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3 1293 010

This is to certify that the
dissertation entitled

NONLINEAR RANDOM VIBRATION OF
COMPOSITE LAMINATED PLATES

presented by

Mohamad Khaled Naja

has been accepted towards fulﬁllment
of the requirements for

Doctor of Philosophy degreein Civil Engineering

 

 

ﬂﬂéwéeaﬂ.

Ronald Harichandran

 

Major professor

Date 11/18/93

 

MSU i: an Afﬁrmative Action/Equal Opportunity Institution 0-12771

 

. LIBRARY
M'Chigan State
University

 

 

 

PLACE IN RETURN 80X to MOI. thin checkoutﬁom your record.
To AVOID FINES return on or before date duo. 7

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NONLINEAR RANDOM VIBRATION OF COMPOSITE
LAMINATED PLATES

By
Mohamad Khaled Naja

A- DISSERTATION

Submitted to
Michigan State University

in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1993

ABSTRACT

NONLINEAR RANDOM VIBRATION OF COMPOSITE LAMINATED PLATES

By

Mohamad Khaled Naja

Structural components in space vehicles, aircraft, automobiles, submarines, etc., that
are made of ﬁlamentary composite laminae are usually subjected to stochastic loads. Com-
posite laminae have strongly anisotropic properties and display significantly nonlinear
elastic behavior when loaded in shear or along directions different from those of the ﬁla-
ments. While such components have been used for over a decade, their design has predom-
inantly been performed using deterministic methods. In this study, the method of
equivalent linearization is used in conjunction with the ﬁnite element method to perform
nonlinear random vibration analysis of laminated composite plates. An approximate, but
sufficiently accurate, sen'es representation of the nonlinear shear stress-strain law is used
to facilitate the formulation. Classical laminate theory that accounts for the coupling be-
tween extensional and bending responses is used, but higher-order shear deformation ef-
fects are not considered in the course of this study. Four-noded elements with ﬁve degrees-
of-freedom per node are used to discretize the plate. The displacement, strain and stress re-
sponses are computed at different excitation load levels. The results indicates that the effect
of nonlinearity on the responses for any given load level depends on the ply-arrangement,

and as expected becomes more signiﬁcant for higher loads.

ACKNOWLEDGMENTS

The author wishes to express his deep indebtedness to Dr. Ronald S. Harichandran
for his guidance, encouragement, support, and valuable input, without which this work

would not have been possible.

Grateful acknowledgments are due to Professors R Wen of the Department of Civil
and Environmental Engineering, N. J. Altiero of the Department of Materials Science and
Mechanics and G. Ludden of the Mathematics Department, for serving on the author’s
doctoral committee. Gratitude is also due to the Mathematics Department for their contin-

uous ﬁnancial supports during the period of this study.

Thanks are also due to the director of the Minority Business Program Dr. E. Betts

and his assistant Ms. L. Gross for their support and encouragement.

In addition, the author would like to express his sincere appreciation and thanks to
his parents in Lebanon, especially his mother Myassar for her prayers and encouragement,

and the sacriﬁces that she has suffered, during the long course of this study.

This work was funded by the State of Michigan Research Excellence Fund adminis-

tered by the Composite Materials and Structures Center at Michigan State University.

Finally, the author would like to thanks his students at Michigan State University and

at Davenport College for their patience and understanding.

iii

TABLES OF CONTENTS

1. Introduction l

1.1 General ........................................................................................................................ 1

1.2 Objectives ................................................................................................................... 2

1.3 Literature Review ....................................................................................................... 2

1.3.1 Physical Nonlinearity ................................................................................ 2

1.3.2 Shear Deformations .................................................................................. 9

1.3.3 Stochastic Dynamic Analysis ................................................................. 10

2. Finite Element Formulationl9
2.1 General ...................................................................................................................... 19

2.2 Constitutive Equations .............................................................................................. 20

2.3 Finite Element Formulation using Classical Plate Theory ....................................... 25

2.3.1 In-plane formulation ............................................................................... 25

2.3.2 Out-of-plane formulation ........................................................................ 28

2.3.3 Formulation of the linear elemental stiffness matrix ............................. 31

2.3.4 Formulation of the elemental consistent mass matrix ............................ 34

2.3.5 Numerical integration ............................................................................ 35

3. Equivalent Linearization 36
3.1 Introduction ............................................................................................................... 36

3.2 Formulation ............................................................................................................... 37

3.3 Derivation of the Nonlinear Elemental Stiffness Matrices ....................................... 40

3.4 Random Vibration Analysis ...................................................................................... 48

3.5 Computation of Strains and Stresses ........................................................................ 49

3.6 Laminate Strength Analysis ...................................................................................... 51

4. Optimizing Computational Effort 53
4.1 Introduction ............................................................................................................... 53

4.2 FORTRAN Implementation ..................................................................................... 53

4.3 Optimization Techniques .......................................................................................... 56

4.3.1 Optimizing a and B ................................................................................. 57

4.3.2 Covariance optimization ......................................................................... 59

4.3.3 The indexing scheme .............................................................................. 61

4.3.4 Elimination of zero-valued computations ............................................... 66

4.3.5 Unrolling of DO-loops ............................................................................ 69

5. Numerical Results 72
5.1 General ...................................................................................................................... 72

5.2 In-Plane Loading ....................................................................................................... 72

5.2.1 Extensional loading ................................................................................. 72

iv

5.2.2 Shear loading .......................................................................................... 86

 

5.3 Out-Of-Plane Loading .............................................................................................. 95
6. Conclusions and Recommendations ...... - __ _ - ..... 109
6.1 Conclusions ............................................................................................................. 109
6.2 Recommendations for future work ......................................................................... 110

LIST OF TABLES

TABLE 4.1. Number of terms in each group ............................................................................. 64
TABLE 4.2. Comparison of total times. .................................................................................... 71
TABLE 5.1. First ﬁve natural frequencies from linear (ﬁrst iteration) and nonlinear

(last iteration) analysis for load level so = 12000 lb2 sec ...................................... 74
TABLE 5.2. Number of iterations required for convergence ..................................................... 74
TABLE 5.3. First ﬁve natural frequencies from linear (ﬁrst iteration) and nonlinear

(last iteration) analysis for load level so = 120001b2sec ..................................... 78
TABLE 5.4. Number of iterations required for convergence ..................................................... 79
TABLE 5.5. First ﬁve natural frequencies from linear (ﬁrst iteration) and nonlinear

(last iteration) analysis for load level so = 12000 in2 sec. ..................................... 83
TABLE 5.6. Number of iterations required for convergence ..................................................... 83
TABLE 5.7. First ﬁve natural frequencies from linear (ﬁrst iteration) and nonlinear

(last iteration) analysis for load level So = 4000 lb2 sec. ....................................... 87
TABLE 5.8. Number of iterations required for convergence ..................................................... 87
TABLE 5.9. First ﬁve natural frequencies from linear (ﬁrst iteration) and nonlinear

(last iteration) analysis for load level So = 300001b2 sec ...................................... 91
TABLE 5.10. Number of iterations required for convergence ..................................................... 92
TABLE 5.11. First ﬁve natural frequencies from linear (ﬁrst iteration) and nonlinear

(last iteration) analysis for load level so = 801D2 sec ............................................ 96
TABLE 5.12. Number of iterations required for convergence ..................................................... 96
TABLE 5.13. First ﬁve natural frequencies from linear (ﬁrst iteration) and nonlinear

(last iteration) analysis for load level So = 801b2 sec ......................................... 100
TABLE 5.14. Number of iterations required for convergence ................................................... 100
TABLE 5.15. First ﬁve natural frequencies from linear (ﬁrst iteration) and nonlinear

(last iteration) analysis for load level So = 801D2 sec .......................................... 104
TABLE 5.16. Number of iterations required for convergence ................................................... 104
TABLE 5.17. The signiﬁcance of nonlinearity on the responses of various

loading conditions ............................................................................................... 108

vi

LIST OF FIGURES

 

 

 

 

 

 

Figure 1.1. Typical stress-strain curve behavior of ﬁber-reinforced composite material ............. 3
Figure 2.2. Fit of approximate stress-strain law for shear ........................................................... 23
Figure 2.5. Layer nomenclature for laminate .............................................................................. 32
Figure 4.2. Direct implementation of- ........................................................... 58
Figure 4.3. Number of Permutations (Np) vs. Combinations (Nc). ............................................ 60
Figure 4.4. Derivation of reverse Pascal triangle ........................................................................ 62
Figure 4.5. Combinations for which F is computed .................................................................... 65
Figure 4.6. Pascal triangle and RPT table ................................................................................... 66
Figure 4.7. RPT offset array needed for indexing ....................................................................... 67
Figure 4.8. Segment of code for computing . - .......................................................... 68
Figure 4.9. Code segment illustrating unrolling of Do-loops ..................................................... 69
Figure 4.10. Do-Loop Unrolling vs. Straight Implementation ...................................................... 70
Figure 5.1. One-ply plate loaded in tension ........................... . ..................... 73
Figure 5.2. Variation of absolute r. m. s. shear strain at the center of= element 2

with excitation load level .......................................................................................... 75
Figure 5.3. Variation of normalized r.m.s. displacement at free comer nodes

with excitation load ................................................................................................... 76
Figure 5.4. Variation of normalized r.m.s. strains at the center of element 2

with excitation load level .......................................................................................... 76
Figure 5.5. Variation of normalized r.m.s. stresses at the center of element 2

with excitation load level .......................................................................................... 77
Figure 5.6. Two-ply laminated plate loaded in tension..... ..... - - - ....................... 78
Figure 5.7. Variation of absolute r. m. s. shear strain at the center of element 2

with excitation load level......................... - - - - - ........................ 80
Figure 5.8. Variation of normalized r. m. s. displacement at free comer nodes

with excitation load level .......................................................................................... 80
Figure 5.9. Variation of normalized r.m.s. strains at the center of element 2

with excitation load level .......................................................................................... 81
Figure 5.10. Variation of normalized r. m. s. stresses at the center of element 2

with excitation load level.......................- - - -- - 81
Figure 5.11. Three-ply laminated plate loaded rn tension ............................................................. 82
Figure 5.12. Variation of normalized r.m.s. stresses at the center of element 2

with excitation load level .......................................................................................... 84
Figure 5.13. Variation of normalized r.m.s. strains at the center of element 2

with excitation load level .......................................................................................... 84
Figure 5.14. Twisting due to extensional loading for [30°], [30°l-30°] and

[30°/0°/-30°] at So = 8000 ......................................................................................... 85
Figure 5.15. One-ply plate loaded in shear ................................................................................... 86

vii

Figure 5.16. Variation of absolute r. m. s shear strain at the center of element 2

 

with excitation load level................... ...... - 88
Figure 5.17. Variation of normalized r. m. s. displacement at free comer nodes

with excitation level ................................................................................................. 89
Figure 5.18. Variation of normalized r.m.s. strains at the center of element 2

with excitation load level ...... , .................................................................................... 89
Figure 5.19. Variation of normalized r.m.s. stresses at the center of element 2

with excitation load level .......................................................................................... 90
Figure 5.20. Tree-ply laminated plate loaded in shear .................................................................. 91
Figure 5.21. Variation of absolute r.m.s. shear strain at the center of element 2

with excitation load level .......................................................................................... 92
Figure 5.22. Variation of normalized r.m.s. displacement at free comer nodes

with excitation load level ......................................................................................... 93
Figure 5.23. Variation of normalized r.m.s. strains at the center of element 2

with excitation load level .......................................................................................... 94
Figure 5.24. Variation of normalized r.m.s. stresses at the center of element 2

with excitation load level ......................................................................................... 94
Figure 5.25. One-ply plate with ﬁber orientation or ...................................................................... 95
Figure 5.26. Variation of absolute r.m.s. shear strain at the center of element 2

with excitation load level .......................................................................................... 97
Figure 5.27. Variation of normalized r.m.s. displacement at free comer node

with excitation load level ......................................................................................... 97
Figure 5.28. Variation of normalized r.m.s. strains at the center of element 2

with excitation load level .......................................................................................... 98
Figure 5.29. Variation of normalized r.m.s. stresses at the center of element 2

with excitation load level .......................................................................................... 98
Figure 5.30. Two-ply plate with ﬁber orientation or ..................................................................... 99
Figure 5.31. Variation of absolute r.m.s. shear strain at the center of element 2

with excitation load level ........................................................................................ 101
Figure 5.32. Variation of normalized r.m.s. displacement at free comer node

with excitation load level ........................................................................................ 101
Figure 5.33. Varariation of normalized r.m.s. strains at the center of element 2

with excitation load level ........................................................................................ 102
Figure 5.34. Varariation of normalized r.m.s. stresses at the center of element 2

with excitation load level ........................................................................................ 102
Figure 5.35. One-ply plate with ﬁber orientation (1 .................................................................... 103
Figure 5.36. Variation of absolute r.m.s. strain at the center of element 2

with excitation load level ........................................................................................ 105
Figure 5.37. Variation of normalized r.m.s. displacement at free comer node

with excitation load level ........................................................................................ 106
Figure 5.38. Variation of normalized r.m.s. strains at the center of element 2

with excitation load level ........................................................................................ 106

viii

Figure 5.39. Variation of normalized r.m.s. stresses at the center 0f element 2
with excitation load level .................................................................................... 107

1 . Introduction

1.1 General

During the last two decades, research and development of laminated composite struc-
tures has grown at an extremely rapid pace. It is becoming apparent and that more and more
composite materials will be used in the design of structures, especially for applications in
which the strength to weight ratio is of primary importance. Due to their high strength to
weight and stiffness to weight ratios, advanced composite laminates made a direct impact
in the aerospace industry over the last two decades, and are now slowly being introduced
in automobile, ship building and other industries. For the most part deterministic dynamic
analysis has been used in the analysis and design of composite components. However, re-
cent advances in random vibration analysis allows more realistic techniques to be used
since most composite components are exposed to stochastic dynamic loads. Space vehicle
components made for NASA and the Air Force are now required to be designed for random
loads. Major ﬁnite element program developers have begun to respond to these needs and
have recently incorporated linear random vibration capabilities in their codes. However,
very little work has been done on the random vibration analysis of elements made of com-
posites which exhibit strongly anisotropic and moderately nonlinear behavior. There is an
important need to address this deﬁciency, to develop suitable techniques for the random vi-
bration analysis of composite components and to incorporate these techniques into general

purpose ﬁnite element codes so that they may be widely used in analysis and design.

One of the most important differences that ﬁlamentary composite laminates have
over the traditional materials (such as aluminum and steel) used in aircraft, ships, etc., is
their anisotropic behavior. Another important feature is that the stress-strain relations ex-
hibit signiﬁcant nonlinearities even for modest loads, when the loading is not parallel to the

ﬁlaments or when the loading involves shear (Hahn and Tsai 1973, Hahn 1973). For the

accurate analysis of such structures which exhibit nonlinear behavior arising due to geom-

etry and/or material properties. It is necessary to include the nonlinearity in the analysis.

In this study a procedure for the computation of the nonlinear random vibration re-
sponse of laminated anisotropic plates modeled using ﬁnite elements is developed using the
method of statistical linearization. Geometrical nonlinearity occurs due to structural con-
ﬁguration or large displacement. Only physical nonlinearity is included in this present

study.

This chapter describes the objectives of the present study, and presents a brief litera-

ture review of related studies.

1.2 Objectives

The main objective of the present work is to develop a procedure for the nonlinear
elastic random vibration analysis of laminated composite plates. The exact solution of the
nonlinear stochastic differential equations governing the dynamic response of systems is
possible only for very simple systems (see literature review). For more complex systems,
approximate methods must be used. Due to practical reasons it is crucial that the method be
capable of dealing with structures modeled using ﬁnite elements, and the method of equiv-

alent linearization is suited for this purpose. The excitation is assumed to be Gaussian.

Furthermore, it is hoped that the material presented will stimulate and enhance further
research on the random response of other laminated composite systems. The computer code
written as part of this work can be easily generalized to analyze laminated shells, and can

be modiﬁed to include higher order shear deformation.

1.3 Literature Review

1.3.1 Physlcal Nonlinearlty
The linear elastic theory of ﬁber-reinforced composite materials is well developed

(see, for example, Jones). However, most composite materials exhibit mildly nonlinear

0'1 0'2 1:12

A
E‘ 2 i" I
2
v. r, v <—
>51 >82 >7

12

 

 

 

 

 

 

Figure 1.1 Typical stress-strain curve behavior of ﬁber-reinforced composite material

stress-strain behavior in at least one principal material direction. The stress-strain curve in
the ﬁber direction of unidirectionally reinforced lamina is linear even at high stress level.
However, the stress-strain response for loading transverse to the ﬁbers is often somewhat

nonlinear. Moreover, the shear response is quite nonlinear.

The degree of nonlinearity varies from composite to composite and is due mainly to
the nonlinear matrix material, which signiﬁcantly affects the transverse modulus 15‘2 and
the shear modulus 012 of the composite. The effect of the nonlinear matrix material on the
longitudinal modulus E1 and Poisson’s ratio v12 is shown with micromechanics analysis
to be negligible for normal combination of ﬁbers and matrix materials. Speciﬁc examples
of ﬁber-reinforced composite materials with nonlinear stress-strain behavior include boron/
epoxy with slight E2 nonlinearity but a strong G12 nonlinearity. On the other hand, metal-
matrix composites such as boron/aluminum have strong E2 and G12 nonlinearities.Three-
dimensionally ﬁber-reinforced composites such as carbon/carbon have nonlinearities in all
principal material directions. The nonlinearities for all these materials are more apparent
with increasing temperature and moisture content. Thus, analysis of composites should in-

clude the effect of nonlinear stress-strain behavior.

Various investigators have attempted to include material nonlinearities in the analysis

of composite materials.

Petit and Waddoups (1969) employed a piecewise linear method. According to this
method, incremental stress-strain relations are ﬁrst obtained at each state of strain and then
the over-all behavior of the laminate is calculated by integrating the incremental stresses.
The ﬁrst increment in the laminate strains is calculated with the assumption that the lami-

nate behaves linearly over the applied stress increment, i.e.,

[A2] = lAl,,lAcrl,,+1 (1.1)

n+1

The increment in the laminate strains, As, is added to the previous strains to determine the

current total laminate strain
[2]“, = [ts],,+[Ae},,+1 (1.2)

The individual lamina strains may be computed using rotational transformations. Account-
ing for the orientation of the ﬁbers in each lamina, the lamina constitutive equations are ex-

pressed as follows:

0'1 Q11 Q12 0 81
(T2 = Q12 Q22 0 82 (1.3)
112 n O O Qoo n 712 n

the stiffness matrix for (n+1) th stress increment can then be calculated.

Hahn and Tsai (1973) derived a stress-strain relation which is linear in uniaxial load-
ing in the longitudinal and transverse directions, but nonlinear in shear. Their theory was
based on the strain energy density function which includes a fourth order term:

—al: 1 l l l
W = 55110? + 5S2”: + Slzcrlo2 + 5566th + 2576on2 (1.4)

where W is the complementary energy density function. They have explicitly shown that
only one fourth order constant in needed to account for the nonlinear shear behavior of an
off-axis composite lamina. In their study, Kirchoff’s hypothesis that each lamina in the
laminate is in the same state of membrane strain as that of the laminate was used. The
stress-strain relation which takes account of frequently observed nonlinear behavior in in-

plane shear is

Sr 511 S12 0 Cl 0
{£2} = 512 522 0 {02} +§661f2{ O } (1.5)
712 0 0 S66 112 1:12

Their predictions of strain response under uniaxial off-axis loading agree fairly well with
measurements by Cole and Pipes as well as the theory and experiments concerning the off-

axis behavior.

Hashin, Bagchi, and Rosen (1974) implemented the Ramberg-Osgood stress-strain
relations to represent the nonlinear response of the lamina subjected to transverse and shear
loads. The behavior in the direction of ﬁbers was assumed to be linear. The Ramberg-Os-
good model is quite ﬂexible and able to represent varying degrees of nonlinearities as well

as hysteresis.

Sandhu (1976) used an approximation of stress- strain behavior under biaxial normal
stress states to predict equivalent multiaxial strain increments. The incremental constitutive

relationship used is deﬁned under the following assumptions:
1. The increment of strain depends upon the strain state and the increment of stress.
2. The increment of strain is proportional to the increment of stress.

using these assumptions, the incremental constitutive law can be written as

day. = S (8"!) do" (i,j, r,s = l, 2, 3) (1.6)

ijr:

where deij, do" are strain and stress increments and S if” is a function of strains 31;- Under

generalized plane stress, the above equation reduces to:
do. = So}. (so) dd}. (1.7)

assuming that the lamina remains orthotropic at all load levels (this assumption was justi-

ﬁed experimentally) the above equation can be written as
[do]. = [Cloldelo (1.8)

This model is similar to the one used by Pent-Waddoups (1969) since both methods
use lamination theory to generate the laminate compliance which is used to compute lami-
nate strain increments under applied stress increment. These laminate strain increments are
added to those obtained previously to detDrmine current strains. In the Pent-Waddoups
technique these strains are used to compute the laminate compliance for the next load in-
crement. This technique is essentially a predictor type. In Sandhu’s analysis, the strains are
used to determine the average laminate compliance for the same load increment, and a new
set of laminate strains are obtained. This procedure is repeated until the difference between
two consecutive sets of laminate strains is less than a prescribed tolerance. The results of
this approach are in good agreement with Cole and Pipes’ data and those of Hahn and Tsai

as well.

Jones and Nelson (1976) developed an orthotropic material model in which the non-

linear mechanical properties are functions of the strain energy density,

U 9
0i

where MPi is the mechanical property (e. g. modulus of elasticity) for the i th stress- strain

curve, and for lamina under plane stress U is given by

U = (6181 +02£2+112712)/2 (1.10)

where A o, B,- and C,- are the initial slope, initial curvature, and change of curvature of the
ith stress- strain curve. The quantity U0 is used to nondimensionalize the strain energy por-
tion of the mechanical property equation. The J ones-Nelson model is used in an iterative
procedure which converges to the state of stress and strain corresponding to an equivalent

linear elastic body.

The mechanical properties in this model cannot be deﬁned for strain energies greater
than or equal to a speciﬁc value of strain energy U (value of strain energy where the ap-
proximate mechanical property curve crosses the U axis). The value of U at which the me-
chanical property becomes negative is U = 3"”. Thus, U is largest for stress-strain
curves with low initial curvature and low rate of change of curvature. Accordingly, the
Jones-Nelson model cannot be used as extrapolation for energies as large as U but must be

restricted to energies less than or equal to U = %oé (6' is the maximum stress). This range
is not sufﬁcient to treat practical problems. Hence, modiﬁcations of this model are essen-
tial.

Jones and Morgan ( 1977) have extended the J ones-Nelson model to include treatment
of more pronounced nonlinearities. Their approach is based on the concept that the stress-
strain curve be connected to a straight line with equation 6 = me + 0'0 instead of merely
specifying the slope m. This approach is useful in ﬁtting the stress-strain curves while si-
multaneously considering the statistical nature of failure data which can be described with
such a line. A disadvantage of this approach is that the ﬁt of the actual stress-strain data is

not as good as with other approaches.Their computed strains agree with strains measured

by Cole and Pipes about as well as the theories mentioned previously.

Amijirna and Adachi (1978) presented a simpliﬁed method of predicting the nonlin-
ear stress—strain curves for an unidirectionally orthotropic lamina, and symmetric biaxial
laminates. The analytical procedure is based on linear elastic analysis with the application

of classical laminated plate theory (L.P.T) to the small stress increments of the stress-strain

curve. Because it is assumed that only the nonlinear component of the principal in plane
shear response governs the nonlinearity of the unidirectional lamina stress-strain curves,

the following relation between A712 and on was used

(AUX) [00829 {1 - (611),, ( (511) ,o " (312) ,o) }1
(1.11)

+ [sin20 (516),, ( (311) n - (512) ")1 = (516),,(13712)
The nth stress increment, (on) n, in the loading direction corresponding to the nth incre-
ment of the in-plane shear strain component, (A712) n, is calculated from the above equa-
tion. The complete nonlinear stress-strain curve was predicted for various laminations. The
nonlinear stress-strain curves for various cases could be estimated and the results have con-

siderably good agreement with experimental ones.

Chou and Takahashi (1987) used the stepwise incremental analysis proposed by Petit
and Waddoups (1969) to predict the non-linear stress/strain responses of flexible ﬁber com-
posites. The uniaxial tensile stress/strain relations were obtained for several types of com-
posites containing glass or kevlar ﬁbers in an elastometric polymer. Their theoretical

procedure considered the ﬁber geometric nonlinearity as well as the material nonlinearity.

Chou and Lou (1988) developed a constitutive model based upon the Eulerian de-
scription to account for material nonlinearity for ﬂexible ﬁbers. They used a complemen-

tary energy function to derive the following material nonlinear stress-strain relation

{3:} = {53110)} (Hz)
where
511 = Slr+srll°r+srlll°i S12 = S12 = S21
S22 = 5221'5222021'522220221 816 = 516606’ 816 = 2516606
566 = S66+S6666°% S26 = 562 = 2522666206

the terms S11 , S22, S12 and Sm5 are needed for linear deformations; the terms Sm, S222
are needed for representing the birnodulus behavior in the axial and the transverse direction
respectively; the terms S1111 , $2222 and S6666 are the nonlinear terms. Although this model
resembles Hahn & Tsai’s model, the former has the capability of treating complicated prob-

lems with more than one nonlinearity in the material property.

1.3.2 Shear Deformatlons
Classical plate theory can yield signiﬁcant error for moderately thick composite 1am-

inae because transverse shear deformation is neglected. It is well known that transverse
shear deformation is signiﬁcant for thick plates, and this is especially true for composites
since the shear moduli of polymer matrices are signiﬁcantly lower than the extensional
moduli. While a ﬁrst-order shear theory (Reissner 1945, Midlin 1951) is adequate for plates
made of conventional materials, a higher-order shear theory is usually required for compos-
ite laminates (Reddy 1990, Noor and Burton 1989). Higher-order shear theories are usually
adequate only for global modeling (i.e., prediction of displacements, natural frequencies
and buckling loads), and are not sufﬁciently accurate for stress ﬁeld computations. Local
layer-wise models that represent each layer as a homogeneous anisotrOpic continuum are
usually required for accurate stress computations, but these often greatly increase the size

of the problem.

Numerous higher-order shear theories have been proposed (e.g., Nelson 1974, Whit-
ney and Sun 1974, Lo 1977, Reddy 1984, and many others). All of them use through-the-

thickness displacement assumptions of the form (Reddy 1987)

u (x1,x2,x3) u0 (xvxz) U (x1,x2,x3)
{v(xl,x2,x3) } = {v0 (x1,x2) } +{ V(x1,x2,x3) } (1.13)

w (x1. x2. x3) wo (x1. 21;) W (x1. x22. 1:3)

Where “0’ and v0, and w0 are the displacement components of the reference plane
x3 = 0, and U, V, and W are functions of x3 which vanish at 13 = 0. The different theo-

ries can be identiﬁed by the assumed functional dependence of U, V, and W on x3.

1.3.3 Stochastlc Dynamlc Analysts
General theory

The general problem of random excitation of physical systems was first investigated
theoretically by Einstein (1905) and was generalized and extended by Smoluchowski
(1916) in context of the theory of Brownian motion. In 1931, Kolmogorov derived a precise
mathematical formulation of the equations governing the probability densities satisﬁed by
such processes. Contributions of major importance were also made by Fokker, Planck,
Burger, Furth, Omstein, Uhlenbeek, Kramers and others. A number of important papers
from this era have been collected in the book by Wax (1954).

Stochastically excited linear systems have been studied in great detail and several an-
alytical techniques exist for treating both stationary and nonstationary problems. Unfortu-
nately, many structures of engineering interest cannot be considered linear and the
techniques for analyzing nonlinear systems are not nearly so well developed. Some simple
nonlinear systems can be analyzed exactly by means of the Fokker-Planck equation. How-
ever, for most nonlinear systems exact solutions are not available. A number of approxi-
mate techniques have been applied successfully to simple one DOF systems, but much less
has been done in the analysis of dynamical systems with more than one DOF. Several po-
tentially useful techniques for the analysis of such systems have been developed, but most
of these techniques are either very difﬁcult to apply or their application is restricted to only
a small class of problems. The statistical linearization technique shows considerable prom-
ise in this regard for it is not limited by the restrictions commonly imposed on the other ap-

proaches. In addition, this approach can be made quite direct and relatively easy to apply.

10

The earliest work on the problem of random excitation of nonlinear system was that
of Andronov (1933), who used the Kollnogorov-Fokker-Planck equations (Kolmogorov,
1931) to study the motions of general dynamic system subject to random disturbances.
Kramers (1940) used this technique to study chemical reaction rates. Caughey and Dienes
(1961), Lyon (1961), Klein (1964), and Herbert (1965) have used the Fokker-Planck equa-
tion to study the response of nonlinear dynamical systems to white noise excitation. Barrett
(1961), Merklinger (1963) and Stratonovich (1963) have applied the technique to solve

nonlinear control problems.

In almost all these investigations only ﬁrst order statistical properties were obtained.
While ﬁrst order statistics are important parameters in the description of random processes,
there are numerous applications where additional statistical information is required. For ex-
ample, the spectral density of a random process requires a knowledge of the second order
statistics of the process. A number of approximate techniques have been developed to ob-
tain second order statistics for the response of nonlinear systems to random excitation.
Booton (1954) and Caughey (1959) independently developed the method of equivalent lin-
earization, which is simply the statistical extension of the well known equivalent lineariza-
tion technique of Krylov and Bogolinbov (1937). Crandall (1961) developed a
perturbational method based on classical perturbation theory. Payne (1967), Wong (1964),
and Atkinson (1970) have developed approximate techniques based on eigenfunction ex-

pansions and variational techniques.

Many problems in mechanics and related ﬁelds, involving the response of dynamical
systems to stochastic excitation can be modeled as systems of ﬁrst order differential equa-

tions of the form

:11: = a(t,x) + Z bk(t,x)-g;wk(t) (1.14)
k=l

ll

where x, a, bk, 1: = l, 2, ..., m are m vectors and the wk (1‘) for k = 1, 2, ..., n are inde-
pendent processes of Brownian motion.

In nonlinear stochastic differential equations the structure of the transition probability
density function is usually much more complex than that of linear systems, and cannot be
obtained in a direct fashion. The most common method of obtaining the transition proba-
bility density function for nonlinear stochastic differential equations is through the use of

the Kolmogorov-Fokker-Planck equations:

361 ._
.37 +a5q _ 0 (1.15)

where ax is expressed in the following way
a = 3—(A q) 342—09 -q) (1.16)
4‘ axo 1’ Zaxjﬂx,‘ 1"

q must satisfy the initial condition q (x, to] go, to) = 8 (x - x0)
Exact solutions of the Fokker-Planck-Kolmogorov equations have been found for two
types of stochastic differential equations:
1. systems of linear equations
2. certain ﬁrst order nonlinear equations.
The steady-state probability density can always be obtained for ﬁrst order nonlinear
systems, and has been found for a certain class of coupled nonlinear oscillator problems.

Caughey and Atkinson (1968), obtained the transition probability density function for a

class of piecewise linear systems excited by Gaussian white noise.

The exact steady-state probability density for ﬁrst order nonlinear system excited by
Gaussian white noise can readily be determined by direct integration. The Fokker-Planck

equation for a stochastic differential equation of the type

dx = —f(x) dt+dw (t) (1.17)

12

is given by

where D is a positive constant. Direct integration of the above equation yields the steady-
state probability distribution function q: (2:) providing that it satisﬁes the condition
34

if"

q. (x) = C‘exp[-jf(§) mag] (1.19)
0

where C is the normalizing constant given by

C = j exp[-jf(§) /Dd§]dx (1.20)
0

—00

Exact solutions of the steady state probability density function for nonlinear equa-

tions of second order excited by white noise have been found only for equations of the form

it'+f(H)Ji+g(x) = w(r) (1.21)
E[dw(t) 21 = 2Ddt (1.22)
H = $122+ {3 (man (1.23)

The associated Fokker-Planck equation is easily shown to be

2
gs; = -x§§+§;1(stx) +f(H)i)ql +0373 (1.24)

and by direct integration, the steady-state probability density function is given by

13

r H ‘
exp -(1/D) jf(n)dn
4.0.x) = ,, ,, ‘ ,3 ; (1.25)

I [exp -(1/D)jf(n)dn dxdrt
-- 0

The same technique can be applied to the system of coupled nonlinear equations.

 

 

 

h

The main advantage of the Fokker-Planck equation approach over all of the other ap-
proaches considered here, including statistical linearization, is the exact solution it pro-
vides. However, this advantage must be balanced by the fact that such solutions have only
been found for certain restricted classes of problems. Caughey showed that the stationary
Fokker-planck equation can be solved and the ﬁrst order probability density of the Mark-

ovian response process can be obtained provided:

1. The only energy dissipation in the system arises from damping forces that are pro-

portional to the velocity.
2. The excitation is Gaussian white noise.

3. The correlation function matrix of the excitation is proportional to the damping ma-

trix of the system.
4. The restoring force vector of the system is derivable from a potential.

The solution of the time-independent Fokker-Planck equation under these conditions rep—
resents a very signiﬁcant accomplishment. However, it is a fact that most systems of prac-
tical interest do not satisfy the above mentioned conditions. Since, in general, it is not
possible to obtain exact statistics for the response of nonlinear system excited by white

noise, a number of techniques have been developed to treat a broader class of problems.

One of the approximate techniques based on the use of the Fokker-Planck equation
known as the eigenfunction expansion was used by Wong (1964) and Payne (1967, 1968)
for ﬁrst order system, in which case the Sturm-Liouville theory applies. For many higher

order systems the Fokker-Planck-Kolmogorov equations are of degenerate form. Atkin-

14

son(1970) has used eigenfunction expansion techniques for second order systems excited
by white noise. Unfortunately, for many nonlinear systems of the second or higher order,
the eigenvalue problem cannot be solved exactly. In some cases, perturbation techniques
may be used to extend the class of systems which may be analyzed by this method. This
requires that eigenvalues and eigenfunctions of an associated Fokker-Planck operator be
known a priori, a situation which unfortunately occurs rather infrequently. The Rayleigh-
Ritz method (Mikhlin, 1964) has been widely used to approximate the eigenvalues and

eigenfunctions of the linear differential operators.

An approximate technique that has been used is the normal mode approach for sta-
tionary random response problems. This technique reduces a set of coupled nonlinear sec-
ond order differential equations to a set of equations that include coupling only in the
nonlinear terms, and is applicable for statistically uncorrelated excitations. The reduced
equations may then be subjected to a number of approximate solution techniques such as
the method of statistical linearization. In order to successfully apply this technique to a giv-

en multi-DOF dynamical system, the system must satisfy the following two conditions:

1. The linear system obtained by neglecting all system nonlinearities must possess

normal modes.

2. The correlation function matrix of the excitation must be diagonalized by the same

transformation that diagonalizes the linear mass, damping and stiffness matrices.

While the ﬁrst condition may be acceptable in a number of situations, the second condition
is quite restrictive. In particular, it precludes the application of the technique to all dynam-
ical systems that are excited at only few points in space. However, such systems are of con-
siderable interest physically and thus, the second restriction represents a rather severe

limitation on the usefulness of the normal mode approach.

Another approximate method is the perturbation approach where the Stochastically

excited nonlinear system is treated in the same way as a deterrninistically excited system.

15

A power series expansion in terms of a small parameter which speciﬁes the size of the non-
linearity represents the solution for such systems. Substituting the assumed solution form
into the original equations of motion and equating coefﬁcient of like powers of the nonlin-
earity parameter then yields a set of linear differential equations for the terms in the solution
expansion. A ﬁrst order approximation is obtained by solving two linear systems. The ﬁrst
is the system which is obtained by setting all nonlinearities equal to zero, and the second is
a system having an excitation which is a function of the solution of the ﬁrst system. Prac-
tically speaking, it is almost impossible to extend this procedure beyond the ﬁrst approxi-
mation except in very trivial cases since the probability density of the ﬁrst order correction

term is non-Gaussian.

In their analysis of nonlinear systems, Lyon (1960) and later Crandall (1963) applied
the perturbation approach to a continuous nonlinear system. This proved useful in a variety
of applications since the approach is not restricted to cases of uncorrelated excitation. The
major source of difﬁculty however, arises in the solution of the equation for the ﬁrst -order
correction term. This is because generally, this equation has a non-Gaussian excitation. The
complexity of this can be overwhelming even when a relatively simple multi-DOF system
is considered. A scheme for reducing the computational difﬁculties was presented by Tung
(1967). By using Foss’s complex-mode approach to solve for the ﬁrst-order correction term
the computational difﬁculties were alliviated, but the overall complexity of the approach is

still considerable

A third approximate technique is statistical linearization. This technique was devel-
oped independently by Booton(1954) and Caughey (1959). It is applicable to nonlinear sto-
chastic differential equations. This approach overcomes many of the limitations
encountered in the previous methods for studying the stationary random response of multi-
DOF systems. It is based on the concept of replacing the nonlinear system by an equivalent

linear system while minimizing the difference between the two systems. This concept has

16

been widely used in control theory and in the study of multi-DOF dynamical systems with

stochastic excitation.

The accuracy of an approximate analysis based on statistical linearization is difﬁcult
to predict in general, but it is usually assumed that the approximate solution obtained is ac-
curate to ﬁrst order in the parameter specifying the size of the system nonlinearity. Wan
and Yang (1972) showed that the accuracy of statistical linearization approach appear to be
well within the limits of practical engineering usefulness. When studying a speciﬁc nonlin-
ear system, the accuracy of statistical linearization should be assessed by comparing its re-

sults with results obtained by Monte Carlo simulation.
Applications to composites

Much of the recent work related to analysis of composites has been on the develop-
ment of suitable ﬁnite elements for composite plates and shells. Studies considering static
loading far outnumber those that consider dynamic loading. There have been only a few
studies considering stochastic dynamic loading, and some of these are briefly discussed in

this section, along with their limitations.

Witt and Sobczyk (1980) were the earliest researchers to study the random vibration
of laminated plates. They used an analytical series formulation and modal analysis to study
the stochastic bending response, but considered only linear behavior. Witt (1986) later ex-

tended the formulation to include transverse shear deformation in the plate.

Cederbaum, Elishakoff and Liberscu (1989) used an analytical formulation that in-
cluded a ﬁrst-order shear deformation theory together with modal analysis to study the lin-
ear random vibration of composite plates. Cederbaum (1990) later extended this to include

viscoelastic material behavior.

Mei and Prasad (1989) appear to be the only researchers who have considered the
non-linear random vibration of composite plates. They included transverse shear deforma-

tion, but considered non-linearity arising from large deformations and not from the consti-

17

tutive laws. They used an analytical formulation together with the method of equivalent
linearization, and considered only a single modal response equation obtained through an

approximate Galerkin approach.
There are several limitations in the works cited above:

1) None of the investigations include non-linear material behavior, which is quite im-

portant for ﬁlamentary composites.

2) They all use analytical formulations, rather than ﬁnite element based formulations,
and therefore are readily applicable only for plates with simple geometries and boundary

conditions (e.g., rectangular plates with simply supported or ﬁxed boundaries).

3) They cannot be easily used in existing large-scale computer programs which are

predominantly based on the ﬁnite element approach.

Harichandran and Hawaaari (1992) performed nonlinear random vibration analysis of
ﬁlamentary composites loaded in extension. They approximated the nonlinear shear stress-
strain law in terms of a power series. The result of their work has shown that the non-lin-
earity in the constitutive law results in signiﬁcant increases in the shear strains, but does not

signiﬁcantly affect the normal strains.

18

2. Finite Element Formulation

2.1 General

The advent of computers has opened new horizons in the ﬁeld of engineering design.
In the realm of analysis for engineering design the ﬁnite element method has emerged as a
powerful tool for modeling and analysis of solids and structures of complex geometries and
variable material properties. Although the original applications of the ﬁnite element were
in the area of stress analysis, its usage has spread to many other areas having similar math-
ematical bases such as heat transfer, fluid flow, electric and magnetic ﬁelds and several oth-
ers. The ﬁnite element method is a numerical procedure which enables a problem with an
inﬁnite number of DOF to be converted to one with a ﬁnite number in order to simplify the

solution technique.

The primary objective of the use of the ﬁnite element method in the analysis of struc-
tures is to calculate approximately the displacements, strains, stresses and other responses
of the structure. The power of the method resides mainly in its versatility. The method can
be applied to a variety of structures with arbitrary shape, loads, and support conditions. The

ﬁnite element mesh can mix elements of different types and physical properties.

Today, the concept of ﬁnite elements is a very broad one. A most important formula-
tion, which is widely used for the solution of practical problems, is the displacement-based
ﬁnite element method. Due to its simplicity, generality and good numerical properties, this

formulation has been used in major general-purpose analysis programs.

The basic process of the displacement based ﬁnite element method is that the com-
plete structure is idealized as an assemblage of individual structural elements. The element
stiffness matrices corresponding to the global degrees of freedom of the structural idealiza-
tion are calculated and the total stiffness matrix is obtained by addition of element stiffness
matrices. The solution of the equilibrium equations of the assemblage of elements yields

the element displacements which are used to calculate the strains and stresses.

l9

In two-and three-dimensional ﬁnite element analyses, we basically use the Ritz anal-
ysis technique with trial functions that approximate the actual displacements. the result is
that the differential equations of equilibrium are not satisﬁed in general, but this error is re-

duced as the ﬁnite element idealization of the structure is reﬁned.

2.2 Constitutive Equations
In the elementary theory of plates, certain assumptions are made regarding the stress

distribution. Certain stresses are assumed to predominate, while others are neglected. In the
theory of bending of beams and plates, the normal stress, 0'3 , which is perpendicular to the
beam or plate midplane, is assumed to be negligible in comparison to the normal stresses,
61 or 02. In other words, due to the geometry of the plate, the magnitudes that 0'3 can as-
sume are several orders of magnitude less than the values of 01 and 0‘2 which are induced
by bending. Also, the assumption is made that any line perpendicular to the plate midplane
before deformation remains perpendicular to the midplane after deformation, and it suffers
neither extension nor contraction. As a result, the shear strains, 713 and 723 , and the normal
strain, 83, are zero. The shear stresses, '1:13 and 1:23 , are also neglected. These assumptions,
which are made in the classical theory of plates, are termed Kirchoff’s hypothesis. For thin
plates, the hypothesis results in the existence of a plane stress state. Thus one pertinent as-
sumption in establishing the constitutive or stress-strain relationships for the laminae of a
laminated composite is that the laminae, when in the composite, are in a plane stress state
- which is not to say that the interlaminar shear stresses, 113 and 123, will not be present
between laminae once they are placed in the composite. However, these stresses may be
neglected in establishing the laminae constitutive relations on which the laminae stress-

strain relations will be formulated.

It is commonly known that most uni-directional ﬁlamentary composites display
orthotropic characteristics, and behave essentially linearly when loaded parallel to or per-

pendicular to the ﬁber directions. However, when loaded in shear, they exhibit signiﬁcantly

20

nonlinear behavior. Fig. 2.1 shows the global coordinate system x-y and material coordi-

nate system 1-2 for a typical uni-directional composite element.

In view of the advantages and the disadvantages of the various nonlinear stress-strain
models described in chapter 1, the model proposed by Hahn and Tsai seems to obtain an
explicit relation for e in terms of the loading and known material parameters including the

nonlinear shear term. Hence, this model is adopted in this study.

Based on experimental results, Hahn (1973) proposed the following strain-stress (or

inverse) law for plane stress problems:

81 5 ll S12 0 or 0
Y12 0 0 S66 112 112

in which 81 and (II are the normal strain and stress in the l-direction (i.e., along the ﬁber
direction), 62 and 0'2 are the normal strain and stress in the 2-direction, 712 and 112 are the
engineering shear strain and shear stress corresponding to the material coordinates 1-2, and

the square matrix on the right hand side is the linear compliance matrix [S] . The cubic

         

    
 
 
  

    

 

"/////V

\\

.s\\\

Figure 2.1 Coordinate systems

     

21

variation of 712 as a function of 1:12 in Eq. 2.1 describes the softening behavior of ﬁlamen-
tary composites loaded in shear, and is sufﬁciently accurate. In ﬁnite element application,
the stress-strain law is preferred. The inverse relation of the cubic shear strain-stress law

may be written as 112 = g (712) , where g (712) is the real solution for 1:12 of the cubic

3 _ _
5*66‘121'366112 712 ’ 0

, s (712) . .
letting f (712) = T— - Q66, the stress-strain law may be wrltten as
12
61 Qll Q12 0 £1 0
{02} = Q12 Q22 0 {£2} +f(712) {0 }
1:12 0 0 Q“ 712 712
(2.2)
Qll Qrz . 0 £1 0 0 0 81
= Q12 Q22 0 {£2} +f(712) [0 0 0] {82}
0 O Q66 712 O 0 712
or, more compactly as
{0'} = [Q] {8'} +f(712) [diag(0.0.1)l {e'} (2.3)

in which [Q] = [S] '1. In terms of elastic moduli and Poisson’s ratios, the elements of

[S] and [Qlare
511 = l/E1,S22 = 1/52, 512 = -v12/E2 = —v21/El,and 566 = l/G12
Qll = El/(l-V12V21),Q22 = E22/(1—v12v21)
Q12 = "1215'1/(l "V12V21) = v2152/(1'V12V2l)

For Eqs. 2.1 and 2.2 to be exact inverse relations, the function f ( 712) is related to the
solution of the cubic equation mentioned earlier, and contains terms involving fractional

powers of 712. This poses computational difﬁculties in the method of equivalent lineariza-

22

tion used to perform the approximate random vibration analysis. In order to facilitate the

analysis, it is assumed that the function f (712) may be approximated by

 

11% f 1 ﬂ r ' I ' I ' I ' T
10000

— Hahn and Tsai
------ Fifth Order Polynomial

 

ITFTI
r.

8000
7000 -

o
o
o
o
C
o
O
o
.0
a...
o
.0
0"
0..

5000-

Shear Stress (lb/ inz)

3000-
2000‘-
1000- -

l E J l 441 A

l

L

 

 

l n l n l n l 1 l n l n l 1 l n J 1

0 n
0.0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
Shear Strain

 

Figure 2.2 Flt of approximate stress-strain law for shw

fIle) = a1 12+027i2+m+0n7i§ = 2017i; (2.4)

i=1

Note that the nonlinear term in eq. 2.2 is f (712) 712, which is therefore being approximated

by an odd-powered series starting with the cubic term.

By suitable choice of the parameters a5, the stress-strain law can be made to approx-
imate the law in Eq. 2.1 for realistic values of 712. Fig. 2.2 shows the shear stress-strain law
given by Eq. 2.1 for Boron/ Epoxy Narmco 5505, and the approximate curve using Eq. 2.2
and 2.4 with n = 2. The material pr0perties used were: S u = 3.32 x 10"8 in2 lb"1,

.122 = 3.48x10‘7 in2 lb'1,S66 = 1.25x10'6in21b"1,S66"' = 1.53x 10'14 in6 lb'2

23

a1 = -2.259x109 lb/inz, a2 = 3.505x1012 lb/inz. The ﬁfth-order approximation

is satisfactory for strains up to 0.02.

The stress—strain law in terms of the global coordinate system, x-y, may be written as
{0} = [-Q-l{€}+f([T31{€})[7*l{8} (2.5)

inwhich to} = lo,o,o,,1T. {e} = legion 101 = [71"[Q1171’T.and['1'l

is the orthogonal rotational transformation matrix given by

cos 20 sin 29 2 sine c050
[T] = sin 29 cos 20 -2 sinOcosO (2.6)
-sin0 c030 sin0cosO COS 29 - sin 29

[T3] is a row matrix consisting of the third row of [T]"T,
andlr'l = lTl"ldiag(0.0.l)] [71'T.Notethatf([T31 {2}) =f(T,2) inEq.2.s.
since 916 = Q26 = 0. A convenient form for the transformed lamina stiffnesses, a, has

been given by Tsai and Pagano (1968):

Q11 = U1 + Uzcos (20) + U3cos (49)
Q22 = U1+ Uzcos (20) + U3cos (40)

Q12 = U4-U3cos (40)

(2.7)
Q66 = US-U3COS (46)

where

24

u1 = %(3Q11+3Q22+2Q12+4Q66)

U2 = $49.. 4222)

U. = gramme-201240..) (2.3)
U4 = %(Qll +Q22+2Q12—4Q66)

Us = éIeri’sz‘lez‘i‘iQoo)

Note that only three of the invariants U1, U 4 and U5 are independent. That is, these terms
remain costant regardless of the angle 0. Thus, examining Eq. 2.8 reveals that each of the
ﬁrst four terms (independent terms) is composed of a constant term plus terms which
change with the angle 0. Therefore, these are inherent lamina properties which are only de-
pendent on the material being used and do not change with orientation of the lamina. This

“invariant” concept is very useful in design with composite materials.

2.3 Finite Element Formulation using Classical Plate Theory
Numerical methods, such as the finite element method, are necessary in practical ap-
plications as they are able to model general geometries, boundary conditions, loading and
materials. In this section, the derived expressions needed for the evaluation of the in-plane
stiffness and mass matrices of the isoparamerric four-node rectangular element shown in
Fig. 2.3 , and the plate bending stiffness and mass matrices of the Melosh-Zienkiewicz-

Cheung (MZC) noncomforrning four- node rectangular element shown in Fig. 2.4.

2.3.1 ln-plane formulation
Isoparametric ﬁnite elements are used for the in-plane extensional response. The el-

ement displacements are interpolated as

4x = N'qul '1’” 2912*”. 34x3 ‘1‘” 4%: (29)

q), = N‘lqy1+N‘2qy2 +N'3qy3 +N‘4qy4 (2.10)

25

where qx and q), are the local element displacements at any point of the element, and q”.
and qyo, (i = l, . . ., 4) , are the corresponding element displacement at the element nodes.
The interpolation functions N‘ ‘- are deﬁned in the natural coordinate system of the element,
which has variables g and n that each vary from -1 to +1. The fundamental property of the
interpolation function N ' i is that its value in the natural coordinates is unity at node i and is
zero at all other nodes. Using these conditions the functions N ' ,- corresponding to a speciﬁc

nodal point layout could be solved for in a systematic manner.

 

 

 

Figure 2.3 Nodal displacements for isoparametric plane stress element

As shown in Figure 2.3 the nodal displacements are:

{q.e} = {qquyi} (i=1,2,3,4)

26

the in-plane displacements at the reference plane of a laminated composite plate can be ex-

pressed as
qx(C,n) _ [M1 0 N2 0 M3 0 N4 0 ] {q' }
quCrTI) O N'1 0 M2 0 N3 0 ”‘4 e

in which {q'e} is a vector of eight element nodal displacements, and N‘ o are the interpo-

lation functions:

N'1(§.n) = 31-(1-é) (l-n) (2.11)
N'2(§.n) = 71(1+§)(1-n) (2.12)
N'3(§,n) = §(1+§)(1+n) (2.13)
N'4(§.n) =-,-1,(1-§) (1+n) (2.14)

The strain ﬁeld within the element may be expressed as
{8} = [3'] {q'.} (2.15)

where the strain-displacement matrix is given by

~a 0. ”la 0 ‘
U; a (la—E 1 a Mix 0
[3'] = 0 33' [N'] = 0 5871' [N'] = 0 Ni, (2.16)
.9. 8 la 1 a Mr, N‘...
_ y 37c. 1,3; 03';

 

 

 

 

where N‘ i, x and N ' o, y represent partial derivatives of N' o with respect to x and y.

i’ 3): a i’ (I ’ ’ ’ ) (. )
N -—i--1N‘ --1234 218
ﬂy a I 3"." (I 9 r 9 ) (° )

27

"‘57 or :. .unl

or more explicitly,

or” = 71150-11) Mo, = -ﬁ(l-§)
or“: alga—n) N'o,,=-Zl-5(l+§)
N'.,. = 3‘;th N3, = gonad)
~'.,.=--41—a(1+n) N'.,,= 21-50—51

2.3.2 Out-oi-plane formulation .
Consider the plate-bending element in Fig. 2.4. It was originally developed by

Melosh , Zienkiewicz and Cheung (Melosh 1963, Zienkiewicz and Cheung 1964). As with

other elements of this type, it has only one displacement in the z-direction.

This element is said to be nonconforming because it does not have normal-slope com-
patibility at the edges. There are some practical reasons for considering nonconforming

plate-bending elements. General-purpose structural analysis programs usually permit six

 

e1;
2 D
N
t

T

QIH

aw,
3x

Figure 2.4 Nodal displacements for rectangular plate-bending element

28

”ﬂ

ﬁr:- f

nodal degrees of freedom, three displacements and three rotations. Compatible ele-
ments(C1 elements) are capable of preserving interelement continuity of the ﬁeld and its
first derivatives at the nodes, but not interelement continuity of all second derivatives of the
ﬁeld. Such elements posses an additional nodal degrees of freedom Ge. 39%;) which does
not ﬁt well into a typical general-purpose program. On the other hand, several incompatible
plate elements exist with only nodal displacements and rotations as degree of freedom.
These elements ﬁt well into general-purpose computer codes and permit the analysis of a
variety of general structures. Such elements are particularly appealing because they can be

used with plane stress elements to model plates and shells.

The displacement function chosen for this element is a complete cubic of ten terms
plus two quadratic terms.
w = C1 + c2§ + C3T] + c4§2 + c5§n + c6112 + 0,? + csﬁzn

2 3 3 3 (2°19)
+ c9§n + c101] + cu§ t] + c12§n

Since the element does not preserve normal slope continuity between adjacent elements
boundaries, it violates one of the conditions for convergence to the exact solution with a
reﬁned mesh. Convergence, however, has been proved by Walz, Fulton and Cyrus(l968),
and numerical results obtained by Zeinkiewicz (1977) demonstrate the convergence rates
and accuracy for both displacements and bending moments. It should be mentioned that
rectangular elements are the most straight forward type of plate-bending elements. Trian-
gular and quadrilateral elements are more versatile for general structural analysis. Zienk-

iewicz presents an excellent discussion of other plate-bending element formulations and
types.

The displacement shape functions for the above element can be expressed as follows

Mo = (1+to) (1+no) (2+§o+no-§2-n2) (2.20)

OOH-

29

. r71
it

19; r-

N".-2 = --;-bn,(1+§o)(l-no) (1+no)2

N"a = gnarl—to) (1+no) (1+§o)

whet-tel;0 = ﬁg and no = non (i=1. 2. 3. 4)

The curvatures at any point within the element are given by

xx
{19} = VZW'] {q".} = 13"] {4",}

K
xy

where the generalized strain-displacement matrix [B"] is

1v"t'l,xx N"

[3"] { 32 32 2 321 [N"l N' N"
i = 2 2 a a i = i1.” 1’2.”
3): By J: y 2N" 2N‘

12, xx

 

t'l,xy i2,xy

The elements of [B"‘.] are:
H l
N il,xx = 2? (l +7191) (€2-3Sgg" 1)

N" =0

i2,xx
Nni3,xx = '4—lo,§?(1+II,-Tl)(1+3§t§)

_1_

Woo, = 4b2(1+§,-§)(hf-311,114)

mm, = -§n?(1 +§,.§) (1 +1191) (1 412112)

N‘ 0

i3, yy =

N"
N"
2N‘

13, xx
i3, yy
0

1'3, xy

2N3“, = halt—b [n,§o(4 - 352 - 3712) + 25,71,- (§o§+n,-n) —2 (ht-+11%,”

30

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)

(2.30)

(2.31)

F—mﬂm“f1

, l
2- ..., = 751131.11 +1151) (1 -3n,-n) (2.32)

2N»... = 4—1,,§%n,<1+§,§) (1—3tot) (2.33)

The full out-of-plane strain-displacement matrix [8"] can be written as

[3"] = 41127411931 13".] [3'31 [19"..113x12 (2.34)
where
3:11-me 0 <1-3§)(1-n)ab2
18".) = 3n(1-§)a2 -(1-§)(l-3n)a2b 0
(4—3§2-3112)ab -(1-n)(1+3n)ab2 -(1-§)(1+3§)a’b
—3(1-n)§b2 0 -(1-n)(l+3§)ab2
13",] = 3n(1+§)a2 -(1+§)(1-3n)a2b 0
-(4-3§2-3n2)ab -(l-n)(1+3n)ab2 -(1+§)(1—3§)a2b
-3§(1+Tl)b2 0 (1+n)(l+3§)ab2
[3"3] = -3n(1+§)a2 (1+§)(1+3n)a2b 0
(4—3t2—3n2)ab -(1+n)(1-3n)ab2 (1+§)(1-3§)a2b
3§(1+n)b2 0 (1+n)(1--3§)ab2
13".] = —3n(1—§)a2 (1-§)(1+3n)a2b 0
—(4-3§2-3n2)ab (1+n)(1-3n)ab2 (1—§)(1+3§)a2b

2.3.3 Formulation of the linear elemental stiffness matrix [K,]

Composite laminates are constucted by bonding two or more laminae. Laminate de-
formations are assumed to be small with respect to the laminate thickness. Using the theory
of thin laminates, strains are considered to very linearly through the thickness and interlam-

inar deformations may be considered small at interior regions.

31

For combined in-plane and out-of-plane behavior, assuming plane section remain

plane after bending, the strain at any point in the plate is
{e} = {8°} +z{x} = [18'] +le"]] {4.} = [B] {q.} (2.35)
The strain in material coordinates is
{2'} = [Tl'TiB] {q.} = [TJ’TilB'HziB'Tl {q.} (2.36)
The shear strain in material coordinates (i.e., the third element of {e'} is
“foo = [T3][31{q.} = [T3] [13'] +ziB"l] {4.} (2.37)

in which [T3] is the third row of [T] “T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.5 Layer nomenclature for laminate

As has been described previously, the plate element used in the formulation has 20

DOF: three displacements and two rotations at each of the four nodes.

The linear elemental stiffness matrix of a laminated plate with N laminae is

32

1K.1 = 1 1317151 [BldV
V

N ab Z;

= 2 H j tlB'lT+le"1T1télllB'l+le"l1dxdydz (2.38)

k=100z

= 2 H I [[B'lT+z[B"]T] [0'1 [[B']+z[B"ll|J|d§a‘ndz

k: 1-1-12._l

where N is the number of laminae, zk is the height from the reference plane to the bottom
of the kth lamina (20 being the height to the top surface of the laminate as shown in ﬁgure
2.4), and Ill = STUFF—l)- (x, y) is the determinant of the jacobian matrix for transformation
from the x-y coordinate system to the E- 11 system. The integral with respect to z in Eq.
2.39 may be performed in closed-form, and the double integral with respect to 5,. and n is
normally computed by numerical Gauss integration. Expanding Eq. 2.38 the elemental

stiffness matrix can be written as

1V
[Kol = 2 llK',1 +1K",l+ [K”’,] + 1K"",11 (2.39)
k==l
in which
1 1
We] = (a-z.-.) j j lB'lTlél [B']|J|d§dn (2.40)
—l-l
zi-Zi—l l 1 —
[K21 = *7— j j lB'lTlol [3"]llld§d|‘l (2.41)
-l-l
Zi-Zz_1 l 1 _
1K"',1 = ——§—— 1 ] 18"17191 [8'1 lJldtdn (2.42)
-l-r
4-44“ —
1K"",l = ——3— j [8"]TIQ] [B"lllld§a‘n (2.43)

-l-l

33

The Jacobian matix [J] is

(2.44)

2.3.4 Formulation of the elemental consistent mass matrlx

The consistent mass matrix for the plate element is

[M]

e

I pNTNdv
V

1v ab)“

2 II I pthT+le"lTl [[N'l +zIN"]]dxdydz

k=looz‘_l

N’ 1 1 Q
2 I I p[[N]T+z[N"]T] [[N’] +z[N"]]IJld§drldz
k = 1’14“” (2.45)

Performing the integral with respect to z analytically, the elemental mass can be ex-

 

pressedas
.N
[14.1 = z ltM',l + [M”,] + lM'".l +1M"”,11 (2.46)
k==1
inwhich
l l
1M',1 = (are.-.) I I lN'lTl‘Q‘l 1N1 lJldtdn (2.47)
, -l-l
ZI'Zi-l l 1 —
[M”el = 2 I [N'lTlQl [N"]lJ|d§dn (2.48)
-l—l
ZI'Zi-l 1 l —
.[M”’,l = -—2— I I lM'lTlQJ [N']|Jld§dn (2.49)

-l-1

3 l l

3—
1M"",l = ff—z’iﬂ I I [It/"17121 [N"]lJld§dn (2.50)

3
—l-l

2.3.5 Numerical integration

In the integration of ﬁnite element matrices, a subroutine is called to evaluate the un-
known function <I> given in Eq. 2.52 at given points, and these points may be anywhere on
the element. A very important numerical integration procedure in which both the positions
of the sampling points and the weights have been optimized is Gauss quadrature. Hence,

for the two dimensional plate bending problem we have

1 l
I I ¢(§.n)d§dn~2W.-W,-¢<§,njl (2.51)
h}

-l-l

where W; and W]. are the integration weights. The above mentioned scheme is directly ap-
plicable to the evaluation of matrices of rectangular elements in which all integration limits

are -1 to +1.

The order of numerical integration to be used in the evaluation of stiffness and mass
integrals depend on the degree of precision that the order of the numerical Gauss integration
must satisfy. Since the shape functions have up to ﬁfth order terms, we must satisfy the con-

dition 2n - 1 2 5. Therefore, the order of integration is taken to be n = 3 x 3 .

35

3. Equivalent Linearization

3.1 Introduction

Random vibration analysis of mechanical systems has become an important subject
in recent years for various engineering applications. Forces due to earthquakes, turbulence
in air or water, storm waves, and the forces experienced by a vehicle traversing rough ter-
rain are examples where an understanding of random vibration theory is essential to the
successful design of the structure. Composite materials used in high speed ﬂight vehicules
are usually exposed to ﬂuctuating loads caused by the ﬂow of turbulent air or rocket en-

gines.

A common feature of such problems is that the excitation is often so complex that is
can be described only statistically. In addition, most physical systems behave in a linear
manner only for a limited range of the excitations, and since under random excitation larger
responses can be expected, at least occasionally, it is often necessary to study nonlinear re-

sponses due to random excitations.

Exact solution of non-linear random vibration problems is possible only for simple

systems. For realistic engineering problems an approximate method must be used.

One of the most widely used approximation techniques for nonlinear random vibra-
tion problems is equivalent linearization (or statistical linearization) in which the original
nonlinear system is replaced by an effective linear system. This approach has proved quite
useful for a broad range of engineering problems. Other methods such as Gaussian closure
and energy balance are closely related and generally give similar response results as equiv-
alent linearizaan while approaches such as the perturbation method may give somewhat

different results.

The essence of the method of equivalent linearization is to replace a given non-linear

system by a linear system in such a way that the difference between the two systems is min-

36

imized for all possible solutions of the associated linear system. The solution of the linear
system is then taken as an approximate solution of the original nonlinear system. The min-
imization of the difference in the equations of motion with respect to the linear system pa-
rameters does not necessarily guarantee that a minimization of the difference in the
corresponding solutions will be achieved. It has been shown that the method of equivalent

linearization gives good results even for strong nonlinearities.

3.2 Formulation

Consider the response to external load of a nonlinear multi-DOF vibratory system:
Generally, The dynamic equations of motion of a composite plate discretized into ﬁnite el-

ements may be written as
[M] {ii} + [C1 {C1} + [K] {(1} +¢(q. <1. it) = {Q(t)} (3.1)

in which [M] is the consistent or lumped mass matrix obtained by assembling the element
mass matrices, [C] is the damping matrix (usually speciﬁed indirectly through modal
damping ratios), [K] is the linear stiffness matrix obtained by assembling the linear ele-
ment stiffness matrices, {4)} is a vector of nonlinear terms obtained by assembling the el-
ement load vectors {the} , {q} , {t1} , and {a} are the displacement, velocity and
acceleration vectors and {Q (t) } is an excitation vector. It is assumed in this study that

{Q ( t) } is a zero mean Gaussian vector random process.

In the case when (I) (q, q, a) is non-zero and nonlinear, attention is directed toward
techniques of approximate analysis. Equivalent linearization offers a systematic and readily

automated method for generating an approximate solution of Eq. 3.1.

In the process of obtaining an approximate solution of Eq. 3.1, let us consider an aux-

iliary (equivalent) system which is described by a linear differential equation of the form
([Ml + [M.l) {ii} + ([C1 + (Col) {<1} + UK] + [K.]) {(1} = {Q (0} (3.2)

where [M e] , [C e] , and [Kc] are deterministic mass, damping and stiffness matrices.

37

The difference {5} between Eq. 3.1 and 3.2 may be written as (Spanos, 1976; Spa-

nos and Iwan, 1978)

{6} = Mq+Cq+Kq+¢— (M+Me)21'+(—(C+Ce)t1)-(K+Ke)q

= ¢(q: 4’ q) TMeq- Ceq-Kgq

(3.3)

In order to determine the matrices [M e] , [Ce] and [K e] of the equivalent linear

system it is necessary to establish a criterion for the minimization of {5} , based on a suit-

able norm of this vector. Here the Euclidean norm deﬁned as
II {61113 = {6} WE}
will be used as a measure of {5} .
The minimization of 5 is performed according to the criterion
E[ {5} T{5}] = E{5f+5§+ +53} = minimum

where 51], 52,... 5" are the elements of 5.

Using the linearity of the expectaan operator E [ I , Eq 3.5 can be written as

n
2D;2 = minimum
i=1
where i = 1, 2, 3, ..., n and D‘- is deﬁned by

n 2
vi = 316%} = EH" 2 (mirir+c?jér+k34r>] I

i=1

(3.4)

(3.5)

(3.6)

(3.7)

Minimization is with respect to the class of functions of t which are solutions of Eq. 3.2.

The necessary conditions for Eq. 3.7 to be true are

ieEltsthsn = low?) = 0
8m”. 29ij

38

(3.8)

iEliWiz‘tll = 597w?) = o (3.9)

Be; cu
—a;El{5}T{5}l = 497(1),?) = o (3.10)
where mfj, cf]. and [:3 are the (i, j) elements of the matrices [M e] , [Ce] and [K e] , re-

spectively.
The expansion of Eqs. 3.8, 3.9 and 3.10 gives the following
[16“.] T

E{¢,-?1} = 13121217} [CZ-1T (3.11)
[MAT

where [Mr] , [Cf] and [KI-I] are the irh rows of the matrices Me, C, and Kc, respec-
tively, and
{a} = lq. q. a] T - (3.12)

So far the method of statistical linearization has been presented irrespective of the
particular probability density which is used in computing the expectations appearing in Eq.
3. 1 1. The formulation is facilitated by assuming that the excitation of the original nonlinear
system is Gaussian. Therefore, the response of the equivalent linear system to this excita-

tion is also Gaussian. Utilizing Kazakov’s formula (1965) for Gaussian random vectors:

Etftnln} = “1111715inth (3.13)

we obtain

39

‘
ﬂ
I...
I
J

.83:
0Q
-~-1

(3.14)

o
O
V

Bloom] = 5112} {2105+

#3:
Q
we

 

 

 

 

Q.)
m
1Q
we

b d

Comparison of Eqs. 3.1] and 3.14 shows that the elements of the matrices [M' ] , [C' ]

and [K' ] are given by the simple expressions

3(1).

* _ r
m tj -- E{r{q’j}} (3-15)
* - E L" 316

M.

at- _ 1

These results are due to Kazakov (1965). They were first used for stationary nonlinear ran-
dom vibration analysis by Atalik and Utku (1976). Subsequently, Spanos (197 8, 1980)

pointed out that they are also applicable for non-stationary problems.

For the laminated composite plate, the nonlinear force vector is only a function of the
generalized displacement coordinate q. Thus the nonlinearity will affect only the stiffness

matrix and consequently, [C'] = [M'] = [0].

3.3 Derivation of the Nonlinear Elemental Stiffness Matrices

Consider virtual displacements {54¢} . The elemental virtual work done is
5W = {54,} T {R.}

where {R e} are the nodal loads on the element. The internal virtual work is

5U = I{52}T{o}dv

setting 5W = 5U and using the strain displacement relations
{58} T = {561,}7181T

and the constitutive law in Eq.2.5 yields

{64.17%} ItSq.}TlBlT{ 151 {a} +f(712) 17*] {a} 14V}

(3.18)

{6a,}TI I [817151 {£}dV+ I 13leth [7‘1 {e}dV]
V V

since Eq. 3.18 must hold for any arbitrary virtual displacements {89¢} T, the following

equation must hold:

{R.} = I I [BlT{ [51 131cm] {4.} + I131 Tftto) 17*] [B] {q.}dv
V V (3.19)
= [Kol {4.} + {4.}

where [K e] is the elemental stiffness matrix given in Eq. 2.39 and

{4,} = I 1317mm) [1*] [B] {q.}dv

V
N “b “' (3.20)
= 2 ll lma) twirl [B] {q.}dxdydz

i=1002,-.

is the elemental vector of nonlinear terms.

For the functional form of f (712) given by

fun) = Zuni; (3.21)
1': 1

the nonlinear vector in the above equation may be written as

41

{4,} = 2 {to} (3.22)
1:1
in which
ab 11
11>] = )3 2:: I I I 755131 17*] [B] {q.}dxdydz (3.23)

k: 11': l iOOz,_,

Since the nonlinear vector (I) is a function of the displacement vector {q} only, the
nonlinearity will affect only the stiffness matrix. Hence, c' = m' = 0. The resulting

equivalent stiffness matrix can be written as

N n l l 11:
[16“.] = BIN—La }{¢ }] = 2 Eta-I Ilz‘IdBthT’l [B] [Alllld§dndz (3.24)

k=li=l

and requires the evaluation of
[A] = E[-a———{aq ﬂ}; {qe} I (3.25)

The partial derivative is

My
814. —755{q.} = 755111 +2iiq.}ti5 8’“qu (3.26)

and since 712: [T3] [B] {qe}, Eq. 3.25 becomes
[A] = E{([T3] [Bl {2.112‘111 +2i{q.} ([Tol [B] {q.})2“‘lT31 {Bl} (3.27)
Consider the ﬁrst term of the above equation
5([7'31131 {u.})2‘= ([T3l [[B'l +le"ll {4.})2‘
(3.28)
= ([T31 [13'] {(1.} +zIT31 [B"ll {qo})2‘

Using the Binomial expansion

42

2i .
(a + bz) 2" = 2 (2")a2“’b’z’ (3.29)
(=0

Eq. 3.28 may be expanded as

2i

1511751 [B] {2.1)2i = 2‘, [ﬂag] [3'1 {qo})2""([T3] 13"] {q.l)'z’

(=0
21' 21-1
= 25 52)“ OLE” ZTEU'B H1192, ) (3.30)
I=() j =lk' =1
3 20 I
x [[2 2 T31...B"j..k..qe’k..) ]
= 1k" =1
or in condensed form
. 21
51([T3IIB] {q.}>2*1 = 2a,? (3.31)
i=0

where orI is given by

3 20 3 ﬂ) 3

20
= 2 Z 2 2 2 2 E[qe,k'qe,k2qe.k3" 'tqe km]

j|=lk|=lj2=lk2=l j21=1k21=1
(3.32)

XL!)111731..)(:I:-I:B",..Ic,..)2i 3"”)

=2i- 1+1

For the case where i=2, which corresponds to a ﬁfth order approximation to f (712) ,

the expanded form of Eq. 3.31 is

43

2t 3 20 3 20 3 20 3 20
I I I D
2 alz _ 2 2 2 Z 2 2 2 T3j1811 ‘5 113.128 12k2T313B 13k3T3j4Bk J4l‘4
i=0 11=1kl=1j2=lk2=1j3=1k3=lj4=1k4=l
x E lqe, 12,42, kzqe, we, k‘]
3 3 20 3 20

+42 2 Z Z 2 Z 2 2 2 TsjB' ﬂax-23'} Tsj,B'j3k,TsJ-.B"m

' —1k1=1j2=lk2=1j3=1k3=1j4=1k =1
X E [qe, k'qe, kzqe, ksqe, 1:4]

3 3 20

+622 2 20 2 20 Z 20 2 2 T3jiB'Jik1T3123)“:T3laanak3T3jIan4k4

j,,=1k =1j2=lk2= lj3=lk3= 1j4=1k =1
X E [qe, kl qe, kzqe, k3qe, k4]

3 3 3

+423 2 E 2 20 2 E Z i 731.3 1. kT3sznjzk 2"T3jaB jsk kT3J' Built

jH=lk =1j2=1k2=1j3=1k3=1j‘=1k =1
X E [qe, k‘qe, kzqe, k3qe, 1:4]

4 3 20 3 20 3 20 3 20
+ Z 2 2 2 Z 2 2 2 2 T3; 3 " 731;” "1255133 "131:, T31.” "141:4
j|=1k1= 1j2=1k2= 1j3=1k3=1j4=1k4=l

X E [qe, k‘qe, kzqe, k3qe, k‘] (333)

Now consider the second term in Eq. 3.25 which can be expressed as follows:

E[{q,} ([T3] [3]) {qe} )2“1 [T3] [3]]

2i-1 ,
= E{qe} 2 (Z'I'IJUTal [B'] {qe})2"‘"’([T3] [3"] {qe})’
(=0
x [T31 [[B'] z‘+ [3"12” ‘1 (3'34)

2i-1
2 {5,} [[B']z’+ [B"1z’+‘1

[=0

where {B,} is given by

2i-l

{5,} = E({q.} 2 (2‘;‘)([T31[B'1{q.})2‘-1-'([T31 [8"] {q.})') (3.35)
i=0

The pm element of { B1} is

3 20 3 20 3 20
B113: 2 2 2 Z 2 E[qe,k,qe,k2qe,k3'“qe,1:214]
j,=1k,=1j,=1k,=1 j2,_'=1ku_'=1
(2. I) 2i-1 2i-l—l 25-1 (3.36)
l- o n
x I (m T3}.)( H BIJ.) H 311.!)
=1 m=1 =2i-l

I
resent 2ith-order moments of the zero-mean Gaussian vector {q e} . The higher-order even

where 12') indicates the binomial coefﬁcient. The expectations in Eqs. 3.32 and 3.36mp-

moments may be expressed in terms of the covariances as follows:

E [9,, hqe, k2° ' 'qe, k2,] = 2E [qe, que, k2] E [qe, k3qe, k‘] ° ' 'E [9,, k2,_ lqe, 2i] (3°37)

in which the summation is taken over all possible ways of dividing the 2i variables into i

combinations of pairs [the number of terms in the summation is 1.3.5...(2i-3)(2i-1)].

The effort required to compute the a, and [3, p coefﬁcient by Eqs. 3.31 and 3.34 is
substantial. However, a special algorithm described in chapter 4 that carefully takes into
account the symmetries involved in the summations has been developed, and the computa-

tional effort has been reduced by a factor of more than 20.
Finally the equivalent stiffness matrix can be written as

2a.. J j 1 [3mm [3] [Alllldtdndz (3.38)

n l 1 ll
=1 i-l-IZ.-'

N
[16“.] = 2
i=1

where [A] is given by

45

2i 2i—1
[A] = 2092’ [I] +2: 2 {[3,} [T3] [[B']z’+ [B“]z’“] (3.39)
i=0 i=0

Expanding [B]T[T"] [B], [K'] canbesplit into

[K21 = [K'*.1+[Ic".1 +[K'"*.1 (3.40)
where
N n 11
m2] = Z aJ I [B']T[7*][B'][A1]|J|d§dn (3.41)
k=li=l -1-1

N n 11
mt] = 2 2a.. 1 [WWW] [B"1+[B"]Tn*1 [3'11 [Azlmdzdn (3.42)

k: li=l 1-1-1

N n 11
[K'"*.1 = 2 ad I [B"1’[1*1{B"][A31md§dn (3.43)
k=li=1 _1_1
and [A1] , [A2],and [A3] are
9
[A1] = J [Aldz
2:“ (+1 (+1
_ Zk -zk—l
‘,§.°‘t(——z+1 )m

25-1 21+1-Zic+11 21+2-21+21
+21. 2 {BI} [T3] [13'] (Ti-‘4)+ [3"] (7%)]
i=0

1:
[A2] = j 2 MM:

zk-l

2.‘ z1+2 “21:21
:22)“ °‘:(-——— 1+2 )[I] (3.45)

2i- 1 Zl+2'2£+2 21+3-zi+31
+2i’20{ﬂl} [T3][[B](—T_—2—i)+13"1(-—1T3—-—):l

(3.46)
31
[A3] = I 22 [A] dz
2i 21+ 3 _ 21+ 3
= 31‘” 1174—”) m (347)

23-1 Z£+3 21:31 21:4“ 21+:
+2: 2 {[3,} [T31 [[81 {—13:74} +[B"] {—M—“H
(=0

The general expression for [A q] , q = l, 2, 3 can be written concisely as

47

1:
[Aq] = Izq'l[A]dz

zk-l

2‘ dig-zit",
= 2a‘( Ha: )[I] (3.48)

(=0

2i— 1 Z(+q_zi:ql zik+q+l_z£-:ql+l
+2i 120 {13,} [T3] [13'] (—_l+q )+ [B"l( ”(1+1 J]

 

3.4 Random Vibration Analysis

Since computation of [K' e] requires the mean and covariance of the displacement
responses to be known, an iterative approach must be used, in which each iteration consists
of a linear random vibration analysis. Linear random vibration analysis is well-known and
is only summarized here. Time-domain or frequency-domain techniques can be used for the
analysis, and the main steps using a frequency domain approach are as follows:

1. Using the stiffness matrix [K e] + [K' c] (with [K' e] =[O] in the ﬁrst iteration)
and mass matrix [M] , determine the frequencies, to], and mode shapes, {vi} , for

a chosen number of modes (say n modes).

2. Perform a linear random vibration analysis to determine the covariance matrix of

the nodal displacements {u}. The rsth element of the covariance matrix is given by

Elq,q.1 = 2 2'; w“): 2 way ,,,,. IH(- com (w)S,,,.(m)dm (3.49)

j=1k=1M kl=lm=l

 

in which V”. are elements of the mode shape matrix, M j = {wj} T[M] {Wj}. is
the jth modal mass, H}. (m) = (0)]? - o) + 21101.03).l is the jth modal frequency re-

48

sponse function, and S1," (to) is the cross spectral density function for the excita-
tions P, and Pm. Note that for synchronous loading only the auto spectra are non-
zero, and the double summation over ( and m may be reduced to a single summation.
For certain classes of excitation spectra, closed form solutions can be used to rap-
idly compute the integrals in Eq. 3.49 (Harichandran 1992), while for more general

cases numerical integration must be implemented.

3. Compute the equivalent element stiffness matrices [K' e] from Eq. 3.40 through

3.47, and assemble the global equivalent stiffness matrix [K' ] .

The three steps outlined above are repeated until convergence is reached in the cova-
riances of the nodal displacements. One method of checking for convergence is by using

the nodal displacement variances, and the mth iteration is assumed to have converged if

_ 2
If; (6%“ agnm'l)
O.2
j; 90’"

 

< e (3.50)

in which Gq‘ = ,/E [qiqi] .

The covariances of the strains within an element may be computed by replacing w”.
and v“ in Eq. 3.49 with strains corresponding to modes j and k in the ﬁnal iteration. The
covariances of the stresses may be computed from the covariances of the strains and the

nonlinear constitutive equations.

3.5 Computation of Strains and Stresses

The multidirectional composite laminate consists of laminae of various ﬁber orienta-
tions. In this form the stiffness of each lamina may differ signiﬁcantly from adjacent lam-
inae. Since the strain components in thin laminates vary linearly through the laminate

thickness, discontinuities in the in-plane stress components will occur at laminae interfaces.

49

fe

Tn

and

Hence, it is imperative that the detailed state of stresses be established for each lamina. In
this study strain and stress components are computed within each lamina using the tech-

niques outlined below.

In global coordinates, the strain at any point (x,y,z) in the element is computed as fol-

lows
{8} = [[B'] +z[B"]] {(1,} (3.51)

The covariance matrix of the strains in global coordinates is given by
[28] = [{8} {8} T] = [[B'] +23") [29.] [[B'] +z[B"]]T (3.52)

in which [Zq ] is the covariance matrix of the nodal displacements.

In local coordinates, the strain can be expressed as
{8'} = [Tl'T{8} = [Tl‘TllB'] +z{B"]] {4,} (3.53)
The covariance matrix of the strains in local coordinates is computed through
[28.] = [T] ‘71 [3'] + z [3"11 [th1 [[B'] + z [3"] ] TIT] '1 (3.54)

For stresses, it is easier to compute the stresses in local coordinate ﬁrst and then trans-

fer to global coordinates. In local coordinates the stresses at any point (x,y,z) are given by

O O O
{o'} = [Q]{8'}+f(712)[0 o OJW} (3.55)
0 0 l
The normal stresses in local coordinates are
0' Q11 912 3'11
{ ”l =[ I (3.56)
0'22 Q21 * 922 {8'22}

and the shear stress is

50

=Qoo7'12 +f (712) 112 = Qoo'Y'lz + Z“ 72'2“ (3°57)
From Eq. 3. 54 the covariance matrix of the normal stresses is

Q11 Q12 Q11 912 T
2 . = 2. = Z . T 3.58
[ 0N] [Q21 Q22][ 1|:Q21 Q22] [QM] [ 2”] [QN] ( )

From Eq. 3.55 the covariance matrix of the shear stress can be expressed as follows

E [on] = E [Qggy'f 1 + 2Q6620 E [y2;+2] + 2a .2a 5 [y2;+21'+2] (3.59)

Since 712 is a gaussian random variable, E [7%] can be expanded as follows
wagl =1x2x...x(2k-1)x(o§2)' (3.60)
l

in which , a; = E [ﬁz] . Note that a: was obtained in the computation of the strain in
12 12 "

the local coordinates using Eq. 3.52.

In global coordinate the stresses can be computed easily through

[2,] = [7‘1"[20] O m" (3.61)

0 steel

in which [20.] is the covariance matrix of the stresses in local coordinates, computed by

using Eqs.3.58 and 3.59.

3.6 Laminate Strength Analysis

From a design point of view, it is important to be able to predict failure due to exces-
sive strains or stresses. The maximum strain criterion is one of the failure criterion used in
the analysis of unidirectional ﬁber composites. In this criterion the orthotropic lamina is

characterized by six ultimate strain allowables.

51

If any one of the ultimate strains is exceeded in any lamina, it is deemed to have failed. U1-
timate strains in material coordinates for Narmco 5505 (the material used in the numerical

examples presented in Chapter 5)at room temperature are
81+ = 0.0040 82+ = 0.0027 712+ = 0.01 1
81- = 0.0065 92. = 0.0038 712' = 0.011

As each ply fails, the laminate stiffness is recalculated to reflect the deletion of the failed

lamina. The lamina failure strains can be used to predict of the laminate ultimate strength.

Alternatively, failure in composite laminates have also been expressed in terms of

stresses. The most common ones are the Tsai-Hill and Tsai-Wu criteria (Jones 1975).

52

4. Optimizing Computational Effort

4.1 Introduction

Finite element programs can be computation-intensive. For example, even small
problems of size n may require a computation time in the order of 0 (n4) . In nonlinear
problems the computation time becomes worse because convergence iteration is required.
The nonlinear random vibration analysis outlined in the previous chapter is extremely com-
putation-intensive and unless the computational effort is alleviated, the user will be restrict-
ed to analyze only small problems. In order to investigate a reasonably large problem, there
is an urgent need to drastically reduce the required number of computations and conse-

quently minimize execution time.

In this chapter a novel optimization technique is developed to make use of the sym-
metricity inherent in the mathematical expressions. The improved performance of this tech-
nique is presented later in this chapter and is compared to the straight forward FORTRAN
implementation. A number of simple techniques is also used to reduce the computation

time for the types of formulas that involve operations on multiple matrices.

4.2 FORTRAN Implementation
This section provides a general description of the FORTRAN code deveIOped for this

problem.

The program is logically divided into a number of different modules. Each module
corresponds to a separate logical computation unit as deﬁned in the general ﬂaw-chart of
Fig. (4.1). This logical separation was adopted in order to simplify the implementation and
greatly enhance the readability of the code. The modular design should also greatly facili-
tate future modiﬁcations and extensions to the existing code. This is true even if major
changes and additions are required. The corresponding modiﬁcations may be added be-

tween the units of Fig.4.l or within each unit without affecting other units.

53

C9 (9 C9 C9

@629

Get input parameters

i

Initialize Variables

i

l Compute Linear

 

 

 

 

 

Mass & Stiffness

’i

Compute Eigenvalues

 

 

 

 

 

& Covariance matrix

 

    

Convergence
reached ?

  

 

Compute
Alpha & Beta
V

Compute Nonlinear
Stiffness matrix

 

 

 

 

 

 

 

 

 

 

 

F

' Compute
Strains & Stresses

Figure 4.1 General ﬂow-chart for Plate.f program

 

 

Figure 4.1 shows the general ﬂow-chart for the main program. Each step in the ﬂow-
chart is described in detail below:

1. The user is prompted to input a number of parameters that completely control the
size, variation and behavior of the problem. The implementation is quite automated
and allows for a large variety of problem sizes, loading conditions, support condi-
tions, number of laminae, and different thicknesses and ﬁber orientation angles for

each lamina.

2. All static variables are initialized, and the topography of the plate based on the input
parameters is generated. Elements are generated along a rectangular grid. The val-
ues of Z, and 2,, _ 1 based on the number of laminae are also computed. These val-
ues represent the distances from the mid-plane to the top and bottom of each lamina
L... The signs for Z]: and Zr _1 are determined according to the position of lamina
Li with respect to the mid-plane (i.e. if L,- is below the mid-plane, Zk and 2,, _ 1 will
both be negative). Finally, 3’ and B” are computed according to the formulation
previously introduced in chapter 2. These computations are done for each Gauss

point within an element.

3. The elemental stiffness and mass matrices are computed using 3 x 3 Gauss quadra-
ture. These matrices are then assembled using the topography matrix computed in
step]. The global structural mass and stiffness matrices are then obtained by elimi-

nating the restrained DOF speciﬁed by the user.

4. The eigenvalues are computed using the structural mass and stiffness matrices de-
termined in step 3 for the ﬁrst iteration or in step 6 for subsequent iterations. In the
ﬁrst iteration, only the linear mass and stiffness matrices are used. In subsequent it-
erations, the stiffness matrix includes both linear and nonlinear terms. The eigen-
values are then sorted and the corresponding eigenvectors are computed. The DOF
chosen by the user are loaded with excitations having the user speciﬁed spectral

densities S ((0). Finally, the structural covariance matrix is computed using Eq.

55

3.49 and convergence based on Eq. 3.50 is checked using a tolerance of 0.001. Co—
variance matrices corresponding to elemental DOF are extracted from the structural

covariance matrix as needed.

5. The coefﬁcient on and B, as introduced in Eqs. 3.32 and 3.35, are computed. These
intensive computations are performed for each element and at each Gauss point. and
is the most time-consuming part of the whole program. It is here, that two major op-
timization techniques that drastically reduce the number of computations are intro-
duced. These techniques are described in details in the following sections. The
individual values of the or and B coefficients for any given pair of indices <i, j>
may be computed independently and this will facilitate the parallelization of this

step as discussed in chapter 6.

6. Using the values of or and B, the elemental nonlinear stiffness matrix for each lam-
ina is computed. These matrices are assembled and the restrained DOFs are elimi-
nated to obtain the ﬁnal nonlinear stiffness matrix for the whole laminate. The
nonlinear stiffness matrix is then added to the linear one to obtain the total stiffness

matrix.
7. Steps 4 through 6 are repeated until convergence is obtained.

8. After convergence, the ﬁnal covariance matrix is used to obtain the covariances of
the local and global strains and stresses. These values are computed at the center of

the element chosen by the user for each lamina.

4.3 Optimization Techniques

The time needed to evaluate any given formula is directly proportional to the number
of arithmetic Operations performed by the corresponding code. When computing the non-
linear stiffness matrix by statistical linearization, the execution time was found to be large
especially for computing the or and I3 coefﬁcients.The actual time to execute a given in-

struction that involves arithmetic operations also depends on the precision of the variables

56

involved (i.e., the multiplication of double precision numbers requires about twice the
amount of time needed for the multiplication of single precision numbers). Hence, mini-
mizing the number of arithmetic operations performed by the program can greatly reduce
the total computation time. Since code normally represents the direct implementation of a
given problem, any technique to Optimize such code will have to preserve the integrity of
the implementation. Thus, optimization has to be done at the semantic level. For example,
a computation that involves the multiplication of a number of variables may be totally
skipped if at least one of these variables has a value of zero. Checking for zero values within
a matrix V is only justiﬁed if it is known a priori that V is mostly sparse. The rest of this
section describes in details a number of techniques developed and implemented to mini-
mize the required number of instructions and thus optimize the execution time. Also, a de-
tailed comparison between the non-optimized and the optimized versions of the code is

presented.

4.3.1 Optimizing or and IS
The computation of Eqs. 3.31 requires the evaluation of ﬁve sub-equations of the fol-

lowingtype
3 20 3 20 3 20 3 20
=22222222F (4.1)
j =lk =lj2=1k2 =lj3=1k3= ljg=lkg=l
whereFis
F = 731.3 ' T3123 .12k2T3lsB '1; 3T3jrB M kE [q, 2 q, que *3 q, ’2] (4’2)

A direct implementation of Eq. 4.1 is shown in Fig 4.2, where each Do-loop repre-
sents the corresponding summation in the equation. Based on this implementation the ex-
pression F has to be evaluated approximately N times, where N s 13 x 106. The time
required to execute the code in Fig. 4.2 is approximately Ta‘ = N x C (F) , where C (F)

represents the computation time required to multiply the twelve double precision variables

57

DO J2 = , 3, 1
DO K2 = 1, 20, 1
DO J3 = 1, 3, 1
DO K3 = 1, 20, 1
DO J4 =

l, 3, 1

DO K4 = 1, 20, 1

tttcov = cove(ielem,k1,k2)*cove(ielem,k3,k4)+
+cove(ielem,k1,k3)*cove(ielem,k2,k4)+
+cove(ielem,k1,k4)*cove(ielem,k2,k3)

alpha2(ic,ielem,1) = alpha2(ic,ielem,1) +

+tt3<J1)*tbn2(Jl,kl)*tt3(j2)*

+ tan(J2,K2)*tt3(J3)*tbn2(j3,k3)*

+ tt3(j4)*tbn2(j4,k4)*ttcov

Figure 4.2 Direct implementation of or

in Eq. 4.2. Let :4? represents the time required to multiply two double precision numbers.
The amount of time, Ta' becomes Ta! = 12 x N x tdp which is approximately
1.56 x 107th. Hence the total amount of time required to compute Eq. 3.31 may be ex-

pressed as
5
Q=2Q‘ an
i = 1

Since the computation time, T0,. for each of the sub-equation or, is approximately the
same, Tcl 5 5 x Tat, for all 1'. According to the formulation of the nonlinear stiffness matrix
[K' ] , or must be evaluated at each Gauss point, for each element and in each lamina.

Hence, the overall time spent by the program in evaluating or may be approximated by
2
T=nxmxxmny xTa (4.4)

where n = the number of laminae in the problem, mx x my = total number of elements,

and G = order of the Gauss integration used.

58

To appreciate the magnitude of computing T, assume I: = 5 laminae, mJr = 20,

my = 20 and G = 3. The total time Tthen becomes T = 1.4 x 1013:”.
The computational optimizations for the at and B coefﬁcients involve the combina-
tion of three types of techniques:
1. The reverse Pascal triangle indexing scheme to optimize Eq. 4.2
2. Elimination of zero-valued computations in the evaluation of F in Eq. 4.2
3. Unrolling of compound DO-loops in Eq. 4.1
These techniques are discussed in details in the next subsections, and then the com-

putational time for the optimized and non-optimized computations of the expression T are

compared.

4.3.2 Covarlance optlmlzatlon
The computation of Eq. 4.2 involves the evaluation of

E [qqukzqkaqk4] = E [qktqkzl X E [qksqkA]
+ E [41“qu x E [9129154] (4.5)
+ 5 [41:19:91 X E [9:991:31
Note that Eq. 4.5 requires three multiplications and two additions of double precision

variables, and will be executed N (13 million) times in evaluating Eq. 4.1. Reducing these

number of computations will have a signiﬁcant effect on the overall computation.

Given that the covariance matrix is symmetric, E [qqukz] = E [qkzqh] for any k1
and k2 and thus, rearranging the indices klk2k3k4 in any order in Eq. 4.5 will produce the
same numerical result. Hence, Eq. 4.5 need to be actually evaluated for only one of these
arrangements. All other arrangements may substitute this computed value provided an ef-

ﬁcient indexing scheme can be devised.

59

-
q
d

1
-

c4
-
.1
d
u
-

 

 

 

 

 

 

_— Nc
e "“" 4
200 - / .
—' /
5 /
w /
ﬂ
08 150 " // "
5 /
D
De /
E /
8 100 - / -
“5 //
"‘ /
B /
8 so — // .
9 ./
Z // .
’l/ __.__._—-I
0 . 1 . 1 _._ L___=—-1"".'"’1 - 1 fl . l . l .
0 2 4 6 8 10 12 14 16 18 20 22
Number of DOFs per element

kavtsbMMmlommmmMNpwlhmeMMOQL
The execution of Eq. 4.1 produces all possible permutations Np = 204 of k 11:21.23 and
k4. This requires the evaluation of Eq. 4.5 for all these permutations. Given an efﬁcient in-
dexing scheme, this number of permutations actually reduces to the number of all possible
combinations with repetition (i.e., including combinations such as (1 1 2 3), (3 2 l 3), etc...).

The number of these combinations N c may be expressed as

(4.6)

 

+ -1 (n+r—1H
”ﬁt" ; J: r!(n-l)!

where n and r represent the total number of DOF per element and the number of DO-loops
in Eq. 4.1 respectively. For our problem n = 20 and r = 4. A comparison between the rela-
tive magnitudes of Eq. 4.6 and the number of permutation is shown in Fig. 4.3. The ﬁgure
outlines the major saving in terms of computational time when n increases. Note that for
n = 20, the indexing scheme reduces the overall execution time by approximately a factor

of 20.

4.3.3 The lndexlng scheme
In evaluating Eq. 4.1, all possible permutations (k1k2k3k4) will be generated by the

four k-DO-IOOps. The previous discussion showed that for any given set of indices, we need
only compute the value of one of all such permutations. This computed value may be saved
by the program in a one-dimensional array (offset (N 6)) to be uSed as a special access struc-
ture. Later, any of the remaining permutations may access this stored value by means of an
indexing scheme. This approach is known as dynamic programming. A necessary condition
in dynamic programming is that the cost of the indexing scheme should be less than the cost
of performing the actual evaluation. Next, we present an efﬁcient indexing scheme, show

its correctness and that it greatly outperforms the straightforward implementation.

To illustrate how the scheme works, we provide the following example: Let n = 4 and
r = 4. Fig. 4.4 lists all possible combinations of (k1k2k3k4) . Note that these combina-
tions will be produced in ascending order of indices due to cascading of the DO-loops as

shown below:

DO Kl = 1, 20, 1
DO k2 = k1, 20, 1
DO k3 = k2, 20, 1
DO k4 = k3, 20, 1

END DO
END DO
END DO
END DO

The indexing problem may be outlined as follows: Given any permutation of indices
P = ( k1k2k3k4) , the problem is to locate the corresponding combination of indices, C, for

which the value of F has already been computed and stored in offset NC.

Since the combinations are produced in ascending order of indices, the combination

C is the permutation that has the indices k, sorted in ascending order.

61

G4 33

 

G3

 

E3

l)(lll
(112

(122

(222

2)
2)

(l
(l
(1

(1
(1

(1

(2

(2

(3

E2

NN

NN

WNH

WN

wN

3)

3)

3)
3)

3)

3)

3)

3)

Figure 4.4 Derivation of reverse Pascal triangle

62

(1
(l
(1
(l

(1
(1
(l

(l
(l

(l

(2

(2
(2

(2

(3
(3

(3

(4

E9.

# WW NNN t—Iv—tr—u—u

h we» NNN

030-)

«b At» «bWN waH

h #00 «AWN

Jib.)

4)
4)

44)

4)

In order to locate combination C corresponding to P, the correct index of C in the ar-
ray (oﬁset) has to be computed. To illustrate the indexing scheme, termed reversed Pascal
triangle (RPT), let I: = 4 and r = 4. Fig. 4.4 displays all the combinations produced by the
four k-DO-loops in the code segment presented earlier. Note that in Fig. 4.4, the combina-
tions can be divided into four groups (31, 32, g3, 34) where each group has the same num-
ber in the ﬁrst column. Within each group, the combinations may be split again into
subgroups (31, 32, s3, 34) based on the number in the second column. The division based
on the third column is shown as a separate lines (I 1, 12, l3, 14) within a subgroup and that

based on the 4th column corresponds to entries (e1, e2, e3, e4) within each line.

In the following, we determine the number of combinations in each line, subgroup
and group. We note that 11 will always have one combination since it corresponds to the
last combination starting at 4. 12 corresponds to starting with 3, and thus has two entries (3

and 4). Similarly, 13 starts with 2 and has 3 entries (2, 3 and 4) and 14 has 4 entries.

Subgroup 31 has only one line, 11 , since it corresponds to starting with 4. 32 has 11
and 12 resulting in 3(ll+l2 = 1+2) entries. 33 has 6(ll+12-I-l3 =1+2+3) and 34
has 10 entries (11 +12 +13 +14 = 1+ 2 + 3 + 4) . The same trend holds for the groups and
thus 31 = 1 (s1), 32 = 5(s1+32), etc ..... To reach a given grouplgi (i = l, 2, 3, 4), we
have to skip groups g1, gz, ...g,_1. Thus, we have 1to skip 423 = 2 gj entries. Similarly,
a subgroup 3,- may be reached by skipping e S = 2 sj entries within1 the given group. The
number of entries in each category, as computed Jabdve, is represented in Table 4.7. This is

done to motivate the use of the reverse Pascal triangle (RPT).

Note that the entries in this table correspond to those of the Pascal triangle. This ob-
servation generalizes the indexing scheme since Table 4.7 can now be generated for any

combination of n and r.

63

TABLE 4.1 Number of terms in each group

 

 

l 1 2 3 4
l 1 2 3 4
s l 3 6 10
g 1 4 10 20

 

Next, an example is presented to verify and show how the indexing scheme helps 10-

cate the correct combination C to any given permutation P.

Example: Let P = (3 2 2 3), then by sorting the entries of P in ascending order we ob-
tain the combination C = (2 2 3 3). The pre-computed value of the combination C in the

offset array is stored at location

N c - RPT(1,2) - RPT(2,2) - RPI‘(3,3) - RPT(4,3)
4 ' n + r co) j
=Nc-j=21RPT(J,C0))=Nc-j§1( r—j+l )=35-5-4- 1 - 1 =24
Note that the column (second) indices in the RPT array are the element of C.

The following code segment illustrates how oﬁ'set is computed

OFFSEt = NC
DO J = 1 , r
OFFSET = OFFSET - RPT (J, C (J) )

END DO
Thus, value(P) = oﬁ'set(24). This is graphically illustrated in Fig. 4.5 for the same example.
The corresponding RPT table is extracted from the Pascal triangle in Fig. 4.6. A table that
represents the RPT indexing structure for the speciﬁc problem size treated in this work is
also presented in Fig. 4.10 at the end of the chapter, where n = 20 DOF and r = 4 DO-loops.
Note that the ijth element in the table of Fig. 4.6 or in Fig. 4.7 is given by L" j: T: '1'} J
where n = 4 and r = 4 for the former, and n = 20 and r = 4 for the later, and with [1 Idefmed

m
as being zero when l< m.

‘ Sort
2.

(3.2.2.3)

P:
C:

 

 

 

 

a: a

 

 

 

 

 

s
m. 12342343442343443444 2343443444 3444 4
w. 11112223342223343344 2223343344 3344 4
m11111111112222223334 2222223334 3334 4
m1234sevsgmuummumnmwm unnuxxnnmm any“... a
o

 

 

 

Figure 4.5 Combinations for which F is computed

65

 

 

 

 

 

 

i l l 2 3 4
l 15 5 l 0
2 10 4 l O
3 6 3 l O
4 2 l O

 

 

 

 

 

 

 

 

Figure 4.6 Pascal triangle and RPT table

4.3.4 Ellmlnatlon of zero-valued computations

In the previous section, an indexing scheme to help minimize the number of compu-
tations involved in evaluating Eq. 4.5 was generalized. While the scheme was shown to be
effective and efﬁcient in reducing the computation time of E [qk‘qkzqkaqk‘] , evaluating F
in Eq. 4.1 still requires eight double precision multiplications and has to be executed
N ~ 13 x 106 times. Reducing this total number of evaluations should also improve the

computation time of Eq. 4.1. Next, a technique that aims to reduce N by eliminating all

asses ea 888 .35 see Ex 5. an»;

 

 

 

 

 

 

 

 

N m v m o h w a on S N~ S 3 9 on E w— 3
n m c c— 2 "N ma on 9. mm 00 ms 3 mom OS 02 m9 Kw com
— v 2 on mm on vw cg n2 cam own 3m nmv con 80 05 a8 3: 0mm:
_ n 2 mm 2. 05 SN cmm 3v m: 89 n0m— ouw— ome coon ohwm nvwv nwan was.
3 3 3 3 mm 3 m— N— 2 S a w h o n v m N u

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

67

computations of F that involve a zero-valued variable is introduced, a code segment for the

computation of or is presented in Fig. 4.8 to illustrate the technique.

DO J1 = l, 3, 1
DO K1 = 1, 20, 1
if (abs(tbn2(J1,Kl)) .gt. MinTol) then
L1 = (Kl-1) * 8000
DO J2 = 1, 3, 1
tsl = TT3(J1) * TT3(J2)
DO K2 = 1, 20, 1
if (abs(tbn2(J2,K2)) .gt. MinTol) then
L2 = (K2-1) * 400
bsl = tbn2(J1,K1) * tbn2(J2,K2)
DO J3 = 1, 3, 1
DO K3 = 1, 20, 1
if (abs(tbn2(J3,K3)) .gt. MinTol) then
L3 = (K3—1) * 20
sz = bsl * tbn2(J3,K3)
DO J4 = 1, 3, 1
ts3 = tsZ * TT3(J4)
DO K4 = 1, 20, 1
if (abs(b1(J4,K4)) .gt. MinTol) then
XpVal = ev(ielem,iOffst(K4 + L3 + L2 + L1))
if (abs(XpVal) .gt. 0.d-12) then
alpha1(ic) = alphal(ic)+ts3*XpVal*b52*b1(J4,K4)
end if
end if

Figure 4.8 Segment of code for computing 0..

Given that some of matrices involved in the computation of F are sparse (have a large

number of zero-valued entries), the evaluation of Eq. 4.1 may be optimized as follows:

Within the scope of any DO-loop, for index k,, check the entry of sparse matrix M

corresponding to Ur» k ) . If M (is: ki) 5 Min T01 then, the value of F will be approximately

i
zero in all succeeding DO-loop computations, and hence the subsequent loops may be

ﬂdppui

4.3.5 Unrolllng of DO-Ioope
One of the well known techniques in optimizing FORTRAN programs is the unroll-

ing of DO-loops. Unrolling means to move, completely or partially, the independent por-
tion of the computations outside the affected DO-loops. These computations may involve
actual numerical values, deﬁned constants (it, e, etc.), simple variables or even matrices.
The main criteria necessary to move a computation outside of a given DO-loop is that the
variables in the computation are required to be independent of the given DO-loop’s index.
Unrolling a computation with m arithmetic operations outside a DO-loop with n-iterations

will save approximately m x (n - l) operations.

In this program implementation, the unrolling technique was extensively utilized to
optimize cascaded DO-loops and further enhance the performance of the program. To il-
lustrate the beneﬁts of this technique, the computation of the displacement covariance ma-

trix is shown in the code segment in Fig. 4.8

do I1=1,60,1
do J1=1,60,1
cov(I1,J1) = 0.d0
do 12:1,60,1
do J2=1,60,1
ztmp = 0.d0
do I3=1,ndof,1
do J3=1,ndof,1
ztmp = ztmp + g(I3,I2) * g(J3,J2)
end do
end do
cov(Il,Jl)=cov(Il,J1)+ztmp*z(11,12)*z(J1,J2)*dint(I2,J2)
end do
end do
end do
end do

Figure 4.9 Code segment illustrating unrolling of Do-loops

In the code of Fig. 4.8, the sub-computation

C = Z(i1, i2) XZUpfz) >< dint(i2,j2)

69

 

-— Do-Loop Unrolling / j

 

 

-- Straight Implementation /
5 - / -
/
. / ,
//
4 /
/
/

/ 4

u
\

Actual time in sec (x 100)
N

t—i

 

 

 

 

 

2
Number of elements per row/column

Figure 4.10 Do-Loop Unrolling vs. Straight Implementation.

is totally independent of the DOJoops corresponding to 1'3 and 1'3. Thus, C may be moved
outside these DO-loops resulting in the saving of ops (C) x (ndofz - 1) , where 0PS(C)
is the number of arithmetic operations in the computation. When all the DO-loops for the
above implementation are considered, the total number of operations saved by this simple
technique is approximately OPS (C) x (ndofz - 1) x 604. The actual timing for the
above illustration was measured on a Sun SPARC ELC workstation and is shown in Fig.
4.9 where it is compared to the straight forward implementation for different DO-loop di-

mensions. The computation time was measured using the UND( time command.

Because of its wide applicability, this technique was used throughout the program.
The interested reader is encouraged to go through the program listing to realize the impor-

tance of this technique in reducing the overall execution time of the program.

Table 4.8 presents a ﬁnal comparison between a straight-forward implementation and

the optimized version of the program. Four seperate runs were made with the same loading

70

conditions and geometry while varying the number of layers. Note that the total time for
each run is directly dependent on the number of iterations required for convergence and
hence, in the above table, the time for 5 laminae (4 iterations) is less than the time for 4
laminae (5 iterations). The fourth column of the table, i.e. the ratio of non-optimized to op-

timized time, suggests an average ratio of 40. For this speciﬁc example, (with 4 laminae),

the optimized version takes only around two hours, while the straight implementation takes

TABLE 4.2 Comparison of total times.

 

 

 

 

 

 

Stacking Sequence Optimized (sec) Non-optimized (sec) Non-Opt/Opt
[30l-30] 4100.6 150871 36.8
[30/0/-30] 3734.4 1701365 45.5
[301301-301-30] 6968.4 2992303 43.0
[30/30/01-3OI-30] 5558.2 2801876 50.4

 

 

 

 

 

roughly 3.5 days. All times in the table were measured on Sun SPARC 10 workstations.

71

5. Numerical Results

5.1 General

In this chapter a number of numerical examples of in-plane and out-of-plane respons-
es of laminated cantilever plates are presented. Various types of loading, ﬁber orientations
and stacking sequences are considered, and displacement, strain and stress responses are
computed. In addition, the twisting effect due to shear coupling in unsymmetrical laminat-
ed composites is discussed. The basic units of length and force are taken to be inches and

pounds.

5.2 In-Plane Loading
5.2.1 Extensional loading

To study the effect of material nonlinearity on the response of a composite laminated
plate loaded in extension, a cantilevered plate made of Boron/Epoxy Narmco 5505 was
considered. The plate was modeled with nine ﬁnite elements and excited by the boundary
loads shown in Fig. 5.1, 5.5 and 5.9, with P being a zero-mean white noise excitation. A

white noise has a spectral density function (SDF) that is constant at all frequencies. i.e.,

S((o) = so , —oo<a)<oo
While the r.m.s. of the load, which is the square root of the area under the SDF, is undeﬁned
for white noise, the response can be evaluated in closed—form. The intensity of the load is
therefore characterized by the level of the SDF, So. Note that since the r.m.s. load has units

of lb, So has units of lb2 sec.

The level of the excitation spectrum, S , was increased from 1000 to 120001b2 sec
and the displacement, strain and stress responses computed. The fifth order approximation
shown in Fig. 2.2 was used for the nonlinear shear stress-strain relation. Three cases of ex-

tensional loading are investigated and the results are discussed below.

72

Case I: A single-ply plate with ﬁber orientation or. Two values of or, 30°and 60°

were used for comparison purposes.

4*“

s
s

 

 

 

 

15"

 

71—

Figure 5.1 One-ply plate loaded in tension

Table 5.1 shows the ﬁrst ﬁve undamped natural frequencies of the plate from the ﬁrst
and last iteration of the analysis, for the excitation level S0 = 120001b2 sec. The values from
the ﬁrst iteration correspond to the case where nonlinearity is neglected, while the values
from the last iteration show the effect of nonlinearity. The natural frequencies show rela-

tively small decreases ranging from 2 to 4%.

Table 5.2 shows the number of iterations required for convergence of the root mean
square (r.m.s.) nodal displacements according to the criterion in Eq. 3.50, with a tolerance
s = 0.001. The number of iterations increases with excitation load level, but shows a slight
oscillations for So between 7000 and 90001b2 sec. This is probably due to the slight oscil-

lation in the approximate ﬁfth-order shear stress-strain law in Fig. 2.2.

73

TABLE 5.1 First five natural frequencies from linear (ﬁrst iteration) and nonlinear (last iteration)
analysis for load level So = 12000 lb2 sec

 

 

 

 

 

 

 

 

[30°] [60°] |
Mode First Iteration Last Iteration First Iteration Last Iteration
=1—__IF_—7—5% 24.6 15.8 15.3
2 52.8 49.7 47.6 46.6
3 96.7 95.0 74.5 70.2
4 139.4 134.0 95.2 93.2
5 170.5 163.2 101.5 96.7

 

 

 

 

 

 

 

 

TABLE 5.2 Number of iterations required for convergence

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Load level, so No. of Iterations
(lbzs) [30°] [60°]
1000 3 3
2000 4 4
3000 4 4
4000 4 5
5000 5 5
6000 5 6
7000 5 6
8000 6 7
9000 5 .7
10000 5 6
11000 6 6
12000 6 6

 

 

 

 

 

 

Fig. 5.2 shows the variation of the absolute r.m.s. shear strain in material coordinates
at the center of element 2 of the plate with the excitation level. For S0 = 120001b2 sec, the

r.m.s. shear strain is about 0.0055 for the [60°] and 0.0045 for [30°]. The peak shear strains

74

be expected to be in excess of three times the r.m.s. strain (i.e. in excess of 0.016 for [60°]

and 0.0135 for [30°] .

 

 

 

 

 

0.“)6 r r 1 ﬂ
— a= 30°
-- a: 60° ’J
/// 3
/
/
{5 0.004 - / -
a //
i8 //
o r
/
r?) //
to
50.002 - // -
//
/
* /
0.0 a l l I l L
0 2000 4000 6000 8000 10000 1200(
Input So (lb2 sec)

Figure 5.2 Variation of absolute r.m.s. shear strain at the center of element 2 with excitation load level

The variation of the normalized r.m.s. x-displacement of the front right corner node

with excitation spectrum level, So, is shown in Fig. 5.3. The displacement is normalized by

dividing by the linear response (for which [K‘ ] = [0] ). The ﬁgure illustrates that the

proportional increase in the displacement due to non-linearity steadily increases with the

excitation level.

The variation of the normalized r.m.s. strains and stresses (in the material directions)

at the center of element 2 with the excitation level are shown in Fig. 5.4 and 5.5. Again, the

norrnalizations have been performed by dividing by the corresponding linear responses.

The non-linearity in the constitutive law results in signiﬁcant increases in the shear strains

and stresses, but does not signiﬁcantly affect the normal strains.

75

 

2"
N

 

e—e
L.
at

Normalized RMS displacement
g s

 

 

 

 

1.04
1.0 n I . I n I . I n 1 L4 1 I n l . I I I n l k
0 .1000 2000 3000 4000 5000 6000 7000 8000 9000 10000110001200
Input 50 (1h2 sec)

Figure 5.3 Variation of normalized r.m.s. displacement at free corner nodes with excitation load

 

 

 

 

 

 

 

 

1.2 v r I r r l
— enfora=30° ,,1
—- enfora=60° ”"'
1.16 --- 712 fora= 30° , , ’ .
0 I
m - - ‘71: fora: 60 I ,
é °°°°°° e22 fora=30° ,’
_- _ ° I
”1.12 (22 fora-60 Ill .;
m ’ I I ,z".’ .
I ./
a / .-.,
1.08 I. I I ’ «I:
E I I I z" I
._ i I I I I.’./' 1
$1.04 - , , ’ ,-’ ,— ’ _
I "
C _ I . ."
z ’ {JLO’ ’ ...................................................................
1.0 " wg'#£;.él—.xi;.:; ———————— —--.\‘ _
- ~. N- ‘\
‘ “<1: \\‘\
096 1 r 1 I I I . _I_ - ‘. 1“" T' —'
0 2000 4000 6000 8000 10000 120C

Input 80 (1h2 sec)

Figure 5.4 Variation of normalized r.m.s. strains at the center of element 2 with excitation load level

76

 

_ auf0ra=30° ’a" _.

1.08 ,
—- «ﬂora-60° l”
'—- 1'12 form30° / ’ i
0 I
1.06 -" rufora=60 ” _.
------ an fora=30° , ’ ’ ,’°‘

 

 

:- :-
8 i

I"
O

Normalized RMS Stresses

0.98

 

 

 

0.96

 

Input So (le see)

Figure 5.5 Variation of normalized rm.s. suesses at the center of element 2 with excitation load level

Case 11: Two-ply laminated plate with ﬁber orientation or and -0t in the top and bot-

tom layers, respectively, as shown in Figure 5.6. The total thickness of the plate is identical
to case I.

Table 5.3 shows the ﬁrst ﬁve undamped natural frequencies of the laminated plate
shown in Fig. 5.5. Due to the nonlinearity effect, the natural frequencies show decreases

ranging from 1 to 4%.

Table 5.4 shows that the number of iterations required for convergence increases with
the excitation load level since the nonlinearity becomes more pronounced at higher loads.
For any applied load level, the number of iterations required for convergence for the [30°]
-30°] laminate is less than that for the [60°/-60°]laminate. This indicates that the nonlinear-

ity has a more pronounced effect for the [60°/-60°] laminate.

77

 

 

 

 

 

 

 

The absolute shear strain in the material coordinates at the center of element 2 is

shown in Fig. 5.7 and clearly the response is nonlinear. For any given excitation level the

 

Figure 5.6 Two-ply laminated plate loaded in tension

 

shear strain for the [60°/-60°] laminate exceeds that for the [30°I-30°]laminate.

The normalized r.m.s. x-displacement of the front corner node is shown in Fig. 5.8.
The ﬁgure illustrates the effect of nonlinearity which is signiﬁcant for the [60°/~60°] lami-
nate and negligible for the [30°l-30°] laminate.

TABLE 5.3 First ﬁve natural frequencies from linear (first iteration) and nonlinear (last iteration)

analysis for load level 30 = 12000 lb2.sec

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[30°/—30°] [60°/-60°]
Mode
Flrst Iteration Last Iteration First Iteration Last Iteration
1 n 24.1 HZ 23.8 HZ 15.6 HZ 15.2 HZ
2 74.5 73.9 46.4 45.4
3 92.2 91.4 86.7 84.9
4 152.5 150.3 95.0 92.3
5 215.8 213.4 103.7 100.0

 

78

 

TABLE 5.4 Number of iterations required for convergence

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Lead by“, So No. of Iterations
("’2“) [30°/-30°] [60°/—60°]
1000 r1 3 3
2000 3 4
3000 3 4
4000 3 4
5000 3 4
6000 4 4
7000 4 5
8000 4 5
9000 4 5
10000 4 5
11000 4 5
12000 4 5

 

 

 

 

 

 

The variation of the normalized r.m.s. shear and normal strains and stresses in mate-
rial directions attire center of element 2 with the excitation level are shown in Fig. 5.9 and
5.10. The nonlinearity in the constitutive law results in signiﬁcant increases in the normal

and shear strains and normal stresses for the [60°/-60°] laminate.

79

 

 

 

 

 

 

 

I ' I I ' I ' I ‘ I ' I I I '
— a=30° ,
0.005 __ a=60° ’/’ "'
/
p /// 1
//
0.004 - ,I’ .
r: //
'8 _ /
b //
CD //
50.003 - // -
Q L //
m //
to
50.002 - // -
0.001
0.0 l 1 I 1 l 1 I 1 I n I n l n I 1 I a I a I 1
O 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 1200(
Input So (le see)

Figure 5.7 Variation of absolute r.m.s. shear strain at the center of element 2 with excitation load level

 

 

 

 

 

 

 

I I ' I ' I I I f I I ' I f

1.24 _. (1:30: /’/'
H —- a=60 /// 3
‘3 /
o 12 l- -
a //
o /
o /
..‘Ei /
c1116 - // a
em /
5". /
E l 12 - // '-

/
:g 1.08 r /// 'i
8 //
z 104 i' // "‘
//
/
1.0 QT 1 1 4 4 1 . 1 . 1 . 1 . 1 . 1 1 1 1 1 1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000
Input So (1b2 sec)

Figure 5.8 Variation of normalized r.m.s. displacement at free corner nodes with excitation load level

80

 

 

 

 

 

 

 

 

1.2 r r r r '
— en fora=30° '
1'16 —" C“ {01113600 a:
°—' 712 fora=30: . I. " ..
m 1.12 _ _ 7121.5“: 60 . /_ ’— ..
a 108 ------ e22 fora=30: ‘ ,o ’ ___________ L
g . —- (”£08.03“) ’0 :‘v’ﬁ’.’ .
(01.04
5 1..
goes .
£0.92 :- \\\\ -
0.88 - \\\ g
b \\\ q
0.84 '- \\\\ -1
. \
0.8 . 1 . 1 . 1 . 1 . 1 T‘4
0 2000 4000 6000 8000 1W0 120“

Input 3,, (1h2 see)

Figure 5.9 Variation of normalized r.m.s. strains at the center of element 2 with excitation load level

 

_ a" fad: 30°
_"" 0'" fora= 60° ’0 ‘
-—- r,,iom=30° , I
- - T12 for“: 60° /'
""" 022 fora=30° /
'—" 0221-0111-1600 '

p—
O
.—

 

 

 

Normalized RMS Stresses

 

 

 

 

Input S0 (1b2 sec)

Figure 5.10 Variation of normalized r.m.s. stresses at the center of element 2 with excitation load level

81

Case 111: Three-ply laminated plate with ﬁber orientation of a, 0 and -a in the top,
middle and bottom layers, respectively, as shown in Fig.5.] 1. The total thickness of the

plate is the same as in cases I and II.

 

 

Figure 5.11 Three-ply laminated plate loaded in tension

Table 5.5 shows that the values of the ﬁrst ﬁve natural frequencies from the nonlinear
analysis (last iteration) are very close to those from the linear analysis (ﬁrst iteration). This

indicates that the nonlinearity has very little effect on the three-ply [30°/0°/-30°] and [60°]
0°l-60°] laminates.

The number of iterations required for convergence as shown in Table 5.6 seems to
be invariant with the increasing load level. This conﬁrm that the nonlinearity of the consti-

tutive law has a negligible effect on the three-ply laminated plate shown in Fig.5.9.

The variation of the absolute r.m.s. shear strain in the material coordinates at the cen-

ter of element 2 with excitation load level is signiﬁcantly less than for the two cases dis-

82

cussed previously. Fig. 5.12 and 5.13 show that the nonlinearity has negligible on the

normalized r.m.s. shear & normal strains and stresses

 

 

 

 

 

 

 

 

 

 

 

 

TABLE 5.5 First five natural frequencies from linear (first iteration) and nonlinear (last iteration)
analysis lot load level 30 = 12000 11:2 sec.
[30°/0°/—30°] [60°/0°/-60°]
Mode
First Iteration Last Iteration First Iteration Last Iteration

1 23.2 HZ 23.4 112 17.9 HZ 17.9 HZ

2 81.4 81.6 65.2 65.1

3 101.5 101.6 80.7 80.4

4 146.9 147.8 110.0 109.6

5 260.6 261.3 215.3 215.3

 

 

 

 

 

 

 

TABLE 5.6 Number of iterations required for convergence

 

Load level, S
(lbzs) °

[30°/0°/-30°]

No. of iterations

[60°/0°/-60°]

 

 

1000

 

2000

 

3000

 

4000

 

5000

 

6000

 

7000

 

8000

 

9000

 

10000

 

11000

 

12000

 

wwwwwwwwwmwm

 

 

hhmwwmwwuwwu

 

 

 

83

 

 

 

 

 

 

 

 

1.0001 , . ' . .
_ 0" forms 30°
—- a" fora=60° ,
'—' Tu {“03300 ”—"‘l
8 - " 1'12 fora: 60° ’I”’
a LOWS ...... 022 fora=30° ’l”’ -l
g "' ”W“ 60° ,4,” ...,- .... ----o 1...:
U) I” _ —--" “-- ................... 1
’ .n—I' "'- -----------
m ”’, "t..:.—...u --------------
E 1.0 $6.38 -"'""‘°""" '_ _"__‘_ _'___:
% ~ ~ ‘ ~ ~ ~ ~ - _ cl
o 0.99995 '- ~ ~ ~ ~ ~ _ -
Z
0.9999 8 l ‘ l 1 l
0 20 40 60
Input So (1b2 sec)

Figure 5.12 Variation of namalized r.m.s. stresses at the center of element 2 with excitation load level

 

 

 

 

 

 

 

s l t I ' I t I
— Cu fora=30°
—" £11f0r0= 60° ‘
'_' ‘le fora=30° ,z”
‘0 1'01 " " 712f0fa=60° I’,/ a
.5 ...... ‘22 fm$300 ”’
g -' £22 forar= 60° ’/’ . ’1
(I) l/ ’. ’
031005 '- l” ‘ ’. ” ................... J
. .I’ _____ ...— ...............
E ”’ . -——- :3: ................
8 I’:—’"’.’ --------- .
E 10 4%" "”5".” ‘_ _________J _________________________ _;
E I- ~ ‘ ‘ ~ ~ ‘
0.995 .- ~ ~ ‘ ~ § 5 _.
0.99 l l 1 1 . l 1 1 A ‘ Q
0 2000 4000 6000 8000 10W 120
Input 80 (1h2 sec)

Figure 5.13 Variation of normalized r.m.s. strains at the center of element 2 with excitation load level

0.2\

“5‘ [30°/-30°]

O.1\

   

o.os\ [30°/0°/—30°]
0\
-o.05N
-o.1 \

~O.15\

 

-O.2_,
0

Figure 5.14 Twisting due to extensional loading for [30°], [30°]-30°] and [30°/0°/-30°] at S0 = 8000

The reason for the behavior exhibited by the three-ply plate is the presence of the mid-
dle layer with a ﬁber angle of 0°. Due to its ﬁber orientation, this layer is stiffer than the
others, carries a larger share of the load, and since it responds essentially linearly to exten-

sional load the overall response is approximately linear.

Fig. 5.14 shows the twisting effect which exists in unsymmetrical laminates. This im-
portant phenomena is due to the induced out-of-plane displacements by the extensional
load. As expected the coupling between the in-plane loading and out-of-plane displacement
is observed for cases I and II. Also, the ﬁgure indicates that less twisting is induced in the

three-ply plate than in the two-ply plate.

85

5.2.2 Shear loading
The same cantilever plate used in the in-plane extensional loading with the same ma-

terial properties and the same geometry is considered again, but now the loads are applied
along the in-plane DOF in the y-direction. The white noise excitation spectrum level, So,

was increased from 1000 to 4000 lb2 sec.
Case I: One-ply plate with ﬁber orientation or. Two values of or, 30° and 60° were

used. Figure 5.15 shows the plate and the loaded DOF

 

 

 

 

 

 

 

 

a“
\
s
\\ o
‘1“
N
s _ .
§ 0 — or
\
\
s ..
15
IA
I‘
Figure 5.15 One-ply plate loaded in shear

Table 5.7 shows the effect of nonlinearity on the ﬁrst ﬁve values of the natural fre-
quencies from linear and nonlinear analysis for excitation level So = 4000 lb2 sec. Due to

the effect of the nonlinearity, the values of the natural frequencies dropped by approximate-

ly 3 percent for both [30°] and [60°] one-ply plates.
The number of iterations required for convergence, listed in Table 5.8 increases with

excitation load level since the nonlinearity becomes more dominant for higher loads. Table

5.8 also shows that a very large number of iterations are required at So > 50001b2 see. This

86

is because for very high load levels, the shear response exceeds the range for which the ap-

proximate ﬁfth-order shear stress-strain law is applicable.

TABLE 5.7 ﬁrst ﬁve natural frequencies from linear (first iteration) and nonlinear (last iteration)

analysis for load level so = 4000 11:2 sec.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[30°] [60°]
Mode First Iteration Last Iteration First Iteration Last Iteration
1 n 25.5 HZ 24.7 HZ 15.7 HZ 15.5 HZ
2 52.8 50.6 47.6 46.7
3 96.7 95.5 74.5 72.0
4 139.4 133.2 95.2 94.0
5 170.5 166.1 101.5 98.5

 

TABLE 5.8 Number of iterations required for convergence

 

 

 

 

 

 

 

 

 

 

Lead by“, so No. of Iterations
(lb 3) [30°] [60°]
1000 4 4
2000 5 4
3000 5 4
4000 5 4
5000 3 4
6000 12 4
7000 21 5

 

 

 

 

 

 

The variation of the absolute shear strain in the material coordinates at the center of
element 2 with excitation load level is shown in Fig. 5.16. The responses for the [30°] ply
exhibits more nonlinearity than that of the [60°] ply. For So = 40001b2 sec, the r.m.s. shear
strain is about 0.0062 for the [30°] ply. The peak shear strain would be expected to be in
excess of three times the r.m.s. strain, which places it near the upper limit for which the
ﬁfth-order shear stress-strain law of Fig. 2.2 is applicable. For shear strains exceeding 0.02,

the ﬁfth-order law increases very rapidly, resulting in poor convergence.

87

The variation of the normalized r.m.s. y-displacement of the corner nodes with the ex-
citation spectrum level So is shown in Fig. 5.17. The ﬁgure illustrates that the proportional
increase in the displacement due to the nonlinearity increases with the excitation level. The
apparent increase in stiffness of the [30°/-30°] laminate as So exceeds 4000 lb2 sec is again
due to the fact that the approximate ﬁfth-order stress-strain law breaks down for the high

loads.

Figs. 5.18 and 5.19 show the normalized r.m.s normal and shear strains and stresses
in material coordinates at the center of element 2 with the excitation level. The nonlinearity
in the constitutive law has a signiﬁcant effect on the shear strain for the [30°] ply (about
13%), but it shows negligible effect for the [60°] ply (less than 2%). Nonlinearity has a neg-

ligible effect on all stresses.

 

 

 

 

 

 

0.007 I I r ,
— a: 300 ‘
—- a: mo
0.006 -
a0.005 - d
b I
E0004 - -
a b
”3 0.003 - u
i
0.002 - _
om“ —————J
P ..., —————————————— *
0.0 —-"‘""'1” 4 . 1 . 1 .
0 1000 2000 3000 4000
Input So (“32 sec)

Figure 5.16 Variation of absolute r.m.s shear strain at the center of element 2 with excitation load level

88

G

-
I ll

«50

c. l.
1

F503: at: :4 m2 Va SUN—.1352. 2

Tile

v..C_r..:.r... m2 Hm 30.12.3223le

no

 

1.1

 

 

 

 

 

I I I
— ar=30°
--- 0:600
‘5
o 1.08 - -
E
8
.2
8‘
:6 1.06 "'
2”;
g 1.04 - -
8102 ,.. ”’.—,” ..
2 ’II”
P ””’ 1
1.0 ’II" 1 1 1
0 1000 2000 3000 4000
Input 30 (1h2 sec)

Figure 5.17 Variation of normalized rms. displacement at free comer nodes with excitation level

 

_ Eu {01' a= 30°
—" C" for a= 60°
-—' 7.2 for 0: 30°

 

 

 

 

 

 

 

 

m - - 112 fora=60° ..... —.
.5112 ------ enfora=30° ./-—- —————— _
—- enfora=60° If.
./'
331.08 - ,.z _
1:11.04 -
.8
'3
a 1.0
E
0.96 -
1
0.92 - .
. 1 _ I l 1
0 1000 2000 3000 4000

Input So (le sec)

Figure 5.18 Variation of normalized r.m.s. strains at the center of element 2 with excitation load level

89

 

I

aUaanﬁﬁ $.21 305:.3: .32

int

351'

It)

(111

iii

dis

Iht‘

lei

 

1.002 I ﬁ I I
-— an for at: 30° J
—- 01110! a: 60° ’"
LWIS °—° 7'12 for a= 30° .’,/' d
"' - T12 {“03 60° .’,/"
------ an fora: 30° ’./”
1.001 —' "22‘0”“ 60° ,,-” ............... A

 

 

Normalized RMS Stresses

 

 

 

 

Input So (le sec)

Figure 5.19 Variation of normalized r.m.s. stresses at the center of element 2 with excitation load level

Case 11: Three-ply laminated plate with ﬁber orientation of or, 0 and —or in top, middle
and bottom layers, respectively, as shown in Fig. 5.17. The total plate thickness is the same

as in Case LThe level of excitation was increased from 5000 to 300001b2 see.

Table 5.9 shows the ﬁrst ﬁve undamped natural frequencies from the linear and non-
linear analysis (ﬁrst and last iteration) of the three-ply laminated plate for the load spectrum
level So=30000 lb2 sec. Due to the softening effect of the shear nonlinearity, the natural fre-
quencies show about 3 to 7% decreases. The laminate with the [30°I0°/-30°] arrangement is

stiffer than the [60°/0°l-60°] laminate, as indicated by the higher natural frequencies.

Table 5.10 shows the number of iterations required for convergence of the r.m.s. nodal
displacement. For the [30°/0°l—30°] laminate the number of iterations remain constant with
the excitation load level, but the number of iterations increases with the excitation spectrum

level for the [60°/0°l-60°] laminate. A large number of iterations are required for the [60°/0°!

90

 

 

 

 

 

 

 

\
s
/ 2 / 5 / . /
a
s 1 4 7
‘1‘”
§ 0=ct°
§ e=0°
§ 0=-or°
s
s
15‘

-60°] laminate at so=2s,ooo 1h2 sec. This indicates that the fifth order shear stress-strain law
is not applicable for the [60°/0°l-60°] laminate at any load level about and beyond 25000

 

 

Figure 5.20

Tree-ply laminated plate loaded in shear

 

 

 

 

 

 

 

 

1b2 sec.
TABLE 5.9 ﬁrst five natural frequencies from linear (ﬁrst iteration) and nonlinear
(last iteration) analysis for load level S0 = 30000 lb2 sec
[30°/0°/-30°] [60°/0°/-60°]
Mode
First Iteration Last Iteration First Iteration Last Iteration
1 EL 23.4 HZ 21.8 HZ 18.0 HZ 17.7 HZ
2 81.6 79.7 65.3 64.0
3 101.6 100.4 80.7 78.1
4 147.9 137.7 110.0 106.6
5 261.3 253.6 215.3 213.3

 

 

 

 

 

 

 

91

 

TABLE 5.10 Number of iterations required for convergence

 

 

 

 

 

 

 

 

Load bye], 50 No. of Iterations
(”’2“) [30°/0°/—30°1 [60°/0°/-60°]
5000 4 5
10000 5 6
15000 5 6
20(X)0 5 6
25000 5 9
30000 5 14

 

 

 

 

 

 

Fig. 5.21 shows the variation of the absolute r.m.s. shear strain in material coordinates
at the center of element 2 with excitation load level. The responses are clearly nonlinear.

The effect of nonlinearity is comparable for both the [30°/0°l-30°] and [60°/0°l-60°] lami-

 

 

 

 

 

 

nates.
0.008 . . ﬂ , . , - ,
—- a=30°
0.007 '“ “=60 4
. ,,’:
0.006 - /”’ -
G r /’/’
ea //
1:10.005 " /// '-
m _ /
1a //’
00.004 - / -
£3 /
m D /
m //
50.003 - // a
» /
0.002 - / -
0.001 - .,
. 1
0.0 4 j 1 I a L a I a l a
0 5000 10000 15000 20000 25000 3000
Input 30 (le sec)

Figure 5.21 Variation of absolute r.m.s. shear strain at the center of element 2 with excitation load level

92

 

the

3‘51

re:

00

uchF- firu-u—fhi -¥\

0‘1a

’\€.l U I I-
. EIII!
~l

The variation of the normalized r.m.s. y-displacement due to nonlinearity with exci-
tation level, So, is shown in Fig. 5.22. There is a steady increase in the displacement with
the excitation spectrum level due to the effect of nonlinearity. The fifth order approxima-

tion is not applicable for the [60°/0°l-60°] laminate at load levels above200001b2 sec.

Fig 5.23 shows that the nonlinearity in the constitutive law has a signiﬁcant effect on
the shear strain of the [30°/0°l-30°] laminate and a negligible effect on the shear strain of
the [60°/0°l-60°] laminate. The normalized normal strain in material direction 1 (all) is rel-
atively insensitive to nonlinear effects, whereas the normalized normal strain in material di-
rection 2 (622) shows a signiﬁcant increase for both angles. Fig. 5.24 shows that

nonlinearity is again negligible for all stresses.

 

 

 

 

 

1.08 ’ I V l ' I ' l

-— a=30°

—- a=60°
‘1'?
g 1.06 - -
.9.
O.
.."3 ______ \
'O // \\

/ \
3“” “ ,/ N.
/
//
§ /
.... //
a 1.02 - / a
Q /
Z //
/
/
1.0 n l a L n L A J 1
0 5000 10000 15000 20000 25000
Input 30 (le sec)

Figure 5.22 Variation of normalized r.m.s. displacement at free comer nodes with excitation load level

93

H1..-11.1.li..|.|..l
l. m5..5.5V. m2“ UQNIﬂCF—CZ

1

0. 0. ”Hug 1|: 1|... 1|,“
F1 aOmaUu-W Viz“ VONZQFCLCZ

 

 

 

, I ,
'— Cu fora=30° —_ _______-_:
1.16 —" E“ fora=60° .’././ .
1.14 '_' 712f0ra=30° /././ _
rn "" 712f0fa=60° ./. - ‘
‘3 ° / /- -—- -— ~- ‘
n81.12 ----- ﬁzzfora=3o .I. /. _”.:
1:: ll —- enfora=60° /.< /' ..................... .
(I) - /- / .......... _
m ’ /'/‘ ........ .
1. L /" _______ q
08 _ //../ ......... %
ul.06 - -/. / ..... ..

 

 

 

 

0.96 - “\___ ________ _____——
l l l . 1 . l .
0 5000 10000 15000 20000 25000 30000
Input so (1b2 see)

Figure 5.23 Variation of normalized rrn.s. strains at the center of element 2 with excitation load level

 

E

E

 

 

3"
O

0.9998

Normalized RMS Stresses

 

 

 

 

0.9996 -
l 1 J A l
0 20 40 60
Input So (1b2 sec)
Figure 5.24 Variation of normalized r.m.s. stresses at the center of element 2 with excitation load level

 

5.3 0U1
“'16

the same
z-diI‘ECtil

Ca
used. Fl

/////////////////; ’7”, -

and nc
ply ha
Stiffer

116in t

 

a1 dii

load

5.3 Out-Of-Plane Loading

The same cantilever plate used in the in-plane extensional loading is considered with
the same material properties and the same geometry, but now the out-of-plane DOF in the

z-direction are loaded. The level of excitation was increased from 5 to 80 lb2 sec.

Case I: One-ply plate with ﬁber orientation at. Two values of at, 30° and 60° were

used. Figure 5.25 shows the plate and the loaded DOF.

   

 

 

 

 

 

 

 

 

>1

 

Figure 5.25 One-ply plate with ﬁber orientation 0:

Table 5.11 shows the ﬁrst ﬁve undamped natural frequencies of the plate from linear
and nonlinear analysis for the excitations level corresponding to So = 801b2 sec. The [30°]
ply has higher natural frequencies than the [60°] ply which indicates that the former ply is
stiffer. Due to the softening effect of the shear nonlinearity, the natural frequencies show

negligible decrease for the [60°] ply, and a slight decrease for the [30°] ply.

Table 5.12 shows the number of iterations required for convergence of the r.m.s. nod-
al displacements. The number of iterations remains approximately constant for the range of

loads and comparable for [30°] and [60°]

95

TAE
ter:

 

of e}

give

1th
[on
ma].
chat

resu

 

TABLE 5.11 First five natural frequencies from linear (first iteration) and nonlinear (last
iteration) analysis for load level So = 80 lb2 sec

 

 

 

 

 

 

 

 

[ 30°] [60°]
Mode First Iteration Last Iteration First Iteration Last Iteration
1 p 25.5 24.6 15.7 15.4
2 52.8 51.3 47.6 47.1
3 96.7 95.5 74.5 71.7
4 139.4 135.8 95.2 93.7
5 170.5 166.6 101.5 99.3

 

 

 

 

 

 

 

 

TABLE 5.12 Number of iterations required for convergence

 

 

 

 

 

 

 

 

Lead by“, So No. of iterations
("’2“) [30°] [60°]
5 n 3 3
10 3 3
20 4 4
40 4 4
60 4 4
80 4 4

 

 

 

 

 

 

The variation of the absolute r.m.s. shear strains in material coordinates at the center
of element 2 are shown in Fig. 5.26. The responses are clearly nonlinear. Note that for any
given excitation level, the shear strain for the [30°] laminate is less than that for the [60°]
laminate. The variation of the normalized r.m.s. z-displacement of the comer nodes with
the excitation spectrum level, So, is shown in Fig. 5.27. The nonlinearity is less pronounced
for the ﬁber orientation of [30°] than for [60°]. The variation of the normalized r.m.s. nor-
mal and shear strains and stresses in material directions at the center of element 2 with ex-
citation level are shown in Figs. 5.28 and 5.29. The nonlinearity in the constitutive law

results in a slight increase of most strains and stresses.

96

0.012

Ill
0
0

00:

Atll

]

.It. WU. n. 0
Hum—«haw hdUn—mﬂ met—2N—

thu

2.05.0057—37 r02! UON:-::CZ

 

 

 

 

 

 

 

0.012 r v I ﬁ ﬁ
-— a= 30°
—- a- 60°
0.01 " -l
i- ’I’4
’
P 0.008 - //”’ -
I
V) II/
a //’
00.006 P // '
a /
m
/
50.004 "' "
. /’ l
//
0.m2 " / d
o /
0.0 a I L I A l n
0 20 40 60 80
Input So (lb2 sec)

Figure 5.26 Variation of absolute rrn.s. shear strain at the center of element 2 with excitation load level

§§§

‘é

Normalized RMS displacement
s E

.H
O
H

 

 

 

 

 

 

— a: 30° .
—- a= 60°
/ ————————
///
//
' // u
/
/ 1
/
.. /
/
/
/ 'l
l" -t
/
/
D / d
/
, /
/
I A I A I A
0 20 40 60 80
Input So (1b2 sec)

Figure 5.27 Variation of normalized rms. displacement at free comer node with excitation load level

97

 

 

 

 

 

 

 

1.12 . . . . .
_ €11 fora: 30° ‘
_" Cu fad: 60°
1.1 "" 712 fora=30° .-
rn - — 712 f0fa= 60° l”—____-
'31 03 """ ezzfow=30° I’,’ _
b ' —' £22 for a= 60° //’

106 " // ’ ’ v a’ d
E // ’ ’ ’ .......... .::_ ..g-q
U // ’ I ’ ......... ;.:-...’u ..—— ——
31.04 — // ’ ’ ’ ,lI” "
a r- / I I ........’.o...’.’o

H102 /f’ ’ ..n,"I’ -
° ' _ / I 5" __- _- __ __ _- _- _. -_
Z _ ”4;...57 _.- .. -—-- --
1" ’- ...-
I...
1.0 L
0.98 ‘ 1 n 1 L
0 20 40 60
Input so (11)2 sec)

Figure 5.28 Variation of normalized r.m.s. strains at the center of element 2 with excitation load level

 

 

 

 

 

 

 

1.12 I l
— an fora=30°
—- a“ fora: 60°
11 ’_' Tu fOfO=30° q
§ - - 1'12 for a= 60°
0
“1.08 """ 022 fora: 30° .-
g —- an forar=60 ”‘
m affﬂd” ‘
$1.06 - //// 1
/// ....--°--'::.‘
'° / ----------- ::;.--—-—'
'- / .......... ___..—-
.81“ /// ...o...._:.’.:'.’ ’ ’I' - - ’1
/ ........ 0’ ’ ........ 1
/ ...... ,"’ — a v "’ '—
Ig // ...... -°,". '1 .. -- '
81'02 - // ...... :a‘ .— -- "' " - —_- __ _- __ __ _:
z b // ..vé'fwzkj .——- ——-- ’- ...-
/’ﬂ£- "
10 “—— ..
0.98 ‘ J m I L
0 20 40 60

Input So (lb2 sec)

Figure 5.29 Variation of normalized r.m.s. stresses at the center of element 2 with excitation load level

98

Case 11: Two-ply laminated plate with ﬁber orientation or and -0t in top and bottom

layers, respectively, as shown in Figure 5.30. The total plate thickness is the same as in

P1

Case I.

 

 

 

 

15" _
VI
Figure 5.30 Two-ply plate with ﬁber orientation (1
Table 5.13 lists the ﬁrst ﬁve natural frequencies from linear and nonlinear analysis
for the excitation level So = 80 1b2 sec. Due to the softening effect of the shear nonlinearity,
the natural frequencies show a slight decrease. Note that the laminate [30°/-30°] is stiffer

than that of [60°/-60°] as indicated by its higher natural frequencies.

Table 5.14 indicates that the number of iterations required for convergence of the
[30°/80°] laminate are comparable to those of the [60°/-60°] laminate. The number of iter-

ations is approximately the same at all load levels.

Fig. 5.31 shows that the r.m.s. shear strain exceeds 0.007 at So values of about 75 lb2
sec for the [30°/-30°] laminate and about 45 1b2 sec for [60°/-60°] laminates. As described
earlier, for load levels in excess of these values the approximate ﬁfth-order shear stressj

strain law breaks down.

TABLE 5.13 ﬁrst five natural frequencies from linear (first iteration) and nonlinear (last iteration)
analysis rot load level 30 .. 80 11:2 sec

 

 

 

 

 

 

 

 

[30°/—30°] [60°/-60°]
Mode
First Iteration Last Iteration First Iteration Last Iteration
u
1 24.0 112 23.1 HZ 15.6 HZ 15.2 112
2 74.5 73.3 46.4 45.8
3 92.2 91. 86.7 85.3
4 152.5 148.2 95.0 92.7
5 215.8 212.6 103.7 101.4

 

 

 

 

 

 

 

 

TABLE 5.14 Number of iterations required for convergence

 

 

 

 

 

 

 

 

Load 19"., so No. of Iterations
“b2" [30°/40°] [60°/—60°]
5 3 3
10 3 3
20 4 4
40 4 4
60 4 3
80 4 4

 

 

 

 

 

 

Fig. 5.32 shows that nonlinearity causes the r.m.s. z-displacement of the comer nodes
to increase by about 6% for the [30°/-30°]laminate and about 3.5% for the [60°/60°] lam-

inate at So values of 80 and 60 1b2 sec, respectively.

Figs. 5.33and 5.34 shows the effect of nonlinearity on the normal and shear strains
and stresses. They indicate that the nonlinearity in the constitutive law signiﬁcantly affects

most strains and stresses.

100

 

 

 

 

 

 

0.012 . . , . l f
——agm°
—- a=60°
0.01 " J
’z’I” ‘
G //
130008 '- ///’
a //
30.006 - // -
m ///
VJ //
50.004 - // 1
/
/
/
0.002 l- / .l
o /
00 I 4 n l . I .
20 40 6O 80
Input so (11:2 sec)

Figure 5.31 Variation of absolute rms. shear suain at the center of element 2 with excitation load level

 

 

 

 

 

 

1.07 I l f

—- a=30° .

—- a=60°
5
g t
..‘3 1.05 a
Q h
.m
6 1.04 - ,__~ -
m y”’ “~~“.

/
/
E 1.03 .. // ..
g _ /
o //
Erm- // -
o ' //
Z _ / ..
101 //
P /
1.0 I . I n I .
20 40 60 80
Input So (le sec)

Figure 5.32 Variation of normalized r.m.s. displacement at free comer node with excitation load level

101

 

1008 I ' I ' I
_ C“ fora= 30°
-- enfora=60° .
'—' 712 fmﬁ30°
- " 712 fora=60°

------ e22 fora: 30: O, "'
-- e22 fora=60 /,/

§

 

 

E

I"
O
N

Normalized RMS Strains

 

 

 

1.0

 

Input so (1b2 sec)

Figure 5.33 Varariation of normalized runs. strains at the center of element 2 with excitation load level

 

 

 

 

 

 

 

1.12 I I I q I
— an fora=30°
-- unfora=60°
1'1 '—' 1'12 fora=30° ‘
o
m - ‘- 7121.01'0360 "—~~_:
8 ----- a fora=30° ,I”
”108 22 ” -
o - __ e ”
I: unfora=60 /
m // .— —- " " .- ----- d
0,106 " // r "' ’ ‘- ’ "'
/ , ’
E h / ’ oooooooooooo 2H
/ ’ ’ ......... :2.“ __..——
”10:“ / ,I’ ----------- .21: 531’” .-
E // ’ ’ ’ ..... ;.;;..’.’.
/// I I ...,;“L:"' 1
E102 l- //’ ’ ..o0":...’.:’ .1
E ,”.---I"’ . ...- .—-- —- —- —- —' —- —- _.
- /, .v/°""_..- I" "" '— ‘
’53: '
1.0 4
‘
0.98 a I a L a I
0 20 40 60
Input So (le sec)

Figure 5.34 Varariation of normalized r.m.s. stresses at the center of element 2 with excitation load level

102

 

Case HI: Three-ply laminated plate with ﬁber orientation of at, 0 and -0t in the top,
middle and bottom layers, respectively The total plate thickness is identical to that used in

Cases I and II.

 

 

 

 

 

=I

Figure 5.35 One-ply plate with ﬁber orientation a

Table 5.15 shows the ﬁrst ﬁve undamped natural frequencies of the plate from the ﬁrst
and last iteration of the analysis, for the excitations level So = 80 lb2 sec. The values from
the ﬁrst iteration correspond to the case where nonlinearity is neglected, while the values
from the last iteration show the effect of nonlinearity. Due to the softening effect of the shear
nonlinearity, the natural frequencies show slight decreases. The ply arrangement with [30°/
0°/-30°] is stiffer than the arrangement with [60°/O°/-60°], as indicated by the higher natural

frequencies for the former case

Table 5.16 shows the number of iterations required for convergence of the root—mean-
square (r.m.s) nodal displacements according to the criterion in Eq. 3.50 with a = 10's.

The number of iterations required for convergence increases with the excitation load level

103

TABLE 5.15 First five natural frequencies from linear (ﬁrst iteration) and nonlinear (last iteration)

analysis for load level So = 80 lb2 sec

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[30°/0°/-30°] [60°/O°/—60°]
Mode
First Iteration Last Iteration First Iteration Last Iteration
1 23.2 HZ 22.8 112 18.0 HZ 17.7 112
2 81.4 80.8 65.3 64.6
3 101.5 100.9 80.7 79.4
4 146.9 143.4 110.0 108.6
5 260.6 258.2 215.3 214.4

 

TABLE 5.16 Number of iterations required for convergence

 

 

 

 

 

 

 

 

 

 

 

Load I”... So No. of Iterations
(”’2“) [30°/0°/-30°] [60°/0°/—60°]
5 4 3
10 4 3
20 5 3
40 6 3
60 5 3
80 6 3

 

 

 

 

since the nonlinearity becomes more pronounced for higher loads. For any particular, load
level the number of iterations required for the ply arrangement with [30°/0°l-30°] is more
than the corresponding number for the [60°/0°l-60°]1aminate. This indicates the nonlinear-
ity is less pronounced for [60°/O°/-60°].

The variation of the absolute shear strain in material coordinates at the center of ele-
ment 2 with the excitation load level is shown in Fig. 5.36 and the responses are clearly non-
linear. For any given excitation level, the shear strain for the [60°/0°l-60°]laminate is

signiﬁcantly less than that for the [30°/0°l-30°] laminate.

104

The variation of the normalized r.m.s. z-displacement of the corner nodes with the ex-
citation spectrum level, So, is shown in Fig. 5.37. The displacement is normalized dividing
by the linear response (fer which [K'] = [0] ). The ﬁgure illustrates that the proportion-
al increase in the displacement due to nonlinearity steadily increases with the excitation
level. The effect of nonlinearity is less pronounced for the [60°/0°-60°] laminate than for
the [30°/0°/~30°] laminate. This is because the smaller shear strains in the former case re-

duce the overall level of nonlinearity.

The variation of normalized r.m.s. shear and normal strains and stresses in material
coordinates at the center of element 2 with the excitation level are shown in Figs. 5.38 and
5.39. Again the norrnalizations have been performed by dividing by the corresponding lin-
ear responses. The nonlinearity in the constitutive law results in a signiﬁcant increase in the
shear and normal strains and stresses for the [30°/O°/-30°] laminate, but it does not affect

the normal strains and shear strain for the [60°/0°/-60°] laminate.

 

 

 

 

 

 

0.012 r 1 f I I
— a=30° 4
—- a=60°
0.01 - .
l'
£3 0.008 - ‘
b ”‘
U) ’,-I"
530.“)6 '- ””’ "
/’
m //’
50.004 - /,I’ -
/
I //
//
0.002 " / -
00 a l . l . I .
0 20 40 60 80
Input So (lb2 see)

Figure 5.36 Variation of absolute r.m.s. strain at the center of element 2 with excitation load level

105

1.1

:- t- :-
2 8 8

Normalized RMS displacement
S
N

1.0

 

 

 

 

 

 

 

Input So (lb2 sec)

Figure 5.37 Variation of normalized r.m.s. displacement at free comer node with excitation load level

.—I
O

u—
N

H
U
y—e

f"
O
0

Normalized RMS Strains
'2 §

1 .02

 

— e" fora=30°
—- E" fora: 60°
°—° ‘yufora=30°
- - 7,2 four-60°
'''''' e22 fora=30°
-" £22 fora=60°

 

 

 

I 1
.43'4
.4".
.4"
.4" "
0’s...
.4".
.4"
.I.-'
04.. -
'4'.
.4”
.ﬁ"
./... -

 

 

 

Input so (le sec)
Figure 5.38 Variation of normalized rms. strains at the center of element 2 with excitation load level

106

:- :- r- :-
2 8 8 E 33

b
N

Normalized RMS Stresses

Figure 5.39

 

_ U“ for a= 30°
'_"' U" for a= 60°
Tn for a= 30°
r12 rota= 60°
------ on for a= 30°
022 for a= 60°

 

 

 

O
O
..

 

 

 

107

Variation of normalized r.m.s. stresses at the center of element 2 with excitation load level

lililll
i
rim Frill!

Table 5.17 provides an overall summary of the responses for which nonlinearity is

important for the Narmco 5505 material.

TABLE 5.17 The significance of nonlinearity on the responses of various loading conditions

 

Responses for which nonlinearity is signiﬁcant

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Loading Stacking Sequence Low Moderate High
less than 5% 5 - 10% more than 10%
[30°] 011: 022, 811 : 822 712: 112 ‘1
[60°] 011: 022: 811 : 822 712,112 '1
III-plane [30°/30°] 911: 522: 311: 712: 112: 922: 0
”mm“ [60°l-60°l 022: 1512 712 011: 311: 922:“
[30°/WW] 311: £22: 712: 011: 022: 112: II
\ [60°/0°l'50° 1 811: 1522: 712: 0'11: 022, 112: '1
[30°] s11: 622. 011: 112: 022 V 712
Impm [60°] E11: 822: 712: V: 011: 1312: 022
“W [30°/0940"] 0'11: 112: 022 Bin" 922’ 712
\ [60°/0990"] 811: 712. 011: 1:12: 022 V 822
[30°] 911: 822: 712: w: 0'11: 112: 022
[60°] 822, 012: 022 712: w: 011 811
Out-of- [30°/'3OO] 222,011 8111271322“
plane
[60°/~60°l 811: 822: 712: W: 022 011: 1512
[30°/0°I-30°] 011:311:112:W 022, 822: 712
\ [60°/Wm 011,022, 112: 311: 922: 712: W

 

108

 

6. Conclusions and Recommendations

6.1 Conclusions

A general formulation for the nonlinear random vibration analysis of laminated com-
posite plates modeled using ﬁnite elements and classical plate theory is presented. Only
nonlinearity in the shear stress-strain law, which is most signiﬁcant for ﬁlamentary com-
posites, is considered. An approximate representation of the non-linear shear stress-strain
law in terms of an odd-powered polynomial of arbitrary order results in a tractable formu-
lation that is sufﬁciently accurate for practical purposes. The solution is performed itera-
tively using linear random vibration analysis during each iteration. Although classical
laminate theory is often inadequate for composite laminates, it was used as a starting point
in this study. An overview of the ﬁnite element discretization is presented. The plate ele-

ment considered is a four noded one having 5 degrees-of-freedom per node.

The bending-extension coupling which always exists for unsymmetrical laminates
was investigated. The numerical examples presented indicate that the effect of nonlinearity
on the responses for any given load level depends on the ply-arrangement, and as expected
becomes more signiﬁcant for higher loads. The responses were computed for Narmco 5505
material, and root-mean-square displacements and strains were found to increase as much
as 20% for certain ply arrangements and loads. For other composite materials with different

degrees of nonlinearity, the results could be signiﬁcantly different.

To realize the above mentioned objectives, a computer program was developed and
implemented for the nonlinear random vibration analysis described above. Several optimi-
zation techniques were developed and used, including an efﬁcient indexing scheme using
a variation of the pascal triangle, for the computation of the covariance matrix of the nodal
displacements. The multidimensional symmetricity in the calculation of the covariance ma-
trix was fully exploited to drastically reduce the actual number of computations. The for-

mulation presented herein requires the use of a large number of nested loops. Testing for

109

zero entries of the strain-displacement matrices within the outer loops made it possible to
skip a number of intermediate nested loops and thus eliminate a large number of unneces-
sary computations. While the above mentioned optimization techniques were able to great—
ly reduce the computational time, it may still be unacceptable for large problem. Parallel
processing techniques appear to be well-suited for this type of analysis and are discussed

in the next section in the context of recommendations for future work.

6.2 Recommendations for future work

The amount of computations for the nonlinear random vibration analysis presented in
this work tends to increase rapidly with the increasing size of the problem. For large struc-
tures, and more general formulations involving shear deformation and interlaminar shear
stresses, the calculation of the nonlinear stiffness matrix discussed in Chapter 3 is even
more computation-intensive. Even with moderate size problems, the computations become
excessive on currently available uniprocessor workstations. For practical analysis using
ﬁner ﬁnite element idealization, the most promising computers are supercomputers and
massively parallel machines. While supercomputers can greatly speed up all vector and ma-

trix operations, parallel computers offer great potential in the near future.

There are a number of stages in the analysis described in this work where parallel

computation may be effectively employed. Some initial ideas along this front are explored.

There are three stages during each iteration in which intensive computations are re-

quired. Possibilities for parallelizing each of these stages are discussed below:

1. The use of modal analysis requires the solution of an eigenvalue problem. The
eigensolutions vary only slightly from one iteration to the next, and may be com-
puted from the old ones using the inherently parallel homotopy continuation meth-
od. The solution of the eigenproblem for the very ﬁrst cycle may also be
parallelized using this method (Zhang and Harichandran 1989, Chu 1984).

110

2. The coefﬁcients 01 and B needed in the computation of the nonlinear stiffness ma-
trix must be evaluated at each Gauss point for the numerical quadrature. Since the
coefﬁcients are estimated on an element by element basis, elements can be divided
into as many groups as there are processors, and the computations for different

groups of elements can be performed in parallel.

3. When a large number of modes, u, must be considered owing to their natural fre-
quencies being in the dominant excitation frequency range, considerable time is re-
quired to compute the integrals corresponding to the n (n + l) / 2 pairs of modes in
order to evaluate the r.m.s. responses. Since each integral is independent, the com-

putations can be performed in parallel.

While the techniques outlined above do provide for parallelism, they also require a fair
amount of communication between processors. In step 1 the system matrices must be ac-
cessible to all processors, and in step 2 the nodal covariances for each group of elements
needs to be accessible to the corresponding processors. Therefore, while the proposed
schemes should be efﬁcient for shared memory computers, they may need to be further en-
hanced for scalable distributed memory computers. Research and implementation is needed

to address these questions.

The classical plate theory used in this study can yield signiﬁcant error even for mod-
erately thick composite laminae because transverse shear deformation is neglected. It is
well known that transverse shear deformation is signiﬁcant for thick plates, and this is es-
pecially true for composites since the shear moduli of polymer matrices are signiﬁcantly
lower than the extensional moduli. While the ﬁrst-order shear theory (Reissner 1945,
Mindlin 1951) is adequate for plates made of conventional materials, a higher-order shear
is usually required for composite laminates (Reddy 1990, Noor and Burton 1989). Even
higher-order shear theories are usually adequate only for global modeling (i.e., prediction
of displacements, natural frequencies and buckling loads), and are not sufﬁciently accurate

for stress ﬁeld computations. Local layer-wise models that represent each layer as a homo-

111

geneous anisotropic continuum are usually required for accurate stress computations, but
these often magnify the size of the problem. For random vibration analysis, both global and
local models are of interest since one or the other may be applicable in a speciﬁc situation.
It appears therefore that separate solution schemes using both types of models should be

developed.

Composite laminates are typically used in either plate or shell conﬁgurations. The
analysis using laminated shell elements is much more complicated than that using laminat-
ed plate elements, primarily because their geometry is more complicated. Although com-
posite shells are widely used in the design of aircraft and automobile components, very
little work has been done on the nonlinear random vibration analysis of elements made of
composites. There is an important need to address this deﬁciency, and to develop suitable

techniques for the random vibration analysis.

112

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115

APPENDIX

PROGRAM

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132

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