tables XV - 

XXVI. 

The general form of the postulated models now includes higher order 
terms, so the predictor variables are tested up to the fourth power. 
Symbolically, 

Y = B„ + B, (THETA) + B 2 (FVR) + B 3 (WR) + B„(EFP1) + B 5 (EM») + 
B,(MVR) ♦ B 7 (THETA) 2 ♦ B, (THETA) (FVR) + B, (THETA) (WR) 
B l0 (THETA) (EFP1) + ... + B, , (THETA) 2 (FVR) (EFP1) + ... 
B 12 (THETA) ,, + B 13 (FVR) 2 + ... etc. 



♦ 



117 



The number of tern* possible in a conplete fourth powar polynomial 
expansion beocmes unwieldy for the cases studied. Considering the 
limitation of the size of the predictor matrix in the regression package 
used (100 x 100), the terms are intuitively grouped in the hope of 
eliminating large groups at one tine. The regressions are conducted 
using "unlikely" candidates for admission into a particular model, and 
if no terms are entered, subsequent regressions are conducted without 
those terms. The justification for this approach is not a statistical 
argument, rather an interpretation of the physical principles active in 
any chosen model. The regressions to eliminate terms are merely used as 
a check on what seems intuitively reasonable. 



TABLE XV- LONGITUDINAL HDCULOS (ECU) 



118 



INPUT 



FVR 



INTERACTION MODEL 
TERH5 ACCEPTED 



Nl 


0.3 


THETA , '«FVRJtEFPl 


N2 


0.4 


THETA ,, *FVR*EFP1 


N3 


0.5 


THETA ,, *FVR*EFP 1 


N4 


e.6 


THETA ,, *FVR*EFP1 


N5 


0.7 


THETA* , *FVR*EFP1 


wi 


0.3 


THETA M *FVR*»EFP 1 


U2 


0.4 


THETA U *FVR*EFF 1 


W3 


0.5 


THETA ,| *FVR*EFT , 1 


W4 


0.6 


THETA^FVRwEFP 1 


U5 


0.7 


THETA a *FVR*EFPl 


N6 


0.3 


FVR*EFP1 , EMP 2 «MVR 


N7 


0.4 


FVR*EFPl,FVR a 


N8 


0.5 


FVR*EFP1 


N9 


0.6 


FVR*EFT»i f EMP 2 *WR,WR g 


N10 


0.7 


FVR«EFPi , EMP*MVR 


U6 


0.3 


FVR*EFF1,WR 


W7 


0.4 


FVR*EFFi , 1»R 2 *FVR 


UB 


0.5 


FVR*EFF 1 , MVR 2 *EMP 


U9 


0.6 


FVR*EFP1 


wie 


0.7 


FVR^EFPl , EHP*MVR 


COK1 


VARIES 


THETA ,| *FVR*EFP 1 


COH2 


VARIES 


FVR*EFF1,WR' 1 



84.50 
92.66 
93.76 
94.24 
85.08 
63.84 
89.86 
71.79 
64.37 
55.68 

99.82 
99.83 
99.72 
99.79 
99.79 
99.17 
98.53 
98.99 
99.58 
99.38 

96.48 
99.92 



Tarm XVI- TRANSVERSE ICOULOS (EC22) 



119 



INPUT 



FVR 



Nl 


0.3 


N2 


0.4 


H3 


0.5 


N4 


0.6 


N5 


0.7 


Ul 


0.3 


U2 


0.4 


U3 


0.5 


W4 


0.6 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


Ni0 


0.7 


U6 


0.3 


W7 


0.4 


UB 


0.5 


U9 


0.6 


wi0 


0.7 


C0K1 


VARIES 


C0N2 


VARIES 



INTERACTION 1EDEL 
TERES ACCEPTED 

FVR*EFP2*EWSEFP2 2 *FVR,THETA*FVR*IWR 
FVR*EFP2«H'F,EFP2 2 *FVR,THETA 
FVR*EFP2*EI"F , THETA 2 *EFP2 , WR* 
FVRwEFP2*ErP , TRETA 2 *EFP2 , BFP2 2 *FVR 
*** NEARLY SINGULAR 
THETA 2 *HUR , ETP 2 *EFP2 , 1WR 2 
FVRJtEfP , THETA , EFP2 2 *FVR 
FVR*EFP2*Et-F , THETA , FV^EFP^WR 
THETA 2 *WR, rNR 2 *EFP2 , EWP*WR 
*** NEARLY SINGULAR 

FVR*EFP2*ErF , EFP2 2 *FVR, ETP«WR 
FVR*EFP2*EfF , ETP 2 FVR 
FVR*EFP2»EfF , FVR*EFP2«IWR 
FVR*EFP2*ErF , FVR a «E»* 
FVR* 1 > EFP2*EMP 

FVR*EFP2*E«> , EFP2 , EFP2 2 *Etf» , FVR*EFP2 
FVR*EFP2*EMP , FVR*EFP2*IWR , FVR*WR 
FVR*EFP2*ErP , FVR*EFP2*WR 
FVR«EFP2*E3"P , EFP2 2 *rWR 
FVR*EFP2*EMP , WR 2 *FVR 

*w» NEARLY SINGULAR 
FVR*EFP2*EfF , FVR*EFP2*1WR 



99.19 
99.55 
98.92 
99.22 

93.26 
96.79 
93.49 
88.35 



99.22 

99.07 

98.89 

99.14 

99.23 

98.62 

98.28 

97.93 

98.44 

97.86 



99.79 



tohit VUTT- IN PLflNE SHEfiR IBDOLUS (EC12) 



120 



INPUT 



FVR 



Ni 


0.3 


N2 


0.4 


N3 


0.5 


N4 


0.6 


N5 


0.7 


Ul 


0.3 


U2 


0.4 


U3 


0.5 


W4 


0.6 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


Ni® 


0.7 


U6 


0.3 


W7 


0.4 


UB 


0.5 


U9 


0.6 


U10 


0.7 


CGN1 


VARIES 


CON2 


VARIES 



INTERACTION 1CDEL 
TERTE ACCEPTED 

THETA , FVRHGrF ,THETA' , *FVR»««f > 
*** NEARLY SINGULAR 
THETA 2 , FVR 2 «GM» , GFP12«GtP 
THETA, FVR 2 *IWR 
THETA , FVR 2 «Gtt> , GFP 1 2 
THETA" 1 , FVR M «GMP , FVR 2 
*** NEARLY SINGULAR 
THETA''*FVR*l«R,THETA ,4 *WR,WR*«a«» 

THETA" , FVR 3 *G«P , THETA 
*** NEARLY SINGULAR 

FVR**GtP,lfJR 2 *K3FP12 

FVR«at , ,GFP12 

FVR«GTP , FVR*GFP12 

FVRMGnP ,FVR 2 *GFP12 

FyR^^GW , FVR«GFP12«G»» , WR**GtP 

FVRwGMP , FVR*GFP12 

FVR*G»> , FVR«GFP12 

FVR«GfP , FVRJ«GFP12 

FVR*GfF , FVR 2 «GFP12 

FVR a «GMP , FVR*GFP12«GMP , JWR 2 «GrF 



R J 



97.86 

97.75 
98.01 
98.46 
95.49 

91.04 
96.70 



97.73 
97.97 
96.52 
97.10 
98.90 
96.91 
92.37 
95.08 
97.42 
96.85 



FVR 2 «VVR,VVR»WTO f FVRiKIP,THETA ,, «FVR»Kaf s 99.09 

FVfi 2 «GHP,VVR*GM > l GFP12«GMP 99.54 



TABLE XVIII- LONG. THERHQL EXPANSION fCTEll) 



121 



INPUT 



FVR 



Nl 


0.3 


N2 


©.4 


N3 


0.5 


N4 


0.6 


K5 


0.7 


wi 


0.3 


U2 


0.4 


U3 


0.5 


w 


0.6 


U5 


0.7 


N6 


0.3 


H7 


0.4 


N8 


0.5 


N9 


0.6 


NIO 


0.7 


W6 


0.3 


W7 


0.4 


U8 


0.5 


W9 


0.6 


W10 


0.7 


CONi 


VARIES 


COM2 


VARIES 



INTERACTION M3DEL 
TEBTE ACCfcyiU) 

THETA 2 *nVR,MWR 2 ,FVR*EFPU*fWR,E« ,2 *EFPl 

THETA 2 *WR,THETA M ,EFPl ,, ,EK' 2 *FVR,rWR 2 «EFPl 

THETA 2 *WR f JWR,EW' 2 *WR,EH> 2 *EFPl 

FVR*EFP1 , THETA»FVR*EFP1 ,EMP 2 *WR 

THETA,EfF*WR 

*** NEARLY SINGULAR 

THETA' l ,r«R 2 *EfP 

THETA",EfP 2 *lWR, 

THETA' , ,WR 2 *WR 

THETA M ,FVR 2 *IWR 

MVR 2 «EfF , EFPi 2 *EM> , FVR a 

11VR 2 «EtP , FVR»»EFP 1*IWR , MVR 2 *WR 

IWR 2 *E^P,EFP1,FVR ,, 
MVR 2 «EM? , EFP1 2 *fNR 
MUR 2 *EMP , EFPi«MVR - 
MVR 2 »EfP , FVR*EFF1«*WR 
MVR 2 *ETF , FVR*EFP1*MVR 
MVR 2 *ECF , EFPi 2 *WR 
MVR 2 *EfF,EFT»l 2 *lWR 
MVR 2 *EFP1 ,EHP 2 *HVR 

THETA , MVR 3 , EFP 1 2 « WR , FVR* WR*EFP 1 
MVR 2 *EfP,FVR*EFPl»lWR f FVR M ,FVR 2 «WR. . . 



92.51 
96.38 
97.26 
96.32 
90.66 

80.81 
87.98 
75.2© 
82.97 

99.29 
99.17 
98.94 
98.94 
99.33 
98.35 
98.55 
98.56 
99.00 
98.20 

96.82 
99.84 



TQHUE XIX- TOMB. THERMAL EXPANSION (CIE22) 



122 



inpitt 

Nl 
N2 
N3 
N4 
N5 
Wi 
U2 
U3 
W4 
U5 

N6 

H7 

N8 

N9 

NiO 

W6 

W7 

UB 

119 

W10 

COK1 
COM2 



INTERACTION HDCEL 

FVR TERMS ACOaMfel) 

0.3 THETA 2 *ttVR,WR 

0.4 THETA x *M0R,MVR**FVR f FVR*EFPl*EM» 

0.5 THCTA 2 *Mra,lWR**FVR,FVR" 

0.6 THETA 2 »MVR,MVR 2 «FVR,EfF 2 »VVR,»VR 2 MWR 

0.7 FVR 2 ,THETA,THETA 2 *FVR 

0.3 THETA 2 MWi,Wm,EFPl 2 *WR 

0.4 THETA 2 «ttVR,MyR,FVR*EFPl*MVR 

0.5 TOETA 2 «MUR f WR 2 *FVR t EFPl 2 «VVR 

0.6 •^^ETA 2 *WR,*^^ETA,•^^ETA ,, 

0.7 *** NEARLY SINGULAR 

0.3 FVR.WR 3 

0.4 FVR.IWR* 1 

0.5 FVR,1WR 2 

0.6 FVR,EWP 2 *EFP1 

0.7 FVR,1WR - 

0.3 FVR f FVR*EFPl*EMP 

0.4 FVR.MVR" 

0.5 FVR 

0.6 F^^MyR" 

0.7 FVR,EMP 2 *FVR 

VARIES THETA 2 ««JR 

VARIES FVR, FVR 3 , MVR 2 *EM > 



99. 6® 
99.38 
99.48 
99.73 
99.81 
95.16 
98.71 
95.91 
95.69 



99.70 
99.59 
99.67 
99.70 
99.82 
99.26 
98.97 
98.88 
99.57 
99.29 

99.32 
99.95 



TABLE KK- POISSON RATIO; MAJOR fNUC12) 



123 



INPUT 



FVR 



INTERACTION MODEL 
TERTB ACCKFTfcl) 



Nl 


0.3 


*** NEARLY SINGULAR 


N2 


0.4 


THETA, EFP2*HVR 


N3 


0.5 


THETA.GFPl 12*MVR 


N4 


0.6 


THETA, EFP1*MVR 


N5 


0.7 


. THETA ,FVR*EFF2 


Wl 


0.3 


*** NEARLY SINGULAR 


U2 


0.4 


THETA , THETA ,I *FVR*GFP1 2 


W3 


O.S 


THETA 


W4 


0.6 


THETA , WR*GFP 1 2 


W5 


0.7 


THETA, FVR*MVR 


N6 


0.3 


FVR 


N7 


0.4 


FVR 


N8 


0.5 


FVR 


N9 


0.6 


FVR.FVRhMVR 


NIO 


0.7 


FVR 


U6 


0.3 


FVR 


U7 


0.4 


FVR,WR*EFP2 


UB 


0.5 


FVR 


W9 


0.6 


FVR, EFP1*EFP2 , GFP12*MVR 


Wl© 


0.7 


FVR,FVR*EFP2 


CON1 


VARIES 


*** SINGULAR 


CON2 


VARIES 


MVR , FVR* WR , EFP 1 *MVR 



97.96 
96.71 
98.17 
96.48 

84.73 
89.43 
84.27 
92.10 

97.83 
98.48 
97.77 
98.52 
99.17 
97.32 
96.50 
96.38 
98.41 
96.97 



99.77 



TABLE XXI- POISSON RATIO: IHNOR (NUC21) 



124 



INPUT 



FVR 



Nl 


0.3 


N2 


0.4 


N3 


0.5 


N4 


0.6 


N5 


0.7 


yi 


0.3 


U2 


0.4 


U3 


0.5 


W4 


0.6 


U5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


Nie 


0.7 


us 


0.3 


M7 


0.4 


UB 


0.S 


W9 


0.6 


W10 


0.7 


CON1 


VARIES 


CON2 


VARIES 



INTERACTION MODEL 
TERMS ACCEPTED 

THETA,THETA ,, *FVR*EFP1 

THETA, FVRHEFP 1 

THETA,FVR*EFF1 ,EFP2*<FP12 

THETA, THETA ,, hFVR*EFP1 , EFP2 

THETA,THETA ,, hFVRhEFP1 

THETA,FVR*GFP12 

THETA, EFP2»MVR 

THETA, FVRhEFPI 

THETA 

THETA , FVRHEFP 1 , THETA ,| hFVRhWVR , EFP2*1WR 

FVRhEFP 1 , FVR*EFP2 
FVRhEFPI , FVRHEFP2 
FVRhEFPI ,FVR*EFP2,FVR*GFP12 
FVRhEFP 1 , FVRHEFP2 
FVRhEFPI , FVRHEFP2 
FVRhEFPI ,GFP12*WR 
FVRHEFP 1 , FVR*EFP2 , FVRHMVR 
FVRhEFPI, EFP2 
FVRhEFPI , FVR»EFP2 
FVR»EFP 1 , FVR*EFP2 

THETA,FRVH«T12,EFF2 f THETA ,, HFVR*WR, . . 
FVRHEFP1 , FVRHWtfR, EFP2 , WR»GFPi2 



91.69 
94.66 
95.10 
97.15 
95.82 
91.16 
89.52 
92.06 
92.53 
95.60 

95.48 
94.69 
95.52 
92.85 
91.77 
87.83 
86.48 
84.36 
89.84 
91.55 

98. 7© 
98.35 



TPrag XXII- LONGITUDINAL TENSILE STRENGTH (SCXXT) 



125 



INPUT 



FVR 



Nl 


0.3 


K2 


0.4 


K3 


0.5 


N4 


0.6 


N5 


0.7 


Ul 


0.3 


U2 


0.4 


W3 


0.5 


W4 


0.6 


W5 


0.7 


N6 


0.3 


N7 


0.4 


N8 


0.5 


N9 


0.6 


Nie> 


0.7 


W6 


0.3 


W7 


0.4 


U8 


0.5 


U9 


0.6 


W10 


0.7 


C0H1 


VARIES 


C0N2 


VARIES 



INTERACTION MODEL 
TERMS ACCfcKim 

THETA a *FVR*SFPT 

THETA' 4 *FVR«SFPT f IWR" 1 

THETA a *FVR*SFFT 

THETA ,, *FVR«SFFr , FVR 2 *EMP 

THETA g *FVR*SFPT 

FVR*SFPT, FVR*EFP 1*IWR 

FVR*SFPT,FVR 2 *MVR 

EFP 1*SFPT, EMP*MtfR 

EPP 2 *MVR , THETA 2 *SFFT» MVR 

THETA 2 *FVR*SFFT 

FVR*SFPT, FVR 2 *EMP 
FVR*SFPT , FVRwEFT* 1 
FVR*SFPT, WR 2 *FVR 
FVR*SFPT 
FVR*SFPT, FVR*EMP 
FVR*SFFT , MVR 2 *EFP 1 
FVR*SFPT 
FVR*SFPT 
FVR«SFFT 
FVR*SFPT 



17.72 
47.65 
27.65 
44.67 
45.35 
39.18 
42.87 
33.97 
45.09 
32.56 

52.95 
64.41 
39.12 
47.13 
27.43 
49.71 
25.19 
32.16 
34.06 
35.09 



THETA u «FVRHSFPT f FVR»VVR*IWR,FVR»EFPl*lWR 81.20 
FVR*SFPT,FVR*WR,IWR 2 *SFFr 84.79 



TABLE XXIII- L0NGITt3DINAL COffRESSIVE STRENglH (SCXXC) 



126 



input 



FUR 



INTERACTION IBDEL 
TERIS ACCEPTED 



HI 


0.3 


SFPChSIFC 


N2 


0.4 


FVR«JNR 


K3 


0.5 


NONE 


H4 


0.6 


SFPOK3T 


N5 


0.7 


FVRNGFP12 


Ul 


0.3 


WR" 


U2 


0.4 


THETA ,, 


U3 


0.5 


EfP*SW»C , THETA 2 *EMP 


U4 


0.6 


THETA"* 


U5 


0.7 


NONE 


N6 


0.3 


SFPC , CFPi2»Srf , C 


N7 


0.4 


NONE 


K8 


0.5 


NONE 


N9 


0.6 


NONE 


N10 


0.7 


GFP12*EMP 


W6 


0.3 


FVR*1WR 


U7 


0.4 


WR 


UB 


0.5 


WR 


W9 


0.6 


CTP12" 


W10 


0.7 


NONE 


CON1 


VARIES 


FVR*WR 


CON2 


VARIES 


FVR*WR,SFPC 



R 2 

12.53 
19.45 

9.81 

10.20 

10.40 

9.32 

23.32 

9.20 



20.04 



14.96 

11.91 

10.76 

9.85 

9.10 



46.48 
44.44 



127 







INTERACTION MDDEL 




INPUT 


0.3 


TERTE ACCEPTED 




Nl 


HVR^SMT 




N2 


0.4 


MUR 2 *STFT 




N3 


0.5 


EW^SIFT 




N4 


0.6 


FVR 2 *1WR 




N5 


0.7 


NONE 




Ul 


0.3 


FVR 2 *WR f SreT 




W2 


0.4 


SfFMWR 


• 


U3 


0.5 


WR 2 *FVR 




U4 


0.6 


FVR*WR*EFF2 , EfP 




U5 


0.7 


FVR*S1FT 




N6 


0.3 


FVR*1WR 




N7 


0.4 


WR 2 *FVR 




N8 


0.5 


FVR*EFP2*STFT 




N9 


0.6 


STFT 2 *!*** 




N10 


0.7 


FVR*EfF 




U6 


0.3 


JTO 2 *E«P 




W7 


0.4 


WVR 2 *WR 




U8 


0.5 


SIFT 2 *WR 




U9 


0.6 


FVR 2 *EFP2 




W10 


0.7 


wr 2 «smt 




C0H1 


VARIES 


THETA* , «SWT*WVR, FVR*EFF2*WR , *S 


VF\ 


C0N2 


VARIES 


SIf»T 2 *11WR, FVR*WR*IWR 





31.60 
37.23 
9.61 
47.59 

25.39 
43.94 
16.32 
24.10 
30.29 

10.47 

8.94 

13.54 

9.40 

9.13 

35.13 

19.34 

12.89 

29.27 

36.77 

73.42 
76.40 



128 



- rows- »W- TRANSVERSE COffRESSIVE STRENGTH (SCYYC) 







INTERACTION KJCEL 


INPUT 


FVR 


TESTE ACCEPTED 


Nl 


0.3 


SlfC*WR 


N2 


0.4 


FVR 2 *EM» 


N3 


0.5 


NOME 


N4 


0.6 


FVR**I»R 


N5 


0.7 


NONE 


Ul 


0.3 


FVR**WR 


W2 


0.4 


FVR*»EMP 


U3 


0.5 


SMPC***WR 


U4 


0.6 


FVR«WR*EFP2 , EM 1 


US 


0.7 


SMPOIWR 


N6 


0.3 


sit>c 2 «wm 


N7 


0.4 


EFF2»IWR 


N8 


0.5 


FVR*EFP2 


N9 


0.6 


NONE 


N10 


0.7 


S»»C 2 *1WR, FVR 2 *1WR 


U6 


0.3 


INR^EfF 


W7 


0.4 


EFP2*1WR 


UB 


0.5 


NONE 


U9 


0.6 


MWl 2 «SJf»C 


Ul® 


0.7 


IWR - 


COK1 


VARIES 


THETA a «SM»C»«WR, FVR M 


C0N2 


VARIES 


WR 11 , FVR 2 *WR, IWR 2 «SB 



33.39 
32.99 

42.31 

26.24 
43.86 
21.13 
25.75 
18.63 

11.57 

9.03 

9.87 

19.07 
32.50 
14.58 

32.85 
35.79 

76.43 
75.59 



129 



TAHLE XXVI- IN PLANE SHEAR STENGTH (SCXV5) 





FVR 
0.3 


INTERACTION MODEL 


INPOT 


TERMS ACCEPTED 


Nl 


FVR*GFP1 2*G»» , THETA" 


N2 


0.4 


FVR*GFP12*Erf> 


N3 


0.5 


THETA 


N4 


0.6 


THETA a «0 : P12 , SWSJtfNR 


N5 


0.7 


NOME 


Ul 


0.3 


THETA , FVR* WR*EMP , THETA^SHPS , F 


U2 


0.4 


THETA ,, *FVR , THETA ,, «<jFP 1 2 


U3 


0.5 


THETA' 1 


m 


0.6 


THETA, FVR* WR 


U5 


0.7 


THETA a *FVR 


N6 


0.3 


NONE 


N7 


0.4 


SMPS«IWR 


N8 


0.5 


NONE 


N9 


0.6 


SPPS f Sl*»S a 


N10 


0.7 


FVR 2 «IWR 


U6 


0.3 


SnPS*tNR,GnP ,, 


u? 


0.4 


FVR«a : P12*J1VR 


U8 


0.5 


NONE 


119 


0.6 


FVR*GFP12*MVR 


Wl© 


0.7 


SHPS 


C0N1 


VARIES 


THETA^FVR , FVR 2 *SMPS 


CON2 


VARIES 


FVR*WR, 1WR" , FVR**WR 



27.64 
13.51 
14.97 
30.84 

52.20 
26.58 
12.89 
22.33 
10.72 



11.24 

16.14 
11.40 
28.58 
8.28 

19.20 
17.73 

36.74 
61.46 



CHAPTER IV 
DISCUSS ION 



A. Overview 

■Hie numerical simulations conducted show that certain assumptions 
about the statistical distribution of local nonuniformities in fiber 
composites lead directly to quantifiable variations in material 
properties. The advantages inherent in the stochastic characterization 
are numerous. The development of quality control and reliabilty 
measures for composites is crucial to their acceptance in aircraft 
designs. The reduction in needed experimental data achievable through 
judicious simulation of the wide variety of available composite material 
systems could significantly lower the costs of material selection and 
acceptance testing. In the results of this study, the confidence 
intervals calculated can be interpreted as the product of an 
experimental program, specifically designed as an analog of the physical 
processes which occur in real materials. 



130 



131 



B. Histograms and Distributions 

Data cases 1, 2, and 3 demonstrate the differences between a 
deterministic base case and random cases with narrow and wide dispersion 
of input data about the base case. 

In Fig. 30, it is apparent that the deterministic case 1 value of 
1575© ksi. for longitudinal modulus falls near the mean of the case 2 
data. However, the case 3 sample appears to have a mean slightly lower 
(approximately 15C00 ksi.). It should be noted that the size of the 
interval over which the sample occurs is noticeably larger in the widely 

distributed case 3 run. 

Transverse modulus, (Fig. 31) demonstrates a higher mean value for 
the wide distribution than for the narrow, which is greater than the 
deterministic value of 1065 ksi. reported in Table II. The increased 
transverse modulus is related to the added stiffness available in fibers 
with high misalignment relative to longitudinal direction. 

Shear modulus, (Fig. 32) is measurably changed by nonuniformities. 
The deterministic value of 516 ksi is exceeded by the case 2 value of 
approximately 62« ksi, which is further exceeded by the case 3 value 
near 900 ksi. Fiber misaligment has a significant effect in shear 
modulus variation. 

Poisson's ratios (Fig. 33, 34) show similar trends in location of 
sample means and relative dispersion of the sample for the data studied. 
Poisson's ratios generally increase with fiber misaligment and volume 
fraction changes. 



132 



The coefficients of thermal expansion (Figs. 35, 36) for the sample 
studied reflect the longitudinal contraction of graphite fibers when 
heated. The longitudinal coefficient of thermal expansion for 
flS-graphite fiber is -0.550 x 10~V F, while the transverse coefficient 
is 0.560 x 10~ 5 / F. The offset orientation of crystal lattice planes 
in graphite fibers can explain this behavior. These values, the fiber 
misalignnent, and fiber volume ratio near 0.5 all contribute to the 
occurrence of a negative longitudinal coefficient of thermal expansion 
for the composite. At higher fiber volume ratios, the values calculated 
would be less than in the present case, because of the contolling fiber 
behavior for high fiber volume ratio. 

The longitudinal strengths (Fig. 38, 39) are significantly reduced 
when nonuniform! ties are present. The deterministic case 1 value of 203 
ksi. for tensile strength is compared to a mean near 160 ksi for case 2 
and a mean near 130 for case 3. In compression, the deterministic value 
of 165 ksi. compares to means near 100 ksi. and 80 ksi. for the narrow 
and wide distributions, respectively. The failure mode in compression 
varies in the random samples. 

Transverse strengths (Fig. 40, 41) show sensitivity to the 
variations assumed. Misalignments, volume fraction nonuniformities, and 
constituent strength variations all contibute to reduction in the 
strength values. Sub-ply shear failures occur, which undermine the 
already low transverse composite strengths. 

In plane shear strength (Fig. 42) values decline from 10.01 ksi. 
for case 1 to a mean near 8.0 ksi. for case 2. However, case 3 shows a 



133 



value of a mean near 8.© also. It appears that the added shear strength 
due to fiber misalignment is balanced by the reduced strength due to 
variable fiber volume fraction. 

C. Confidence Curves 

The effects of various shape parameters of fiber strength are shown 
in Figs. 43 and 44. The higher weibull distribution shape parameter of 
20 produces a narrow distribution of fiber strength values. The 
composite that has few weaker fibers is expected to be .stronger, and 
Fig. 43 denonstrates this for lonitudinal tensile strength. However, 
compressive failure (Fig. 44) is a nore complex phenomenon. In the 
region of low fiber volume ratio, the 'rule of mixtures' failure 
criteria for a subply can control the failure node. At higher fiber 
volune ratio, however, compressive failure can be initiated by 
delamination, or by a shear failure in a sub-ply. The mixture of 
failure nodes in compressive failure is not well understood, but can 
explain the seeming inconsistency of the intersection of the curves in 
Fig. 44. At a fiber volune of 0.7, the weakest fibers (a = 10) are in 
the strongest composite, when strength is normalized with respect to 
fiber compressive strength. 

The effects of various shape parameters for matrix strengths are 
studied in Figs. 45, 46, and 47. Transverse tensile and compressive 
strengths show expected reductions for lower matrix strengths. In-plane 
shear strength shows lower dispersion at a large fiber volume of 0.7, 
and also declines in general for higher fiber volume. 



134 



The fiber misalignment effects are studied in Figs. 48-57. 
Longitudinal modulus (Fig. 48) shows narrow intervals and slight 
reductions for greater misalignment. Transverse modulus (Fig. 49) and 
in plane shear modulus (Fig. 50) are enhanced by fiber misalignment. 
Longitudinal tensile and compressive strengths are degraded by 
misalignment (Figs. 51, 52). Transverse tensile and compressive 
strengths are enhanced (Figs. 53, 54). In-plane shear strength shows 
total separation of confidence intervals between curves with different 
degrees of misalignment. Poisson's ratios (Figs. 56, 57) increase for 
high fiber misalignment values. 

The fiber stiffness effects (Figs. 58-67) are very small for the 
distribution parameters studied. 

D. Examination of Regression ftodels 

The regression models for thernoelastic properties denonstrate 
resonably high predictive capability in the simple models assumed. 
Ifarginal improvements are achieved in expanding the models to include 
higher order interaction terms. Further improvement is gained by using 
sample data from the wide range of volume percent values. The higher 
multiple correlation coefficients of these models may be due to the 
additional information available in the sample size of 1TO that was 
used. The nearly singular predictor matrices which occur in the higher 
order models indicate that terms must by selectively removed to 
eliminate linearity between assumed predictor terms. The regression 
results support the use of the simple models for thernoelastic 



135 



properties, because improvements in predictive capability in the higher 
order node Is for the same data are small. 

Strengths are not node led well by the sinple or the interaction 
node Is. The predictors chosen are average properties, whereas the 
strengths are based on the weakest points in the material. Even the 
unidirectional cases (N6-N10, W6-W10) present data that the interaction 
node Is have considerable difficulty in accomodating. Somewhat greater 
predictive value is gained by using the expanded data for strength model 
prediction. Using fourth order algebraic functions, values of the 
multiple correlation coefficient square approach 857. for longitudinal 
tensile strength. The other strengths generally have poorer results. 



CHAPTER V 
CONCLUSIONS 



A tractable, constituent based, probabilistic analysis procedure 
for fiber conposites has been developed using the ICAN program as a 
basis. Within the limitations of the mechanics of material model, 
properties and strengths of a variety of composite material 
configurations can be simulated. 

This study quantifies the thermoelast ic and strength properties of 
a graphite/epoxy ply subject to assumed uncertainties for fiber 
misalignment, constituent volume fractions, and constituent properties. 
The results show several advantages of probabilistic characterization of 
this material. These include the identification of unforseen variations 
in composite material properties, and the mechanical effects of local 
nonuniformities. The relative importance of the various fabrication and 
material variables on composite properties is identified, and the 
resulting behavior quantified. 

The advantages of a probabilistic formulation of composite material 



136 



137 



properties over a deterministic one are numerous. Conparison of the 
results of this study with test data oould reveal sone souroes of 
previously unaccounted scatter in the data. Expected value ranges oould 
be predicted for experimental results. Since the simulations provide 
data that is analagous to experimental data at lover cost, laboratory 
classification, material selection, and acceptance testing of composites 
can be guided by the information made available by these methods. 

Although the method presented provides results for only the basic 
ply, extension of the simulation to include lamination angle variations 
in a general layup is feasible. Since finite element material property 
cards are generated, structural analysis of components with randomly 
varied properties defined at a number of points in the body can supply a 
more realistic description of the random nature of structural response 
due to material inhomogeneity. 

The stochastic formulation of material properties is generally 
recognized as one requirement of failure theories for materials. 
Although the failure criteria in the models used in this study are 
conservative, progressive failure of fiber composites could be modeled 
by incorporating load redistribution and material property recalculation 
in the vicinity of failed material. 



REFERENCES 



1. Harter, H. Leon: A Survey on the Literature on the Size Effect on 
Material Strength. AFFDL-TR-77-11, Ik-ight Patterson AFB, April, 
1977. 

2. Griffith, A. A.: The Phenonena of Rupture and Flow in Solids. 
Philosophical Transactions of the Royal Society of London A, Vol. 
221, pp. 163-198. 

3. Murthy, P. L. N. and Chamis, C. C. : Integrated Composites Analyzer 
(ICAN) User's and Progr aimer ' s Maunual. NASA TP 2515, March 1986. 

4. Flaggs, D. L. : ADVLAM- An Advanced Conposite Laminate Analysis 
Code. Lockheed Missies and Space Company, Inc., 1983. 

5. Chamis, C. C. and Sinclair, J. H. : Micromechanics of Intraply 
Hybrid Composites: Elastic and Thermal Properties. NASA TM 79253. 

6. Kural, M. H. and Min, B. K.: The Effects of Matrix Plasticity on 
the Thermal Deformation of Continuous Fiber Graphite/fetal 
Composites. J. Coup. Mater., v. 18, Nov. 1984, pp. 519-535. 

7. Bolotin, V. V. : Statistical Methods in Structural Mechanics 
(trans. S. Aroni). San Francisco, Holden Day, Inc. 1969 

8. Sobol, I. M. (USSR Academy of Sciences): The Monte Carlo Method. 
Chicago, The University of Chicago, 1974.' 

9. Nance, R. E. and Overstreet, C. : A Bibliography of Random Number 
Generation. Computing Review, Oct. 1972, pp. 495-508. 

10. Lehner, D. H. : Mathematical Methods in Large-Scale Computing 
Units, Proceedings of the 2nd Symposium on Large-Scale Digital 
Computing Machinery, Cambridge, Harvard University Press, 1951 pp. 
141-146. 

11. Haimersly, J. M. and Handscomb, D. C. : Monte Carlo Methods. 
London, Menthuen I Co. Ltd., 1964, pp. 28-31. 

12. Oh, Kong P.: A Monte Carlo Study of the Strength of Unidirectional 
Fiber-Reinforced Conposite Materials. Journal of Composite 
Materials, Vol. 13, p. 311. 

13. Box, G. E. P., and Mueller, M. E. : A Note on the Generation of 
Random Normal Deviates. Annals of Mathematical Statistics, Vol. 
29(2), 1958, pp. 610-611. 



138 



139 



14. Howell, L. W. and Rheinfurth, M. H. : Generation of Pseudo-Random 

Numbers. NASA TP 2105, 1982, pp. 7-8. 
13. Welbull, W.: Statistical Tneory of Strength of mterlals. (In 

English) Ingenoirs Vetenskaps Akademien Handlinger, Vol. 151, p. 

16. Chamis, C. C: Simplified Composite Micronechanics Equations for 
Strength, Fracture Toughness, Impact Resistance, and Environmental 
Effects. NASA Tfl 83696, 1984. 

17. Ryan, Thonas A., Jr.: mNITAB. The Pennsylvania State University 
Computation Center, Jan 15, 1981. 

18. Draper, N.R. and Smith, H. : Applied Regression Analysis. New York, 
John Wiley * Sons, Inc., 1981. 

19. Ginty, C. A.: unpublished data. 



APPENDIX A 



140 



„ „„,,„ ,»»»»»»»•» «■ » 

c c :::::::::::::::::::.: ""rT^T^ - 

C PROBABILISTIC INTEGRATED COMPOSITES AMLYZERC PI C»H> J J 
£ > rnnPUTERCODE TOR AHALYSI5 OF PROBABILISTIC VARIATIONS IN »"• 

iaaSF Si f S j s E r RO s E si: P ;? D o^!;iiST5oH S H ?o Y oB?«H 

C CoSJSsiTE T pHOKllTuI »HS"0E5METR?V WHICH ARE THEH INPUT TO 1CJIN. » 

C ^O^J^'EUSFSSSk* OF ICJHUHICH ARE NMIW » 

g :::::::::::...».:.--«. 

c 

E . tut* T<t I tUSTER PROGRAM FOR "ICHH" WHICH ALLOCATES 

C»- DVHM1XCRLW "™riCKNT STOAAOE FOR THE ARRAY VARIABLES — 

C— IH "ICRH" HMD "PIC AN" CODES. 

cSn /PS"'/ nAXlEN.NUM) 

mxLEH • moo 

CALL SPIHIT 

STOP 

END 



C «M» 8 IHPUT T MT«lET N tO KWIM If PAOAAAILISTIC ANALTSIS IS DESIRED 

COmON /IUII4/ INHWI.OUTr.INr.IHPr.IHDS.IDBK 

loSlC*t lSTiT. ) »HOWV.WMTV.riMTV 

^ C ^».S8 U HS..UN N O.OUTr 
DATA PIN/51/ 

SS IKh:!J!J! fife.™ 

ir <.MOT. B5TAT) 00 TO JM 

C KM IPIH.llll) RUNS 

C SR UP POINTERS TOR WASTER ARRAT 

lII! - iiil ♦ Nuns 
us) > u») ♦ Hurts 
uo • ui) ♦ Nuns 
us) ■ u*> * Nuns 
Ut> • us) ♦ NUtlS 
1(71 > Lit) * HI 
L(«l ■ t(7) ♦ ML 
II • Ml) 
11 ■ LI2) 
LS • US) 
L« " LIO 
IS • US) 
It ■ U«) 
17 ■ ID) 

C WOP*'RUNs' , TI«ES THROUGH DATA CREATION AND ICAN ROUTINE 

M cUt"uP0»T^!MW?.MLJ).»tL»).MlS).»<U..»(t7).»(L.).Nl.NU f lS) 

•CUIND IDBK 
CALL ICAHHN 
ENOriU OUTf 

111 CONTINUE 

RENIN* OUTP 
CO TO lilt 
SI* CONTINUE 
CALL COPT ^ 
CALL ICANHN , 

i!,, &S! i.H:!S!. x .».».u) 

ins roimm*) ^ 

»••• continue 5 

AETUAN 
END 



i- 



fMTraER ML.MLC.HHS.WT.I».IHPr 

mean tu.inm.wM* 

CMMCTKMt ►"";»"« 

C0W10M /SEtO/ ISEED 

is» MP'"®*' 

l «» IH UNirOMI MNMM HOME. OEHERRTOR SEED 

c remind xsuor 

S»D(ISEEDr.*> ISEEO 
C REWIND IMPr 

HRITEIIHW.?) (OECK(I). !■».») 
C inura.ll IDEHTUl.ia.NlC.NnS 

iriNi.n.MM) m to SI 

WRITE (Wr.MI 
STOP 

c si MiXTE fmrr.iii imiTiii.iit.Mic.wB 

C mO(MH.U) C0«S»T.»M0lEV.nMTV.VOI.m.COMW 

KlTE "iHPF.in CMM 
ICftD tPIM.ll> ■ 1Dt 
URITE (lHPr.U) BIDE 
REM IPIN.lU MHDV 

MinuNrr.iiinNM 

C •5K B l Jp".l"VMT(»).TO.TCU.Pt(72.1).PL(».» 
Sf <I0tH?(»>.EQ.PW) GO TO II 

SSS iSS* 1HWICII 



STOP 

«• IF CRNOIEVI 00 TO I* 
DO it IR ■ I.Ht 
THET»«IR> ■ THtlU 
•2 COHTINUE 






CO TO III u , 

&t Ki"«.THnU.TH5IC.t, 
THETMH) • » 

1«S CONTINUE 

c ir criMT»» oo to n« 

DO 111 X* " '• HMS .. 

wrrcim • vrwu 

lit CONTINUE 

00 TO 1M , „,„ 
114 DO its I* • i«hhs 
119 CIU UMHDIXI 

CALL USMMHXll „_„„, u - MTa yi 

"W.2t: x : JWtS 8 ft" 

ir a .5. ••»•> co to in 
.vrrait ■ v 
12S continue 

c it* xr ivoimi «o TO !«• 

DO 111 X* " I*""* 

vvr<i«> • *»'"" 

IM 8W8. 

wwpam - vvrciM'iM. 

Itl CONTINUE 

c "•.ssHiEiSin^ii^i-.w^; 1 i "- mil "- 

21 S CONTINUE 
C «E»0 W»DINO CONDITIONS 

UHITEHNPr.m iMJJTlViSSii IB I HBSCi.ilO.tlBStJ.lM 

SttTM»!i , .rBaTi l .wai? i i!i..!i*. , ..*i 






c 
c 
c 



c 
c 



JM CONTINUE 
REM OUTPUT OPTIONS 

IEMXPIH.2I) IDENTtil.IOUT 
URITE UNPr.Jl) IDEMNSl.IOUT 

INCREMENT »N0 PEEILE SEED rOR FUTURE RUNS 

ISEED " ISEED ♦ II* 

REMIND ISEEDr 

WRITE t ISEEDr. »> ISEEO 



1 

4 
1 
• 
I 



roantT ux.iiHiDEHTm •.»•> 

rORIMT II*» , 

foEuT {"THERE IS » MX UP IN THE UYER PROPERTIES CARD') 
, FORIUT UI.JIII 

i« roRtuT m.m> 
u rORtUT <u) mmAt 

!! SSSS iK:Sri8J.p 4 i».«*«.r».>».«.».*.M.i« 

u foriut (M.aM.ux.M«>srt-si 
17 roRn»T m.m.ar*. 2. *»«.»?••*' 
it roRnkT ui.ira.o 

it FORtUT CM.7FB.O 
21 FORIUT (M.II) 

21 FORMAT (M.Itl 

22 format <«» .. 

21 rOMMT (*E1I.J» mmmM 
2 J FORMAT C INPUT EBROR. 

RETURN 

END 



NttS MUST BE SET EQUAl TO NL.') 



U1 



SUBROUTINE URRNDIEJ «H*anM«« •• »»»*»»»» 



DkU Z/l/ 
INTEGER *. X 
C0W10H /SEED/ I5EED 
ir IX .EQ. •» 00 TO I 
X • • 

n ■ !"■»• 

rn • n 

X ■ ISEED . 
A ■ >""!• ♦ S 

I t ■ IWBtMX. MI 

rx • x 

z ■ rx/rn 

RETURN 
END 






51»5Kl!;j55.2S5!J!il!ii55;»i"S"!iliJ 

C OMirORH MNDOtl VARIABLES Ji.^S^-Bi-oxx"" »•««•»■••« 

° REAL M.ltU.SIOItt.Xl.Xl.t 

?ViSIc!iAMi"2-uoo«xin-'«.»)-<cos(2»pi»xz))..nu 

RRURH 
END 



•b 
•o 



c . Ha R8S88SSSS.nK!HS!8;!&SS """m:^^ 

C SUBROUTINE TOR OCNCRkTIHO OMf« VRRIATES WITH PARAMETERS 

C RLMIDk AND K. H|(|M ,„ M , H „o»HMMMH „»*»*»**»»»»*» 

DIMENSION U(IH) 

oincNSioH pum> 

COntlON /SEED/ I3E» 
DO »• X • I.K _ 

31 CAU UMHO(U(I)» 

pen ■ od) 

DO It* I • 2>K 

in pti> • uu> « pcx-n . .. 

I ■ <-l.l/AtAt1DA» » MM(»CK» 

RETURN 

END 



00 



C THIS ROUTIHE GENERATES THE DESIRED UEIBULL MSTMBUTtD MHDOH 

C VARIABLE DISCAIPTIOHS 
C ALPHA ■ SHAPE PARAMETER 

S K l\ I UHirOWt?*DIST5lSUTED RAHDOH VARIABLE OH 11.11 
C 1 • UEIBULI. DISTRIBUTED RANDOM VARIABLE 

C USt IS HADE Or THE HEIBULL DISTRIBUTION rUHCTIOH 

C NX) ■ 1 - EXM - <X/BET»> •» ALPHA) TOR X .OE. ZERO 

c 

^IEtI « ( l -»UKJCOHXD) "« Cl/ALPHA) 

RETURN 

EHD 






SUBROUTINE COPT ,***»»»»•»»*•»*•••• • 

I "^"^m ;»««»« IHFUT DRTR INTO THE FILE TO RE R«D 

PinCHSION IDEMT(S) 

C ^!£rL E S:H.?^!.I» S .«HPV.HOHUOr 

IHTEOER NL.NLC.HNS.INT.IR 

INTEGER MM. POUT 

DATA PIN/41/. POUT/J/ 

RERL TU.TCU.NIS.niS 

DM* PtT/' P"'' 

c k>d(pin.4) (decmii.im.m) 

WRITE (POUT. I) (DECK(I).I-l.tR) 

c K»D(HH.t» iBtHUli;?lf; H i!t , K!2 nhs 

WRITE (POUT.H) IOCHTIII.Hl.HLC.MIB 

C ■£»»( PXM.lt I COHMT 

HRITC (POUT. ID CWSRT 

itiotPiM.in cs»h; 

MUTE (POUT.U) CSKNt 
RERO (PXN.lt) IIDE 
WRITE (POUT. 11) MOE 
RERO (PIN.lt) RXNDV 
HRITE(P0UT.1S)UNDV 
RERD(PIM.tt) NONUOr _ 
WRITE (POUT.ll) NONUOr 

C HERD tRTER DRTR 

11. mO (PIN.l*) IDEKT(t).INP(»).IPnR».TU.TCU.PL(«.IR).THET»(IR). - 

, ir , (i0EHT(t).NE.PW)00TOH5 
CO TO 1*4 m% 
1M WRITE «POWT.«) „_._,,, 
URITE (POUT.t) XOEMT(J) 

m MRITE (POUT.U) IOEHT(t).INP(III)."«I»>•■"'• tCU • P, • < "•" , • 
ITHETRUM.PtCT.IR) 
5r (Xt.M.NL) 00 TO 1M 
IR«IR«l 
CO TO lit 

C RERD MRTERIRL DRTR Jt 

C 5 

II* IR'« 



"'S:i:w!S!:K:A!:K»M""" • 



URITE CPOUT.IM IDENT(4».(CODES(l.J.IR).J«l.2).VrP(IR>.VVP(IRI. 
ir IM.Efl.HHSI CO TO 121 

oo to m 

C RERD tOkDINO CONDITIONS 

lit !»■• 

1S ' ■rln'lPTN 111 IDEHT(J>.HBS(1.IRJ.HBS(2.IR>.NBS<J.IR>.THCS 
SSlTEPOut I ) SDWT?J).HBs}ia ? ).NBS^ 
■MD (PIN. It ) IDENTI J) .HBS( I . IK) .n«S< 2 . IR ) .I1BS( S . IB ) 
URITC (POUT.lt) I0ENT(J).nBS(l.IBI.nBS(2.W).t1BS(J.I«) 
mD»PIN.IB I0ENT(1».(0BS(I.IB).IM ; * 
WRITE (POUT.M) IDENT(J>.(DBS(I.IR).W.O 
IP UR.EQ.IOC> 00 TO HI 
00 TO 111 

141 CONTINUE 

C RERO OUTPUT OPTIONS 

C RERD(PIH.tt) IDEMT(J).IOUT 

URITC (P0UT.2I) IDENT(5).I0UT 
2 rORMT (IX,1IHIDEHT(2) •.»•> 
t rORtUT (2IR4) 

i fS5S*T ("THERE IS R MX UP IN THE LRTER PROPERTIES CRRDM 

« format us. mi) 

11 rORHRT (ftl.SIl) 

12 rORH»T (Li) 
IS FORtUT (I*) 

is roRtuT (ki.2ii.sra.)) 
i» roRn»T (M.2ii.sra.]> 

It rORWRT (M.2»*.2r«.2.2R«.Jrt.2» 

17 rORtUT (RB.2fc4.2rB.2.2k4.jri.2) 

IB FORtlRT Ul.7ri.4l 

It rORHRT (Rl.7rt.4l 

21 rORIHT (Rl.lt) 

21 rORMRT (Rt.IB) 

22 rORHRT (15) 

2) rORtIRT (4E1I.S) 
RETURN 
END 



U» 



subroutine vrrconi prp . prs . mr . wis . codes . wis i 



C "sUiioOTIHETO SOPPU W»M»TIOMS IM CONSTITUENT PROPERTIES 

g M .S5.555S!5!.!X.I!!5.1l55!.21I.S!:!:ilI.!:i:2^5?2!:ii!l5i 

INTEGER PIN 
DRTR PIN/4?/ 

DMENSioSi ount( iti .pfmn . 1 1 .pps( si . n . pupc u . n .pns( it . i » . 

C VAM McFpROPMT* MHICH COMESPOMDS TO A BOOLEAN UITM VALUE 'TRUE' 

C DO 9t J • l.MMS 

C OENERRTE rilER PROPERTIES 



C 



RERBI PIN. 11*1 > BOOL.SHERN.STDEV 
IM. NOT. ROOt I 60 TO S 

CRLL URAND(Xl) 

CALL UR»ND(X2) 

CILL NOMKXI.Xt.SNERN.STDEV.CrPtn 

PFPU.J) ■ EPPI1 

» RER»(PIN.t«tl> BOOL.SnEM.STDEV 
XPC.NOT. tOOLI 00 TO « 
CRLL URANDIXll 

CRLL SSom?XI.Xt.SnE»N.STDEV.ErPI2) 

prp(«,J> • Err« 

4 READ! PIN. I MM BOOL.StlERN.STDEV 
in. NOT. ROOD 00 TO 1 
CRLL URRND(X1> 
CALL URRHDIX2) 
CRLL N0RmXl.X2.SHERN.STDEV.crPI?) 

rrru.Jt • omt 

7 READ! PIN. 1(»1> B00L.3NCAN.STDEV 
tN. HOT. BOOL) 00 TO • 
CRLL URRHDIXI) 
CRLL URRNO(XII 
CR LL NORM X I . X* . SHERN , STDEV . Or P2 J ) 

pppir.ji * orris 

• ■ERDIPIN.IRII) ROOL.BETR.RLPHR 
irt.NOT. ROOt) CO TO * 
CRLL URRND(XI) 
CRLL UEIR(XI.RLPHR.BETR.SrPT) _ 

prp(i«.J) » srpT „ 

» RERD<PIM.IIII> ROOL.BETR.RLPHR 
ir(.NOT. BOOL I CO TO II 
CRLL URRNO(XI) 
CRLL HEIRIXI.RLPHR.BETR.SrPO 



prp(i».J) • srpc 
c 

II CONTINUE 

C GENERATE HATAIX PROPERTIES 

° 2* UtD(l>IN,lMI) BOOl.StlEAN.STDEV 
IM.HOT. BOOL) GO TO 21 
CALL UAANDtXl) 

^tNoSmil'xi.SnEAN.STDEV.EMNPl 
PdP(S.J) « ENNP 

° 21 REAO(PIN.lMl) BOOL.BETA. ALPHA 
ir(.HOT. IOOL) GO TO 22 

CALL UMHD(Xl) __ . 

CALL WEIBU1. ALPHA. BETA. SMTP) 
PHP(*.J) ■ SNTP 

° 22 IEAD<PIN.1MI> IOOL. BETA. ALPHA 
in. HOT. IOOL) GO TO 21 

S& Kxil.LPHA.BETA.SnCP, 

pnp<n.j) - shcp 

C 2S REAOJPIN.1M1I BOOL. RET A. ALPHA 
in.HOT. BOOL) GO TO 24 
CALL URAHD(Xl) 
CALL WEIRCX1. ALPHA. BETA.SI13P) 

ptipcii.J) • snsp 

24 CONTINUE 

REWIND PIN 
St CONTINUE 
lilt rORHATCl«X.LI.2E2(.tl) 
RETURN 
END 



C 
C 






1 



APPENDIX B 



154 



155 



This appendix outlines the theories and equations in the ICAN 
program that are used in this project. In the first section on 
composite micronechanics , the elastic and thermal properties of a 
composite ply are defined with respect to its principal material axes. 
The next section, demoted to laminate theory, contains the 
transformations and summations of ply properties used to arrive at 
laminate properties. The last section contains a brief discussion of 
the failure criteria. 

1. Composite micromechanics 

The theory for calculation of the properties of a unidirectional 
fiber composite ply based on the properties, volume fractions, and 
orientation of its constituents is known as composite micronechanics . 
In this section, the subscripts f , m, v, and / represent fiber, matrix, 
void, and laminate, respectively. The symbolic notation and the 
equations used are summarized below. 
Volume fractions: 

k„ + k ♦ k = 1 
£ m v 

Longitudinal Ifedulus: 

=711 = Wil + k m E m 
Transverse Modulus: 



m 



E /22 = E /33 



i-^i* - vw 



Shear Moduli: 



156 



7 12 



1 - ^ (1 - G n /G £12 ) 



723 



i - JT £ (i - Q m n m ) 

Poisson's Ratios: 

W /12 = W /13 - W m * M W £12 ~ *J 



U /23 - k f U f23 + K 



"712 
m E /ii **2 



Coefficients of therm 1 expansion 

/ll n L * m m ill' *11 J 
a, 



711 



1 + k (E /E. f , - 1) 
m* m /ll ' 



a /22 = a .« x ' *r > 



1 


+ 


VAu 


I E /il 


+ 


k A - E flO 1 



+ W/ 



a 33 = a /22 



157 



2. Laminate Theory 

This section describes the nethods which are used to calculate the 
elastic properties of laminates from the properties, orientation, and 
distribution of individual laminae. The elastic properties are then 
used to predict the response of the laminate to external loads. The 
methods used to predict stresses in the laminae under application of 
external loads are also described. Failure loads can be predicted by 
using these methods; as described in a following section. 

a. Generalized Hooke's Law 

The stresses acting at a point in a solid can be represented by the 
stresses acting on the planes normal to the coordinate directions, or 
equivalents, on the surfaces of an infinitesimal cube as shown in Fig. 
B-l. The stresses (a. .) on each face are resolved into three 
components: one normal stress and two shearing stresses. The first 
subscript refers to the direction normal to the plane in which the 
stress acts and the second subscript to the direction in which the 
stress acts. The stress components shown on the faces of the cube are 
taken as positive and can be taken as the forces (per unit area) exerted 
by the material outside the cube upon the material inside. A stress 
component is positive if it acts in the positive direction on a positive 
face of the cube. Thus normal tensile stresses are positive, and normal 
compressive stresses are negative. Mine stress components must be used 
to define the state of stress at a point, nam-ly «r n , o^ C33, " 23 . 

n and a . There are nine corresponding strain 
*31» a 12' °32' a 13' *^ 21* 



1S8 



components, following the sane subscript convention. 

For bodies in which each strain conponent is a linear function of 
all six stress conponent s, the generalized Hooke's Law can be expressed 

a ij = E ijkl *kl 

where E is a fourth order tensor of elastic constants. For nine 
ijkl 

stress components and nine strain conponent s, there mist be 81 elastic 

constants defining E i ^ Certain reductions in the number of 

independent constants for an anisotropic body are due to symmetry 

properties of the tensor E. .... By considering nonent equilibrium about 

the center of the cube, it can be shown that at any point a^ = a^, 

a 31 = a 13 , and a i2 = a^. Thus, E. jkl is symmetric with respect to the 

first two indices. Second, because the strains are symmetric (that is, 

* = e ), E. .. . must be symmetric with respect to the second two 
ij ji 7 ijkl 

indices. This reduces the number of elastic constants to 36. Further 
reduction to the final 21 elastic constants for a general anisotropic 
material is accomplished by assuming the existence of a strain energy 
density function, such that 

with the property 

J 32 - - •-■ 
d *ij 1J 

From the generalized Hooke's Law, 

-22_ - E 5 
de. . ijkl kl 

Partial differentiation with respect to * fcl yields 



159 



as. 



[a* J = E ijki 



c kl l *"iJ 
Since the order of partial differentiation is imnaterial, 



as 



kl 



[ m 1 a 



au 



df 



kl 



and the subscripts can be interchanged to yield 

a 



3s, 



[as J = ^lij 



kl l_ "ij 
so that 

E ijkl = ^lij 
Thus the first pair of subscripts in E. jfcl can be interchanged with the 

second pair without any change in the values. The nunter of elastic 
constants is thus reduced to 21. 

b. Lamina Constitutive Relation 

Several sinplif ications to the generalized Hooke's Law can be node 
for the special case of a thin orthotropic material, which approximates 
a unidirectional fiber composite lamina. By considering the invariance 
of elastic properties under coordinate transf orimt ion for planes of 
symretry, the tensor E. jfcl can be reduced to the following nine 

constants: 

E U11 E 1122 E 1133 

E 1122 E 2222 ^233 

E 1133 E 2233 E 3333 



'ijkl 















It is now convenient to make the following notation changes: 



160 



*11 " °l 



'22 



'33 



a 23 = T 23 = a 4 



a 13 = T 13 = a 5 



a 12 - T 12 * °6 



e li " e l 



! 22" *2 



e 33 * e 3 
^23 = Y 23 * U 
2e 13 = V 13 " *5 
2e l2 " ¥ 12 " *6 



The generalized form of Hooke's Law can now be written 

6 
a . = T C . £ . for i ,J = 1 , . . . , 6 
x >1 '" 

The matrix C. . is known as the stiffness matrix, and s. are the 

engineering strain components. In matrix form Hooke's Law is witten 



a l 




°2 




a 3 


_ 


T 23 




T 31 




T 12 





C ll C 12 C 13 * 



C 12 C 22 C 23 
C 13 C 23 C 33 


















C 







44 
0- C 



55 
C 











66 



'23 
'31 



r 12 



vfcere the coordinate axes coincide with the symmetry axis of the 
material. For laminae that are assumed sufficiently thin, the through 
the thickness stresses are zero. Thus ° 3 = a 4 = ^5 = ®» for P lane 
stress. It is apparent that e^ = fg= 

The stress strain relations for a thin unidirectional lamina are 
written 



r 12' 



'11 



'12 







*12 



< 2 2 











2Q 



'66* 



161 



'2 

l 
TV 



12 J 



using the tensorial strain T * 12 instead of the engineering strain Y^. 
The Q terns are known as reduced stiffnesses, i.e. 



Ej 



*11 = C ll - 



*12 - C 12 



^22 ' C 22 



1 - Wi2«21 

V 12 E 2 
1 - V 12 U 21 

E 2 
1 - u 12 u 21 



u 2 ,E, 



1 - wia«2i 



*66 



l 
= T 



< C 11 " C 12) = G 12 



where E lf E 2 , v l2 , v 21 , and G 12 are the ply elastic constants, neasured 
with respect to the natural material system. It may be noted that only 
four of these constants are independent. 

The stress- strain relation above shows that there is no coupling 
between tensile and shear strains, as long as the applied stresses are 
coincident with the principal material directions. However, coupling 
appears when a lamina is tested at arbitrary angles with respect to the 
principal material directions. The general form of the stress-strain 
relation for any angular orientation of a lamina is considered next. 

c. Stiffness matrix transformations 

A lamina is loaded along a coordinate system x-y oriented at soma 



162 



angle * with respect to the principal material directions as shown in 
Fig. B-2. Since stress and strain are second order tensors, they are 
transforned by 



= [T] 



12 



xy 



and 



'2 
l 



12 



- [TJ 



. T* 



xy 



where [T] is the transformation matrix for plane stress and plane strain 

transformed by clockwise rotation about the (3,z) axes, given by 

cos 2 * sin 2 * 2 sin© cos* 
sin 2 * cos 2 * -2 sin* cos* 
-sin* cos* sin* cos* cos 2 * - sin 2 * 



[T] = 



Inversion and substitution yields 



xy J 



= [T] _1 [Q][T] 



y 

i 



xy 



which is the stress strain relation for a lamina referred to arbitrary 
axes. For simplicity, the notation [ Q ] is introduced 

[Q] = [T] _1 [Q][T] 
where [Q] is called the transforned reduced stiffness matrix. 

Using the approach outlined above, it is possible to obtain 



163 



expressions for the elastic properties referred to the x-y coordinate 
system. 

d. Elastic properties of laminates 

A number of assumptions are made in laminate theory to obtain 
theoretical predictions. These are: 



1. the lamina are perfectly bonded and do not slip relative to 

each other 

2. the bond between the laminae is inf initesimally thin 

3. the laminate has the properties of a thin sheet 

These assumptions allow the laminate to be treated as a thin 
elastic plate. The classical hypothesis of Kirchhoff is applied to 
derive the strain distribution throughout the plate under external 
forces. Because the laminate is composed of laminae oriented in 
different directions with respect to each other, the stress-strain 
equation for each layer (k) is defined as 

*11 *12 *16 



xy 



*12 Q 22 Q 26 
I *16 ?26 *66 l 



l 
lT * 



xy 



Thus for a given strain distribution, the stress in each layer can be 
determined. The strain at any point in a laminate undergoing 
deformation must be related to the displacements and curvatures of its 
midplane. The discussion which follows assumes that the laminate is 
thin. Kirchhoff plate theory is used in this formulation. 

The deformation of an arbitrary section of a laminate is showi in 
Fig. B-3. It is assuned that lines straight and perpendicular to the 



164 



midplane before defornation remain so after deformation. This is 
equivalent to neglecting transverse shearing deformations. Comparing 
Fig. B-4(b) with Fig. B-4(a), in which the normals to the midplane 
remain perpendicular after deforrsBticn, it is seen that the upper and 
lower surfaces of the plate must not shift their relative positions. It 
is obvious that the resistance of a thin plate to such deformation is 
large, much larger than its resistance to deformations perpendicular to 
the midplane. 

It is assumed that the point B at the midplane undergoes 
displacements u 0f v„, and w along the x, y, and 2 axes, respectively. 
The displacement u in the x direction of a point C located on the normal 
ABCD at a distance z from the midplane is given by 

u = u - za 

where a is the slope of the midplane in the x direction, 

dw 

The last two equations can be used to obtain the displacement u of an 

arbitrary point at a distance 2 from the midplane as 

dw 



Similarly, 

dw 

v - v ° " z "aT 

Since the strains normal to the midplane are neglected (plane 
strain) , the displacement w at any point is taken equal to the 
displacement w at the midplane. The strains in terms of displacement u 
and v are 



165 



du du d 2 Wo 

£ x ~ Ox = 0x z 0x a 

3v dv 3 2 w 

*y " dy = dy~ ~ Z 3y z 

3u dv du dv d 2 *to 

xy dy dx dy 3x axdy 



In terns of midplane strains and plate curvatures, the strains in a 
laminate wary linearly through the thickness, 



£ 
X 




f «° 1 

X 




k 

X 


£ 

y 


= 


£° 

y 


+ z 


k 

y 


y 

1 xy J 




y° 
1 xy J 




k 

1 xy J 



where midplane strains are given by 



x 

"y 

xy 



du 

ax - 

dv 

ay" 

du 

ay~ 



dv ( 

aiT 



and the plate curvatures by 







a 2 u 


k 

X 




ax 2 
a 2 w 


k 

y 


= •*■ 


ay 1 " 


k 

. *yJ 




a*w 


dxdy 



The stresses in any (k) lamina can be obtained by substituting the 
previous equation into the stress strain equation 



a 

X 






a 

y 




= 


1 xy J 


k 





$11 ^12 Q 16 
Q 12 °22 Q 26 
$16 $26 $66 1 



x 

y 

xy J 



+ z 



*y 



166 



e. Laminate Stiffness ffetrix 

Classical laminate theory provides a method of oaloulating the 
resultant forces and moments per unit length acting on the laminate by 
integrating the stresses acting in each lamina through the thickness (h) 
of the laminate. Resultant forces are obtained by 




The moment resultants are obtained by integration through the thickness 
of the corresponding moments of stresses about the mi dp lane: 



f ' 

X J- 

r ■ 

xy J ., 



h/2 

azdz 
h/2 X 



h/2 

a z dz 

y 



-h/2 

h/2 

t z dz 
h/2 ** 



The units of N , N , N are force per unit length and n^ H y , n^ are 
moment per unit length. The sign conventions are showi in Fig. B-5. 

Using the resultant force and moment relations, a system is defined 
that is statically equivalent to the laminate stress system, but applied 



167 



at the midplane. Tnus, the external loading has been reduced to a 
system that does not contain the laminate thickness or z coordinate 

explicitly. 

For a laminate consisting of n laminae (Fig. B-fi). the resultant 
force-romsnt system acting at the midplane can be obtained by adding 
integrals representing the contribution of each layer by 



N 

X 




* h/2 


a 

X 


N 

y 


= 


-h/2 


a 

y 


N 
1 xy 






T 

1 xy ' 



n r 



dz = 



I 



k=l J 



Vi 



xy 



dz 



r m l 

X 




* h/2 


a 

X 


n 
y 


= 


-h/2 


a 

y 


1 xy ' 






T 

1 xy J 



z dz = 



n 

I 

k=l 





a 




X 


\ 






a 


Vi 


y 




T 




1 xy J 



z dz 



Using the expressions for the stresses in the k-th lamina derived 
earlier, and noting that the midplane strains and plate curvatures are 
constant not only within the lamina, but for all laminae, it is apparent 
that they can be taken outside the integral sign. TTie stiffness matrix 
[Q] is constant within a lamina so it also can be taken outside the 
integration to give 



[ H 1 

X 




n 


N 

y 


= 


I 

k=l 


N 
1 xy J 




, 



*11 «12 °16 
*12 ^22 *26 



1^16 Q 26 Q 



66 



I 



Vj 



dz 



x 

y 
xy 



n 

1 

k=l 



*1I *i2 Q 16 
«12 *22 ^26 
«16 *26 *66 






z dz 



xy 



168 



11 

X 




n 


y 


= 


1 


M 
1 xy J 




k=l 



*ii Q 12 ^16 
Q 12 Q 22 <? 26 

*16 *26 *66 ik 



I z dz 



x 

y 

xy 



k=l 



*il 9 12 ^IG 
*12 *22 *26 
*16 *26 *66 'k 



T^ 2 „ 
I z dz 

J Vi 



xy 



Three new matrices, A. ., B , and D ., are defined, where 



k=l 

k=i 
i n 



k=i 
These new matrices, A, B, and D, simplify the resultant force and nonent 

relations, and are know as the extensional, coupling, and bending 
stiffness matrices, respectively. The total plate constitutive equation 
is then 



M 

M 


= 


A B 
B D 




k 



It may be recalled that in an orthotropic lamina with arbitrary 
orientation the shear stress is coupled with the normal strain and the 
normal stresses are coupled with the shear strain. In general, a 
resultant shearing force on a laminated plate produces midplane normal 
strains in addition to the expected shearing strain. Similarly, a 



169 



resultant nornal force will induce shear strains in addition to midplane 

normal strains. 

The nonzero coupling natrix B in the plate constitutive equation 
explains the coupling between bending and extension of the laminated 
plate. Thus, normal and shear forces at the midplane induce not only 
midplane deformations, (and hence, midplane strains) but also twisting 
and bending, producing plate curvatures. Similarly, resultant bending 
and twisting nonents induce midplane strains, 
f. Lamina stresses and strains 

The aim of the analysis of a laminated composite is to determine 
the stresses and strains in each of the laminae forming the laminate. 
These stresses and strains are used with failure criteria to predict the 
loads for failure initiation for a laminate. The failure criteria are 
discussed in the section devoted specifically to that purpose. 

The strains in a lamina caused by external loading are a function 
of laminate midplane strains and plate curvatures, as previously 
discussed. Once the lamina strains are known, lamina stresses can be 
found using the lamina stress-strain law. Thus, the starting point for 
calculating lamina stresses is the determination of laminate midplane 
strains and plate curvatures in terms of the applied loading. The plate 
constitutive equation given previously can be inverted to give the 
midplane strains and plate curvatures explicitly in terms of the 
resultant external forces and nonents. ThB result of the inversion 
process is 



17© 



f° 




A' 


B' 




N 




A' 


B' 




M 


k 


= 


C 


D* 




n 


= 


B» 


D« 




II 



where A', B' , and D* are simplified farms of the inversion process 
results, and are functions of the A, B, and D matrices of the original 
form of the plate constitutive equation. 

It is now apparent that with these equations, an analysis of a 
laminate subjected to external forces and moments can be conducted: 

1. calculate midplane strains and plate curvatures 



k 



A' B' 
B« D' 



N 

M 



2. calculate lamina stresses in global (x-y) system 



xy 



<>il Q 12 Q 16 
Q 12 Q 22 Q 26 
*16 ^26 %G 



k 


X 

s* 

y 

1 xy J 


+ z 


k 1 

X 

k 

y 

k 

l Xy J J 



3. calculate lamina stresses in natural (longitudinal and 
transverse to fiber) system. 



- [T] 



'12 



«y 



The strain variations in a lamina are calculated in an analagous 
manner. Tne stress-strain variation is compared with the allowable 
stresses and strains in each lamina. Thus the load at which failure is 
initiated in one of the lamina can be calculated, as long as a strength 
criteria exists in terms of the lamina natural axis system. The 
formulation of lamina failure criteria is discussed in the next section. 



171 



3. Strength TTieories 

It is assured that the strength of a laminate must be related to 
the strengths of the individual laminae. A simple failure criteria 
consists of evaluating the lamina strengths in their principal material 
directions subject to induced stresses or strains at the boundaries of 
the lamina. In this context, it is assumed that the lamina and its 
constituents behave in a linear elastic manner to failure. The strength 
analysis described here assumes that the behavior of each lamina in an 
arbitrary laminate is the sane as the behavior observed in the natural 
axis system **en the lamina is part of any other laminate under the sane 
stresses or strains. In other words, it is assured that the strength 
criteria for a lamina in plane stress is valid for any orientation of 
the lamina in a laminate. In the ICAN program, the lamina strengths are 
calculated using the expressions given below. 
Longitudinal tension 

s /iit " s rr < k f + WW 

Longitudinal compression: 

Tne longitudinal compressive strength mist be computed on the basis 

of three different criteria: 

a. rule of mixtures 

S /iiC = S fC < k f + WW 

b. de lamination 

S /11C - < 13 S /12 + S -C> 



172 



c. fiber microbuckling 
S 



r 2°. 



/11C - 1 - k £ (l - G B /G fl2 ) 



Transverse tension 

S /22T - S mT (FACT/DEl«H) 
Transverse conpression 

S /22C " S „C ' DBBM 
Transverse shear 

[( F i- 1+ V G n 2 ) F 2 G /i2 S ^ 

c = — FACT 



where F. and F 2 are given by 



F -1 * 


1 1 " 4 <Ol £ 


r a .x- y 


4k 

V 


irk 



The variable EEHOM is introduced for convenience: 

EEHOM = [1 - w*J(l - E m /E f22 )] >l 1 ♦ T{T - 1) ♦ */,(* - 1) ; 



tfiere 9 is given by 
F. - 



E 

HI 



1 " E [1 - *,(1 - E^/E^Jj 



V22 
9 = 



F l-' 



173 



The variable FACT is used to correlate the strengths of KB and Kevlar 
fiber conposites with the experimentally observed values. Since neither 
of these fibers is used in this work, FACT takes the value unity. 



174 





a 


33 t 










/ 


a 32 


/ 






/ a 3l/ 




/ 




°23 




°13j 




^ 


>a22 


J 


V 
a 12 


°2lJ 















Fig. B.l- Components of Stress acting 
on elemental unit cube. 




Fig. B.2- Rotation of coordinates from 1-2 to x-y. 



175 



r 

Zc 

1_J 



■♦X 




•> INITIAL CHOSS-SCCTION »> DEFOaMCD C«OSS-SECTlOM 

Fig. B.3- Bending geometry in the x-z plane. 





a. D«A«cf«rf bar witkmvt aSaor b. OtlbcM bar wtffc chaor aV- 

•Wrenaafiant farwafioiM 

Fig. B.A-Shearing force deformations on straight cross section. 



-uojiusAuoo uoi^eiou xapuj ajeuiuiei -9*9 *6ij 




s^ueainssj juauioui pue ssajjs sjeu -S'g 'Btj 




9£T 



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1. AGENCY USE ONLY (Leave blank) 



REPORT DATE 

January 1996 



3. REPORT TYPE AND DATES COVERED 

Final Contractor Report 



4. TITLE AND SUBTITLE 



Probabilistic Fiber Composite Micromechanics 



6. AUTHOR(S) 



Thomas A. Stock 



5. FUNDING NUMBERS 



WU-505-63-5B 
G-NAG3-550 



7. PERFORMING ORGANIZATION NAME(S) AND ADDRESSES) 

Cleveland State University 
Fenn Tower 1010 
1983 E. 24th St. 
Cleveland, Ohio 44115 



8. PERFORMING ORGANIZATION 
REPORT NUMBER 



E-10082 



9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESSES) 

National Aeronautics and Space Administration 
Lewis Research Center 
Cleveland, Ohio 44135-3191 



10. SPONSORING/MONITORING 
AGENCY REPORT NUMBER 



NASA CR-198443 



11. SUPPLEMENTARY NOTES 

This report was submitted as a thesis in partial fulfillment of the requirements for the degree Master of Science in Civil 
Engineering to Cleveland State University, Cleveland, Ohio 441 15. Project Manager, Christos C. Chamis, Structures 
Division, NASA Lewis Research Center, organization code 5200, (216) 433-3252. 



12a. DISTRIBUTION/AVAILABILITY STATEMENT 

Unclassified -Unlimited 
Subject Category 05 

This publication is available from the NASA Center for Aerospace Information, (301)621-0390. 



12b. DISTRIBUTION CODE 



13. ABSTRACT (Maximum 200 words) 

Probabilistic composite micromechanics methods are developed that simulate expected uncertainties in unidirectional 
fiber composite properties. These methods are in the form of computational procedures using Monte Carlo simulation. 
The variables in which uncertainties are accounted for include constituent and void volume ratios, constituent elastic 
properties and strengths, and fiber misalignment. A graphite/epoxy unidirectional composite (ply) is studied to demon- 
strate fiber composite material property variations induced by random changes expected at the material micro level. 
Regression results are presented to show the relative correlation between predictor and response variables in the study. 
These computational procedures make possible a formal description of anticipated random processes at the intraply level, 
and the related effects of these on composite properties. 



14. SUBJECT TERMS 

Computational simulation; Uncertainties; Probabilistic distribution; Monte Carlo; 
Fiber content; Void content; Misalignment; Graphite fibers; Epoxy matrix; Micro level; 
Unidirectional: Random processes 



15. NUMBER OF PAGES 

192 



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