NASA Technical Reports Server (NTRS) 20010046859: Free Vibration of Uncertain Unsymmetrically Laminated Beams

12 Pages·0.61 MB·English

by  NASA Technical Reports Server (NTRS)

AIAA 2001-1317
FREE VIBRATION OF UNCERTAIN
UNSYMMETRICALLY LAMINATED BEAMS
Rakesh K. Kapania and Vijay K. Goyal
Department of Aerospace and Ocean
Engineering
Virginia Polytechnic Institute and State
University
Blacksburg, VA 24061-0203
42nd AIAA/ASM E/ASC E/AHS/ASC
Structures, Structural Dynamics, and
Materials Conference and Exhibit
April 16-19, 2001/Seattle, Washington
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344
AIAA 2001-1317
FREE VIBRATION OF UNCERTAIN UNSYMMETRICALLY LAMINATED
BEAMS
Rakesh K. Kapania* and Vijay K. Goyal t
Department of Aerospace and Ocean Engmeemn9
Virginia Polytechnic Institute and State University
Blacksburg, VA 24061-0203
Monte Carlo Simulation and Stochastic FEA are used to predict
randomness in the free vibration response of thin unsymmetrically lam-
inated beams. For the present study, it is assumed that randomness in
the response is only caused by uncertainties in the ply orientations. The
ply orientations may become random or uncertain during the manufac-
turing process. A new 16-dof beam element, based on the first-order
shear deformation beam theory, is used to study the stochastic nature of
the natural frequencies. Using variational principles, the element stiff-
ness matrix and mass matrix are obtained through analytical integration.
Using a random sequence a large data set is generated, containing pos-
sible random ply-orientations. This data is assumed to be symmetric.
The stochastic-based finite element model for free vibrations predicts
the relation between the randomness in fundamental natural frequen-
cies and the randomness in ply-orientation. The sensitivity derivatives
are calculated numerically through an exact formulation. The squared
fundamental natural frequencies are expressed in terms of deterministic
and probabilistic quantities, allowing to determine how sensitive they
are to variations in ply angles. The predicted mean-valued fundamental
natural frequency squared and the variance of the present model are in
good agreement with Monte Carlo Simulation. Results, also, show that
variations between ±5 ° in ply-angles can affect free vibration response of
unsymmetrically and symmetrically laminated beams.
INTRODUCTION wanted uncertainties.
In recent years, there has been an increasing de- The design and analysis using conventional mate-
mand for laminated composite materials in aircraft rials is easier than those using composites because,
structures. The main reasons are that the compos- for conventional materials, both material and most
ites posses the following characteristics: lightweight, geometric properties have either little or weft known
cost-effective, and can handle different strengths in variation from their nominal value. On the other
different directions I. However, these materials offer hand, the same cannot be said for the design of struc-
quite a few challenges to structural engineers. Be- tures using laminated composite materials. The
cause of their inherent complexity, laminated struc- understanding of uncertainties in the laminate com-
tures can be difficult to manufacture according to posite structures is highly important for an accurate
their exact design specifications, resulting in un- design and analysis of aerospace and other structures
using composite materials.
•Professor, Associate Fellow AIAA, rkapania_vt.edu These uncertainties are defined as random-
tGraduate Research Assistant, Member AIAA, ness from non-cognitive sources involving physical
vgoyal_vt.edu
stochastic likelihood and human factors 2. This ran-
Copyright (_ '2001 by R.akesh K, Kapania and Vij&y K.
domness can occur in each laver and involve quan-
Goyal, Published by the American Institute of Aeronautics and
Astronautics, inc. with permission,
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titles such as: ply-orientations, thickness, density, laminated beams with uncertainties. In order to
and material properties among others. Such varia- study the stochastic nature of the dynamic response
tions can affect the behavior of the structure. of such beams, a new 16-dof element is developed
What has intrigued many engineers to study using first-order shear deformation beam theo_, to
account for uncertainties. Only an overview of this
structures with laminated composites is the com-
element is presented here, and a more detailed anal-
plexity of these materials. Since it is costly to an-
alyze a composite structure as a three-dimensional ysis of this new 16-dof element will be presented
elsewhere. Using this element the free vibration
solid, analysis of many composite structures can be
analysis is performed for only those uncertainties as-
performed using laminated one and two dimensional
theories, such _ beam theory. A review of vari- sociated with layer-wise ply-orientations.
ous available theories for analyzing laminated beams
is given by Raciti and Kapania 3. Earlier, a 12-dof 16-DOF LAMINATED BEAM ELEMENT
element was developed and formulated for deter-
An overview
ministic symmetric laminated beams to study their
static and dynamic behaviors 4. Later a 20-dof el- The present 16-dof laminated beam element takes
ement (Kapania-Raciti Element) was developed to into account the existence of various coupling ef-
study static, free vibration, buckling, and nonlinear fects, which play a major roll in laminate composite
vibrational analysis of unsymmetrically laminated materials. This element is valid for the analysis
beams 5'6. In both works the effects of uncertain- of both symmetric and unsymmetrically laminated
ties were not considered. beams. The motivation to develop a new beam ele-
ment was to have a formulation consistent with the
In considering uncertainties in a system, one can
first-order shear deformation beam theory (FSDT),
consider three different approaches: (i) probabilistic
able to analyze unsymmetricaliy laminated beams,
methods. (ii) fuzzy set or possibility-based meth-
and that would account for most of the uncertainties
ods, and (iii) antioptimization r. Here, however, only
involved in a thin laminated beam when modelled
the probabilistic approach is considered. The prob-
using FSDT. In the present work. the reference sys-
abilistic analysis can be performed using either an
tem of coordinate is such that the x- axis lies
analytical or a computational approach. An ana-
along the length of the beam and the z - axis is
lytical approach would be most accurate although
placed at the mid-surface measuring the transverse
cumbersome and impractical except for very sim-
displacements. The present work assumes that the
ple systems. However, with the availability of ex-
x - z plane divides the beam in two identical parts:
tremely fast computers, the finite element method
in other words, material, geometry., and loading are
has become a widely used technique for design and
symmetric about the x - z plane. When considering
analysis in engineering. Therefore, a finite element
twisting and ignoring in-plane shear, the displace-
approach using Monte Carlo Simulation is developed
ment field for the first-order shear deformation beam
to take into account the stochastic nature of the ply-
theory, in the defined reference system, can be ex-
orientations.
pressed as follows
Work done by Vinckenroy s presents a new tech-
nique to analyze these structures by combining the
U(z,y,z) = u(x) _ zo(x) (la)
stochastic analysis and the finite element method in
V(z,y,:) = 0 (lb)
structural design. Agrawal et al. 9 used a wavelet-
based stochastic analysis to analyze isotropic beam W(x,y,z) = w(_)- y_(z) (ic)
structures.
Librescu et al. 1° studied the free vibration and Therefore, the finite element formulation considers
reliability of cantilever composite beams featuring eight degrees of freedom at each node: axial dis-
structural uncertainties. They used a Stochastic placement u, transverse deflection w, rotation of the
Rayieigh-Ritz formulation. However, to the best of transverse normals ¢, twist angle r. and their deriva-
the authors' knowledge, no work has been found re- tives with respect to x. These nodal displacements
garding the effect of uncertainties incurring in the are denoted as
ply-orientations on the natural frequencies of thin
unsymmetrically laminated beams using Stochastic {qI=(u,' , , ,}r (2)
U,W,W ,O,O.7.
Finite Element Analysis.
The primary goal of this investigation is to pre- The deflection behavior of the beam element for the
dict the dynamic behavior of thin unsymmetricalIy
first-order theory including transverse twist effect is
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described as follows is identically zero and the coupling between bending
and stretching vanish. However, for unsymmetri-
u(z) = ,% ut + N2u_+ ,V3u2+ N4u_
cally laminated beams the coupling cannot be ig-
w(x) = Nlwl+N2@+N3w_+N4w"
nored and it must be included in the analysis. In the
O(z) = Nt¢_t+ N'2¢,+N3¢_+N4¢'2 - (3)
presence of uncertainties, laminate composite struc-
r(x) = N, r, + N2 v{+ N3"r2+ N4"c_
tures are no longer symmetric and the analysis of
where the shape functions N,(x) are thewellknown unsymmetrically laminated structures is a more ac-
curate one.
Hermitian polynomials.
Laminate Constitutive Relations
Constitutive Material Law
The basic constitutive relation is
In thisinvestigation,alllaminated composites are
considered as orthotropic materials.Also,consider-
{N} = [D]{_} (9)
ing astateofplane stressand eliminatingezz from
the stress-strainrelationship,the reduced material where {N} r is the stress resultant vector, [D] is the
coefficientsare expressed asQu' Since inlaminated bending-stiffness matrix, and {a}r is the stretching
composites each ply may have differentorientation,
and bending strain vector.
the stressesare expressed in terms ofan arbitrary
The displacement field, Eq. (1), suggests that
angle 9 and the transformed stress-strainrelation-
ship takes the form of 11 2e_y = 7_u+z_u=O (10)
Therefore, ,,V.u and M._ are not considered because
they vanish in the strain energy formulation. In ad-
(4)
O'xy = dition, the present formulation assumes:
CYyZ
Myr_ = Nyy =0 (11)
.0"_:z
Qll Qr_ Ql6 0 0 This leads to the following bending-stiffness matrix:
QY2 Q_2 Q26 0 0 eyy
QlG Q_6 Q66 0 0 2 e_
[D]= [ [D[D,,,.,,tI] [[DDHt.z.H,l] ] (12)
0 0 0 Q44 Q45 I 2exevxz }
0 0 0 U745 q55 2e_z
where
where Q,j are the transformed stresses t.
[Dr,t] = AB,n_ DB,tlt ] (13)
[A], [B], [D] Matrices
The extensional matrix [A], the extensional- [Dt,nl = [ AB,t22 OB,,_2 ] (14)
bending coupling matrix [B], and the bending stiff-
ness matrix [D] are of great importance in the
{Dtt,1] = [ +B4,t2_' DB,t2_ I (15)
present work and these are calculated as follows 11
Nta., [Dtt.tt] = [ ,B42__. BO__.'2 ] (16)
a,j = E Q'i (zk+t - z_) i,j = 1,2,6 (5)
k=l The reduced form of the bending-stiffness matrix is
Nla_ calculated as
aij = K E -Qu (z_'+' - zk) i,j = 4,5 (6)
k=l [DR] = fDt,,] - [DLnl [D,t,HI-_IDH,,] (17)
B,) = E -Q'J zk+_ z_ i,j = 1,2,6 (7) Using the above expression, an equivalent
k=l bending-stiffness matrix [D_] for a thin unsymmet-
rically laminated beam is found:
D,j = _ _,j i,j = 1,2,6 (8)
Dc, t Dc,_ 0
k=l
D_,_ D_ 0 0 _zz
where K = 5g, the shear correction factor n, and 0 0 D¢:,_ DC34 t;::
Ntam is the total number of plies considered. When 0 0 Dc_,
(is)
considering symmetrically laminated composites, [B]
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where, The minimization of Eq. (21) results in the element
stiffness matrix for the 16-dof laminated beam ele-
ment:
A = B_2 - AND22
(22)
(K't Jo
Dot, =(A22 Bt22 - 2A12 B12 B22-.
All B_2 + At22 D22 - ,411.422 D22)/A
where bo is the width of the beam, e_ is the length
Dc,_ =(B_2 B__2+ Bll B_2 + A22 Bn Dn-
of the beam element, and [Dc] is the equivalent
An Bn Dr2 - A22 Bu Dn + At2 B12 D22)/A bending-stiffness matrix.
Element Mass Matrix
D_2 =(B_2 Dil - 2Bi2 Bn Dl2+
The kineticenergy for this 16-dof beam element
is
Dcaa --A44
Dc_, =A45
7" = _ p (Y2+ + fiz_"
Dc,, =A,55
V
Note that the availability of symbolic manipula-
tor like MATHEMATICA®Version 4.0 $ has made it where p is the mass density. The mass matrix is
possible to determine the above matrix analytically, obtained by substituting Eq. (1) into Eq. (23) and
By obtaining this matrix analytically, the CPU time then taking the first variation of the kinetic energy:
is saved, a great help in the Monte Carlo Simulation.
Strain-Displacement Relationship 0-/ 0-r
07" 5 /)7".
The strains in the above formulation are related ST= ou + a-zw-6w + "a:o?=":z--O-Su+ -_rOr (24)
to the displacements as follows:
The mass matrix coefficients are obtained as follows
{_} = _ = 0-7 (t9)
7yz --r
(25)
[10L t' N,(x)Nj(x)dx] a,+
Using; Eqs. (2), (3) and ([9) the strain-displacement
relation can be expressed as
{d = [B, ai {q} (20)
where [Bsul is the s_rain-displacement maCrix.
6w i (26)
Element Stiffness Matrix
Using Eqs. (9) and (20), and noting that matrix
IDol has been integrated throughout the thickness,
_,+ (27)
the strain energy for this new 16-dof beam element
becomes
lJ/ 1
U = 7)_{q}V [B_alr [D_l[B_ai dA {q} (21)
5_ (28)
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where where a2 is the variance ofthe random variable: and
Nla_ # is the mean value of the random variable. [n the
Io = bo _ pk (zk+i - zk) (29) present work, the random variables are considered
k=l as independent and are denoted as
= = {el,62,. ,o.} (38)
I, =boE: 2-- (30)
k=/ where 8i's are the ply-angles.
Various methods exist to analyze an uncertain
=boE (31)
unsymmetrically laminated beam by integrating
k=t 3
stochastic aspects into the finite element modelling:
bozN,°. perturbation techniques, Taylor Series, and Monte
j= = ]2 _ pk (zk+l -- zk) (32) Carlo Simulation. These techniques have been used
k=l in the past two decades in fields involving random-
In the present work, analytical expressions for co- ness. Especially, in the last decade there has been a
efficients of both mass and stiffness matrices are growing interest in applying these methods to better
obtained. Once again, this greatly saves CPU time understand laminated composite structures. More-
for the Monte Carlo Simulation as one no longer has over, with the finite element technique becoming so
to perform numerical integration for each case. popular, there has been a new interest in integrat-
ing the stochastic nature of the structure in the finite
FREE VIBRATION ANALYSIS
element analysis 15.
The Hamilton's principle is used to study the dy-
Monte Carlo Simulation
namic nature of the structure. The principle uses
the Lagrangian which is defined as Monte Carlo Simulation, although computation-
ally expensive, methods are quite versatile tech-
/: = T-U-V (33)
niques capable of hand[{ng situations where other
where, methods fail to succeed. The MCS is also used
to verify the results obtained from other methods.
T is the total kinetic energy defined by Eq. (23)
Monte Carlo Simulation methods are based on the
/4 is the total strain energy defined by Eq. (21) use of random numbers and probability statistics to
investigate problems. For purposes of the present
)2 is the potential of the applied loads
work a random number generator is used to generate
The Hamilton's Principle can be represented as 13 possible angle-variations between -5.0 ° and 5.0 °.
A large sample is generated and then using PDF's
g_ = {&T - &lA- 5V} dt = 0 (34) one evaluates the probability of having such values.
1 The larger the number of simulations more the con-
The free vibration analysis is obtained by setting fidence in the probability distribution of the results
6),' = 0 in Eq. (34). This leads to the free-vibration obtained. Therefore, for the present analysis at least
equations of motion for the defined finite element ten thousand realizations of the uncertain beam are
problem: performed, increasing the accuracy of the ply-angle
[M] {/i/} + [K] {q} = 0 (35) distribution fitted to the sample data.
where [AI] is the mass matrix and [K 1is the stiffness
Stochastic Eigenvalue Analysis
matrix. Further assuming a harmonic response, the
solution of these equations results in an eigenvalue The stochastic eigenvalues problem is expressed as
problem:
[g- AM] {0} = 0 (39)
[[K] - w_"[M]] {q} = 0 (36)
where K, M, )_, and 0 are the stochastic stiffness
A PROBABILISTIC APPROACH matrix, mass matrix, eigenvalues and eigenvectors,
An overview respectively.
The present analysis assumes that the random The presence of uncertainties in ply angle orienta-
tions results in certain randomness in the extensional
data is symmetrically distributed. Therefore, it as-
sumes normal distribution14: matrix [A], bending-stretching coupling matrix [B],
and bending matrix IDI. Since the coefficients of
these matrices are present in the equivalent bend-
f(x) = _ exp -_ (37) ing stiffness matrix [D_ I, F.q. (18), the matrix [De]
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will have certain randomness associated as well and of their mean-centered zeroth-, first-, and second-
these uncertainties are expressed as order rates of change with respect to the random
variables as
[Dc] = D_ ° + De s (40) Eigenvalues
where Dc ° is the equivalent bending stiffness ma- A(xt, x2 ..... x,_) = (42)
trix evaluated at the ply mean values and De s is
the equivalent bending stiffness matrix taking into
account the random nature of the problem. As a i=I t----3I=1
result, these uncertainties affect the stiffness ma-
trix. However, the mass matrix is not affected when
only considering those uncertainties involving ply- )_0 = )k z=(m,_2 .....u,_)
orientations. Therefore, the mass matrix M is eval-
uated at the given deterministic values. ,k_ - 0A ::=(m,_,2 .....u-)
Oxi
In problems where uncertainties are considered,
there exists no density function describing the ran- It 02_
"kiJ -- Ox_Ox i z=(u_,u_ ....._,,)
dora nature of the system. The information is lim-
ited to only the mean values of the random vari-
Eigenvectors
ables. In such cases, perturbations techniques are
suggested among other existent techniques 15'16. In
{¢)(x_,z2..... z,,)} = (43)
the free vibrational analysis of the present problem,
the random nature of the stiffness matrix, eigen-
i_2-[ n
values, and eigenvectors are studied using a Taylor
'=1 t=l )=1
series expansion up to second order about the mean
of each random <_ariable. The present formulation
follows the approach followed by Librescu et at. l°
{@} = (0} =(.,,.,.,,.,,..).
Opposed to the work done by Librescu et al., the
present formulation can use results provided by com-
O{,N_=(,,,.,.._..,...
mercial finite element codes, greatly reducing the
number of calculations, and calculates the sensitivity
of the eigenvalues up to the second-order approxima- {o,']=} o_{_}.=(.,.,._..._
Ox,Ox: .u, )
tion.
Therefore, the random nature of the elastic stiff- After substituting Eqs. (42), (43), and (44) into Eq.
ness. K, is expanded in terms of the mean-centered (39), the stochastic eigenvalue problem is expressed
zeroth-, first-, and second-order rates of change with
as
respect to the random variables as
K(xl, x2,. •., x,_) = (41) K ° + K[e, + 72
tml t=l j=l
K°+ _ K[_, + _ K,j =::
_=1 t=l j=l x V°+ 0[_,+_1 _A',., d91,Jt:.''_
_=1 _=1 j=l
K ° = K ==(u1,*,2,..,*,.)
= A°+ x_, + _ --,-,-_
$=L I=I )=l
K_ = __OKo_x=,(u,,u_ ....._,)
" (44)
K [J - O'2K
Oz,Oxj ==(re,u2 .....0-) _=1 t=l 3=1
_i = Zi -- P,i
The uncertainties in the random variables are in
The eigenvalues and eigenvectors are also affected general small. As a consequence, in the applied per-
turbation technique it is sufficient to only consider
by uncertainties in the ply angles. The perturbed
the first and second derivatives of eigenvectors and
eigenvaIues and eigenvectors are expressed in terms
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eigenvalues with respect to the random variables. By where
equating the zeroth order terms of e, in Eq. (44),
N,.. (xq -/_,)2
an eigenvalue problem for the mean-valued system E[d] X: ._.o7:i.
is obtained. From which, the mean-centered zeroth
q=l
derivative eigenva[ues and associated eigenvectors
are obtained E[d4] = NX,o.: (xq -N_,-);-=_(--z-_q/_k)"_
[g °-A°M °] {0 °} =0 (45) q----1
Premultiplying the first and second order terms of _, E[s,2IE[s_] =
in Eq. (44) by {4)°} T and simplifying, the following
expressions are obtained
and Nsam is the number of samples, equal to ten
{_o}_[K_-A;MO{],ot = 0 (46)
thousand in this work.
The standard deviation is calculated as
{:V [_,1I - _li,,M0]{,o} : 0 (47)
_ = (53)
This results in expressions for the mean-centered
first and second eigenvalue derivatives: Calculating derivatives
: { o}[T,<] (48) The above formulation only requires the calcula-
tion of the derivatives of the stiffness matrix. These
{4)o} T [M o] {¢0} derivatives are obtained by taking the derivatives
of the equivalent bending-stiffness matrix, eq. (18).
= {:1 {:t
However, the derivative for the first two rows and
{(;b°}r[M°]{¢ °} columns (reduced bending-stiffness matrix) are more
involved. Various numerical schemes exist to eval-
The advantage of this method is that the eigenvalue uate these derivatives. When using some of these
problem needs to be solved only once. The sen- numerical schemes, ill-conditioning could be a prob-
sitivity analysis is done by using results from the lem. This problem can be avoided by the following
mean-valued eigenvalue problem. This results in a formulation which allows the derivatives to be ob-
great computational saving. tained exactly by numerical multiplication. The
When studying the effect of uncertainties of ran- technique consists in taking derivatives of eq. (17)
dom variables on the fundamental natural frequen-
cies, it is convenient to study their squared value, [DR],_, = [D,,,] _,- (54)
i.e., eigenvalues. Ang 16gives a good description on [D,.,I]z'[Ou.l,]-I [D,r,t] -
statistical analysis.
[DI.11] [Du.n]2 l,[Du,I] -
The mean value of the eigenvalue for random vari-
ables is obtained by taking the expected value of Eq. [D,,,,,] [D,I.,,}-l [Ou.,],_,
(43)
(55)
1_-_ _-_ l' E[e,e,] (50) [DR],=. _, = [D,.,].=. =, -
_-=-i j=l [D,.u] .... , [Du.H] -1 [D,,.,] -
The variance of the eigenvalues is obtained as
[D,.I,] _. [O,l.ll]_: [D,I.ll -
VariA] = Eli _] -/_ (51)
[Dl.ll] z. [D,I.,,]-' [Du.,] ::, -
For symmetrically distributed independent ran- [D_,.],_, [n.,.]_ _.[DH._}-
dom variables,
[DI.II][Du.n]] _,z,[Du.i] -
Var[A] = (52)
_,_, E[_;]+
[D_,u],_, [Du.H] -_ [D_,_] _. -
l [Dr,u] [D,,,.19, [Du.I] _, -
- - E[e,2IE[E_] }
4 [D_.u] [Du.u] -_ [Dn,I] .. _,
i=L k=l
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Dtcn6n.._onles._ Fundamental Eigenvatge 10¢a Oi_l_"lle_ Fu_ntal E_alue for a
90/-4.5/--45/90 L.amtnatecl Beam 94Y.-45/0/25 Lam=nateo Be.=u'n
7 'E \
i
I
I
1
i
1
,\
°'!
°3 14 _s 11 17 _$ ig ¢ _s 4 45 S s$
O_nates= Fun_eamntal E_genvelue D_ Func_mw_ E_wwuu$
Figure I: Dimensionless Eigenvalue distribution for MCS Figure 2: Dimensionless Eigenvalue distribution for MCS
and SFEA for a thin symmetrically 4-ply laminated and SFEA for a thin unsymmetrically 4-ply laminated
beams beams
The derivatives of [Dtt,tt]-1 are calculated using the
Dimensm_le_ Fund_nent, aJE=genvalu_ _or a
following matrix definition 9G/-45,'30/O/_30/-45F_ Lam_atea Beam
lOft,it] -l [Du,u] = [1] (56)
and these derivatives being
[D ]-l (57)
11,11 ,x,
i'i
-[Dtt,u] -l [Dtt,tt _, [D.,u] -l
],t
I
(58) °i i!
-[Du./tl_l[Dll,lt],[D=H,.H] -l -
[D11,11]_t,[Dtt,ttl,z,[Ou,.'t]-'-
O_nens,,o_le_ Fundarz_n_ll E_e_wslue
[Dtt..,/] -t [Dtt,tt] z, _, [Dtl.._l] -_
RESULTS Figure 3: Dimensionless Eigenvalue distribution for MCS
and SFEA for a thin symmetrically 8-ply laminated
The numerical results are obtained for a can-
beams
tilevered thin laminated beam. The plies are as-
sumed to be made of Graphite-Epoxy. The beam's
material and geometrical properties used in the anal- Young's Modulus:
ysis are§:
Materid density: Evu = 1.36x L06psi
E== = 13.75E w
p = t.449 x 10-4 slugs
in 3
Shear Modulus:
Major in-plane Poisson's ratio:
uzv = 0.30 G=v = 0.55E_
G_= = 0.25Evv
All properties are assumed to be uniform throughout the
bEtaJn G=z = 0.25Evv
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Table 1: Statistics analysis
//,)_ a_ x 10-2
Ply-orientations
MSC SFEA MSC SFEA
[90/-45], 1.63468 t.63925 0.41519 0.41928
[90/-45/0/25] 4.10664 4.12107 6.44410 6.67425
[90/-45/30/0]s 3.23110 3.23559 2.79163 2.80235
[90/-45/30/0/
/25/45/-90/-30] 4.53781 4.54736 9.51499 9.69171
Din_ns_onlesS Fun(_'nental Eigenvalull lot a Senm_ ot theFunO,tmental E_JenvaJue for a
90/--4_30/0t25/45/-90/-30 Laminated Beam 90/.-4&'_ LammateO Beam
I
i
_OSt-
04_"
I
°3 _S 4 45 $ $$
O_n,ens_neJess Func_,r_n_ E(_ovaJue
Figure 4: Dimensionless Eigenvalue distribution for MCS Figure 5: Sensitivity of the dimensionless eigenvalue for
and SFEA for a thin unsymmetrically 8-ply laminated a thin symmetrically 4-ply laminated beams
beams
inated cantilever beam with four plies, (iii) symmet-
Beam's dimer_sions rically laminated cantilever beam with eight plies,
(iv) and unsymmetrically laminated cantilever beam
Width : bo = 1.0 in with eight plies.
Thickness : ho = 0.5 bo The natural frequencies are non-dimensionalized
Length : Lbea,n = 30.0 ho with respect to the deterministic quantities as fol-
lows
&n = _nLbea'n P (59)
ho Evv
For both the stochastic finite element analysis
and Monte Carlo Simulation ten beam elements In general, the ply-angle uncertainties are between
were used and four different laminated beams are +2 °. However, results show the effect in considering
considered: (i) symmetrically laminated cantilever twice the uncertainty in ply orientations, i.e., from
beam with four plies, (ii) and unsymmetrically lam- -5° to 5°. The statistical analysis is shown in Table
9oF 11
AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS