Large variation finite element method for beams with stochastic stiffness
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Transcript of Large variation finite element method for beams with stochastic stiffness
Large variation finite element method for beamswith stochastic stiffness
Olivier Rollot a, Isaac Elishakoff b,*
a LaRAMA, Institut Franc�ais de M�eecanique Avanc�eee, Aubi�eere F-63175, Franceb Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA
Accepted 1 October 2002
Abstract
The behavior of beams with stochastic stiffness subjected to either deterministic or stochastic loading is studied via
finite element method. The results are contrasted with exact solution to check the accuracy of the FEM for the case
of large variations. It represents a generalization of the previous study in which the stiffness matrix was decomposed
as a product of three matrices, two of which are numerical ones and the third matrix involves the uncertain stiff-
ness analytically. To illustrate the proposed method, we evaluate the mean and the auto-correlation functions of
the displacement of beams under various boundary conditions. Two statically determinate beams (clamped-free or
simply-supported) and two statically indeterminate beams (clamped–simply-supported or clamped are both ends) are
investigated in this study. The beams are subjected to a deterministic uniform pressure or a stochastic excitation.
� 2003 Published by Elsevier Science Ltd.
1. Introduction
Finite element method in stochastic setting attracted numerous investigators. The recent state of the art is given in
review article by Elishakoff et al. [1], Ghanem and Spanos [2], Schu€eeller and Brenner [3] and Matthies et al. [4]. As noted
in Ref. [1], the most popular method utilized presently is the method of perturbation. Naturally, this method is ap-
plicable for small coefficients of variation of involved stochastic fields. The case of the large variations can be dealt with
the combination of the FEM with Monte-Carlo simulations, as advanced by Shinozuka and Yamazaki [5]. Alternative
analytical methods are now under development. Some preliminary results have been reported by Elishakoff et al. [6,7]
and Ren and Elishakoff [8]. In particular, in Ref. [9], the authors presented a generalization of the Fuchs� technique thatwas originally developed for deterministic optimization problems. The present study is an extension of Ref. [9], which
dealt with stochastic stiffness but deterministic load. We consider both statically determinate and indeterminate beams
under either deterministic or stochastic loading.
2. Description of the method
2.1. New formulation of FE stiffness method
A straight beam element of uniform cross section is shown in Fig. 1. Following classical studies in the finite element
method, the element number i has a constant stiffness Di and length a. The element has two degrees of freedom at each
*Corresponding author. Tel.: +1-561-297-2729; fax: +1-561-297-2825.
E-mail address: [email protected] (I. Elishakoff).
0960-0779/03/$ - see front matter � 2003 Published by Elsevier Science Ltd.
doi:10.1016/S0960-0779(02)00470-8
Chaos, Solitons and Fractals 17 (2003) 749–779
www.elsevier.com/locate/chaos
end (nodal points): a transverse deflection w and an angle of rotation or slope h. Corresponding to these degrees of
freedom, a transverse shear force Q and a bending moment M , respectively, act at each nodal point. The shape function
of this element are given by [10]
N1 ¼ 1� 3xa
� �2þ 2
xa
� �3; N2 ¼
xa
� �� 2
xa
� �2þ x
a
� �3;
N3 ¼ 3xa
� �2� 2
xa
� �3; N4 ¼ � x
a
� �2þ x
a
� �3 ð1Þ
Since it is assumed that the finite element mesh is uniform, i.e. all elements have the same length a, the stiffness matrix
can be written as:
Di
a3Kiqi ¼ Fi ð2Þ
where
Ki ¼
12 6 �12 6
6 4 �6 2
�12 �6 �12 �66 2 �6 4
2664
3775 ð3Þ
where Di is the uncertain parameter associated with element i, qi and Fi are the load and displacement in the nodes of the
element i, respectively. Uncertainty in Di stems from the fact that the elastic modulus constitutes a random field with
known mean function and auto-correlation function. We are interested in finding probabilistic characteristics of the
beam�s response. It is instructive to determine first the eigenvalues and eigenvector matrices of Ki which read,
k ¼ diag½0; 0; 2; 30� ð4Þ
V ¼
�1 1 0 2
1 0 �1 1
0 1 0 �2
1 0 1 1
2664
3775 ð5Þ
The generalized strain ei and stress ti, are introduced as follows:
ei ¼ V �1qi ð6Þ
ti ¼ V TFi ð7Þ
Introducing Eqs. (6) and (7) into Eq. (2), the latter becomes
Di
a3V TKiVei ¼ ti; i ¼ 1; 2; 3 ð8Þ
Since V is the eigenmatrix of Ki so that
V TKiV ¼ k ð9Þ
a
Di
y
x Qi2, wi2 Mi2/a, θ i2a
Qi1, wi1 Mi1/a, θ i1a
Fig. 1. Straight beam element of uniform cross section.
750 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
Eq. (8) can be rewritten as
siei ¼ ti ð10Þ
where
s ¼ Di
a3diag½4; 300� ð11Þ
Eq. (8) defines the local element�s behavior and we relate to the global vectors from the elementary vectors by using the
Argyris matrices [11]. We define the global displacement vector e and the global stress vector T by
ei ¼ ½A�ie ð12Þ
ti ¼ ½A�iT ð13Þ
where ½A�i is the Boolean mapping matrices containing only ones and zero, namely, ðAiÞnm ¼ 1 if the local number mcorrespond to the global n, otherwise ðAiÞnm ¼ 0. Substitution (11) and (12) into (8) leads to
si½A�ie ¼ ½A�iT ð14Þ
The global relation is equal the sum of the elements and we obtain the constitutive law
X2Ni¼1
½A�Ti si½A�i� � !
e ¼ T ð15Þ
which can be rewritten as
T ¼ Se ð16Þ
where
S ¼X2Ni¼1
½A�Ti si½A�i ð17Þ
Analogously, substituting Eqs. (11) and (12), respectively, into the Eqs. (6) and (7), we obtain
qi ¼ V ½A�ie ð18Þ
½A�iT ¼ V TFi ð19Þ
which can be rewritten as
qi ¼ ðV ½A�iÞe ð20Þ
Fi ¼ ðV ½A�iÞT ð21Þ
Summing over all elements we obtain the kinematic equation
u$X2Ni¼1
qi ¼X2Ni¼1
V ½A�i
!e ð22Þ
and the equilibrium condition
F$X2Ni¼1
Fi ¼X2Ni¼1
V ½A�i
!T ð23Þ
The kinematic equation can be rewritten as
e ¼ Ru ð24Þ
while the equilibrium condition is put in the form
F ¼ QT ð25Þ
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 751
where
Q ¼ RT ¼X2Ni¼1
ðV ½A�iÞ ð26Þ
Combining Eqs. (16), (24) and (25) gives the global finite element equilibrium equation
Ku ¼ F ð27Þ
where K is the global finite element stiffness matrix
K ¼ QRS ð28Þ
We now specify the characteristics of the load. In the work-equivalent load method, we set the work produced by the
unknown nodal concentrated force to be equal to the work produced by the actual distributed load. Considering our
beam element, the work done by the nodal loads is [10]
W ¼ 1
2Q1i
M1ia Q2i
M2ia
� w1i
ah1i
w2i
ah2i
8>><>>:
9>>=>>; ð29Þ
On the other hand, the work done by the distributed load is
W ¼ 1
2
Z a
0
qðxÞwðxÞdx ð30Þ
W ¼ 1
2qR a0N1ðxÞdx q
R a0N1ðxÞdx q
R a0N1ðxÞdx q
R a0N1ðxÞdx
� w1i
ah1i
w2i
ah2i
8>><>>:
9>>=>>;; if q is constant ð31Þ
Since these work expressions must be equal, taking into account Eq. (29), we get
F ¼
Q1iM1ia
Q2iM2ia
8>><>>:
9>>=>>; ¼
qa2qa12qa2
� qa12
8>><>>:
9>>=>>; ð32Þ
2.2. Imposition of displacement constraints
To obtain the explicit displacement of u, we needs to invert the global stiffness matrix K which is singular without the
incorporation of boundary condition. After the incorporation of the first two displacement constraints given by the
boundary conditions, Eq. (27) reduces to
K1u1 ¼ F1 ð33Þ
and
K1 ¼ Q1SR1 ð34Þ
where u1 is obtained from u by canceling the two constrained displacements, F1 is obtained from F by canceling the
corresponding two forces. Analogously, Q1 is obtained from Q without two rows corresponding to the constrained
displacements. For statically determinate beams, the displacement solution is obtained immediately from Eq. (33)
u1 ¼ K�11 F1 ¼ ZF1 ¼ Q�T
1 S�1Q�11 F1 ¼ GTS�1GF1 ð35Þ
where Z ¼ K�11 and G ¼ Q�1
1 . For statically indeterminate beams, additional displacement constraints exist and we
suppose that u1 is divided into two parts
u1 ¼uc1uu1
� �ð36Þ
where uc1 is the vector of constrained nodal displacement excluding two previously imposed constraints, uu1 is the vectorof unconstrained nodal displacement. Eq. (35) becomes
752 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
uc1uu1
� �¼ Zcc Zcu
Zuc Zuu
� �F c1
F u1
� �ð37Þ
where F c1 is the vector of unknown nodal forces relevant to constrained nodal displacement and F u
1 is the vector of
unknown nodal forces relevant to unconstraint nodal displacement. Expressing the unknown reaction F c1 from the first
equation of Eq. (32) and enforcing the condition uc1 ¼ 0, we arrive at
F c1 ¼ �Z�1
cc ZcuF u1 ð38Þ
and
uu1 ¼ Zuu
�� ZucZ�1
cc Zcu
�F u1 ð39Þ
Representing the matrix G ¼ ½Gc;Gu� through the sub-matrices Gc and Gu, we obtain
uu1 ¼ GTuS
�1Gu
�� GT
uS�1Gc GT
c S�1Gc
� ��1GcS�1Gu
�F u1 ð40Þ
2.3. Mean and auto-correlation functions
For statically determinate beams, the mean vector of the nodal vector u1 is
E½u1� ¼ Q�T1 E½S�1�Q�1
1 F1 ð41Þ
if F1 is a deterministic vector. The means value E½u1� equals
E½u1� ¼ Q�T1 E½S�1�Q�1
1 E½F1� ð42Þ
if S�1 and F1 are statically independent stochastic vectors. The auto-correlation function of the nodal vector u1 is
E u1uT1�
¼ Q�T1 E S�1Q�1
1 F1F T1 Q�T
1 S�1�
Q�11 ð43Þ
which can be rewritten as follows
E u1uT1�
¼ Q�T1 E S�1Q�1
1 F1F T1 ðS�1Q�1
1 ÞTh i
Q�11 ð44Þ
We define the following matrices:
L1 ¼ S�1Q�11 ð45Þ
L2 ¼ F1F T1 ð46Þ
L3 ¼ L1L2 ð47Þ
L ¼ L3LT1 ð48Þ
Therefore, the auto-correlation function of u1 can be put as
E u1uT1�
¼ Q�T1 E½L�Q�1
1 ð49Þ
Matrices L1 can be expressed in the component-wise form as follows:
L1ij ¼X2ni¼1
S�1ik Q�1
kj ¼ S�1ii
X2Nk¼1
Q�1kj ð50Þ
Therefore, the elements of the matrix that is the transpose of L1 are
LT1ij ¼ L1ji ¼ S�1
jj
X2Np¼1
Q�1pi ð51Þ
Then
L2ij ¼ F1iF1j ð52Þ
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 753
L3ij ¼X2Nl¼1
L1ilL2lj ð53Þ
Introducing Eqs. (50) and (52) into Eq. (53), we can simplify Eq. (53) to obtain the following equation:
L3ij ¼ S�1ii Fj
X2Nl¼1
Fl
X2Nk¼1
Q�1kl
!ð54Þ
Substituting Eqs. (50) and (52) into Eq. (48), we arrive at an explicit expression for Lij
Lij ¼X2Nm¼1
S�1ii Fm
X2Nl¼1
Fl
X2Nk¼l
Q�1kl
!" #S�1
jj
X2Np¼1
Q�1pm
" # !¼ S�1
ii S�1jj
X2Nm¼1
Fm
X2Np¼1
Q�1pm
! ! X2Nl¼1
Fl
X2Nk¼1
Q�1kl
! !ð55Þ
Finally,
Lij ¼ S�1ii S�1
jj
X2Nm¼1
X2Nl¼1
FlFm
X2Nk¼1
Q�1kl
! X2Np¼1
Q�1pm
!" #ð56Þ
Then, if S�1 and F1 are two uncorrelated random fields, we have the following expression for the mathematical ex-
pectation:
E½Lij� ¼ aE S�1ii S�1
jj
h ið57Þ
with
a ¼X2Nm¼1
X2Nl¼1
FlFm
X2Nk¼1
Q�1kl
! X2Np¼1
Q�1pm
!" #ð58Þ
We notice that this expectation is written as the product of a matrix which depends of the auto-correlation function of
the stiffness, namely, E½S�1ii S�1
jj � in Eq. (57), and a scalar in Eq. (58) which depends of the auto-correlation function of
the load. Such a multiplicative representation is not possible if S�1 and F1 are correlated. Eq. (57) should be replaced
then by
E½Lij� ¼X2Nm¼1
X2Nl¼1
E S�1ii S�1
jj FlFm
h i X2Nk¼1
Q�1kl
! X2Np¼1
Q�1pm
!" #ð59Þ
For statically indeterminate beams, the mean vector and auto-correlation matrix of the nodal displacement uþ1 are,
respectively,
E½uþ1 � ¼ GTuE S�1h
� S�1GcðGTc S
�1Gc�1GTc S
�1iGuF1; ð60Þ
if F1 is a deterministic vector. On the other hand,
E½uþ1 � ¼ GTuE S�1h
� S�1GcðGTc S
�1Gc�1GTc S
�1iGuE½F1�; ð61Þ
Let us discuss this, if S�1 and F1 are independent stochastic vectors. The auto-correlation function of the nodal vector uu1is represented as follows:
E½uþT1 uþ1 � ¼ GT
uE ZGuF 1u ðF 1
u ÞTGT
uZT
h iGu ð62Þ
with
Z ¼ S�1 � S�1GcðGTc S
�1Gc�1GTc S
�1 ð63Þ
To illustrate the proposed method, we study both statically determinate and indeterminate beams.
3. Statically determinate beams
We first study a beam that is comprised of three equal length elements. The two side elements have the stiffness D1
and the mid-segment has the stiffness D2. The beam is subjected to an uniform load q. We consider D1 and D2 as random
754 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
variables and q as a deterministic quantity (Fig. 2). For every statically determinate beam, the mathematical expectation
of the displacement is a linear expression of the mean values of D�11 and D�1
2 , whereas the mean-square value is a linear
function of the mathematical expectations of D�21 , D�2
2 and D�11 D�1
2 . The elemental stiffnesses are represented, respec-
tively, as
D1 ¼ D0ð1þ kaÞ ð64Þ
and
D2 ¼ D0ð1þ kbÞ ð65Þ
where a and b form a joint by random vector, k is a constant. The normalized variables, a and b are assumed to possess
the Pearson type II distribution with the following density function [12]:
pabðx; yÞ ¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
p ; for x; y 2 D; ð66Þ
The domain of variation D in Eq. (66) is x2 � 2qxy þ y2 6 1� q2 (Fig. 3), which can be written as follows:
qx �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� q2Þð1� x2Þ
p6 y 6qx þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� q2Þð1� x2Þ
p; �16 x6 1 ð67Þ
where q is the correlation coefficient between a and b. The distribution in Eq. (65) assures that the values of the bending
stiffness are positive and bounded, in contrast to often used physically unjustifiable assumption of Gaussianity. The
marginal density of a, defined as
paðxÞ ¼Z
R
pabðx; yÞdy ð68Þ
Fig. 2. Beam under consideration.
Fig. 3. Domain of variation of the Pearson type II distribution.
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 755
is given by
paðxÞ ¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
p Z qxþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�q2Þð1�x2Þ
p
qx�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�q2Þð1�x2Þ
p dy ð69Þ
paðxÞ ¼2
p
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2
p; for � 16 x6 1 ð70Þ
In perfect analogy, we find the marginal density of b
pbðxÞ ¼Z
R
pabðx; yÞdx ð71Þ
to equal
pbðyÞ ¼2
p
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� y2
p; for � 16 y6 1 ð72Þ
As expected, the same function is valid for the marginal density for both a and b due to the symmetry of the joint
probability density function pabðx; yÞ with respect to its arguments. The Eq. (41) shows us that the moments of the
displacement are based on the moments of the inverse of the stiffness. Due to the symmetry of the joint density function,
the mean of a3=Di is given by [13]
Ea3
D0ð1þ kaÞ
� �¼ E
a3
D0ð1þ kbÞ
� �¼Z
R
paðxÞa3 dx
D0ð1þ kxÞ ð73Þ
Taking into account the Eqs. (65) and (66), we obtain
Ea3
D0ð1þ kaÞ
� �¼ 2a3
pD0
Z 1
�1
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2
pdx
1þ kxð74Þ
The integration yields the following final result:
Ea3
D0ð1þ kaÞ
� �¼ 2a3
D0k21�
�ffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2
p �ð75Þ
The second moment of a3=Di is given by [13]
Ea3
D0ð1þ kaÞ
# $2" #
¼ Ea3
D0ð1þ kbÞ
# $2" #
¼Z
R
paðxÞa6 dx
D20ð1þ kxÞ2
ð76Þ
whereas the auto-correlation equals
Ea3
D0ð1þ kaÞ
# $a3
D0ð1þ kbÞ
# $� �¼Z
x2�2qxyþy2 6 1�q2pabðxÞ
a6 dxD2
0ð1þ kxÞð1þ kyÞ ð77Þ
The integration of the Eq. (76) results in
Ea3
D0ð1þ kaÞ
# $2" #
¼ 2a6
pD20
1ffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2
p#
� 1
$ð78Þ
The evaluation of Eq. (77) is more involved because of the elliptic domain involved. We modify this equation in two
steps. First, we approximate the integrand by a following Taylor expansion
a6
D20ð1þ kxÞð1þ kyÞ
a6
D20
Xn
i¼0
Xi
j¼0
ð�kxÞjð�kyÞi�j ¼ a6
D20
Xn
i¼0
Xi
j¼0
ð�kÞiðxÞjðyÞi�j ð79Þ
We define the following change of elliptic variables,
x ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
pcos h
�þ q sin h
�; y ¼ r sin h ð80Þ
756 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
so that the integration domain reduces to {06 r6 1 and �p6 h6 p}. The Jacobian matrix
½J � ¼oxor
oxoh
oyor
oyoh
264
375 ð81Þ
becomes, by taking into account Eq. (80)
½J � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
pcos h þ q sin h r q cos h �
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
psin h
� �sin h r cos h
" #ð82Þ
Finally, its determinant equals
jJ j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
pr ð83Þ
Combining Eqs. (79), (80) and (83) into Eq. (77), the auto-correlation function is approximated by the following ex-
pression
Ea3
D0ð1þ kaÞ
# $a3
D0ð1þ kbÞ
# $� �¼ a6
D20
Z p
�p
dhp
Z 1
0
Xn
i¼0
Xi
j¼0
ð
� kÞiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
pcos h
�þ sin h
�jsin hð Þi�j
!riþ1 dr
ð84Þ
Using the additivity property of the integration, we get
Ea3
D0ð1þ kaÞ
# $a3
D0ð1þ kbÞ
# $� �¼ a6
D20
Xn
i¼0
Xi
j¼0
ð�kÞi
pðiþ 2Þ
Z p
�p
ffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2
pcos h
��þ q sin h
�jðsin hÞi�j
dh
�ð85Þ
Finally, a polynomial expansion of the left-hand side in Eqs. (84) and (85) is expressed in terms of powers of k, asfollows:
Ea3
D0ð1þ kaÞ
# $a3
D0ð1þ kbÞ
# $� �¼ a6
D20
1
�þ k2
4ð2þ qÞ þ k4
24ð7þ 6q þ 2q2Þ þ k6
64ð12þ 13q þ 8q2 þ 2q3Þ
þ k8
640ð83þ 100q þ 84q2 þ 40q3 þ 8q4Þ þ k10
1536ð146þ 183q þ 184q2
þ 124q3 þ 48q4 þ 8q5Þ�þ � � � ð86Þ
Comparison of this result with an integration of (79) by a direct Gaussian�s integration, using ten points, is conducted in
Figs. 4 and 5. For each example, we are in position to compare the results by the proposed stochastic finite method with
the exact analytical solution that is obtainable in this case.
3.1. Clamped-free beam
As first example we study a clamped-free beam. The exact solution is based on solving the differential equation
Did4wdx4
¼ q ð87Þ
valid in each segment of the beam. To perform this integration, we have to separate this beam into three separate
segments with uniform stiffness with attendant satisfaction of the continuity conditions. Thus, Eq. (87) must be replaced
by the following set
D1
d4w1
dx4¼ q; for 06 x6 a ð88aÞ
D2
d4w2
dx4¼ q; for a6 x6 2a ð88bÞ
D1
d4w3
dx4¼ q; for 2a6 x6 3a ð88cÞ
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 757
Upon integration we find
w1ðzÞ ¼qa4
D1
1
24z4
#þ a1
6z3 þ b1
2z2 þ c1z þ d1
$ð89aÞ
w2ðzÞ ¼qa4
D2
1
24z4
#þ a2
6z3 þ b2
2z2 þ c2z þ d2
$ð89bÞ
w3ðzÞ ¼qa4
D1
1
24z4
#þ a3
6z3 þ b3
2z2 þ c3z þ d3
$ð89cÞ
with
z ¼ x=a ð90Þ
0 . 9
1 . 0
1 . 1
1 . 2
1 . 3
1 . 4
1 . 5
1.6
1.7
1.8
5 % 1 0 % 15% 20% 25% 30% 35% 40%
Coefficient of variation
V
a r i
a
n c
eo
fth
ein
vers
eo
fth
est
iffn
es Integration by 10Gaussian points
Approximation withk^2
Approximation withk^6
Approximation withk^10
Fig. 5. Calculation of the variance of the inverse of the stiffness by different approximations.
0 %
2 %
4 %
6 %
8 %
10%
12%
5 % 10% 15% 20% 25% 30% 35% 40%
Coefficient of variation
erro
r(i
n%
)
error with k^2
error with k^6
error with k^10
Fig. 4. Calculation of the variance of the stiffness inverse by different approximations.
758 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
representing the non-dimensional axial coordinate. The boundary conditions read
w1ð0Þ ¼ 0; at z ¼ 0 ð91aÞ
w01ð0Þ ¼ w0
2ð0Þ; at z ¼ 0 ð91bÞ
D1w003ð3Þ ¼ 0; at z ¼ 3 ð91cÞ
D1w0003 ð3Þ ¼ 0; at z ¼ 3 ð91dÞ
The continuity conditions mean that, for z equal, respectively, one or two, the displacement, the slope, the bending
moment and the shear force must be continuous. Those conditions are
w1ð1Þ ¼ w2ð1Þ ð92aÞ
w01ð1Þ ¼ w0
2ð1Þ ð92bÞ
D1w001ð1Þ ¼ D2w00
2ð1Þ ð92cÞ
D1w0001 ð1Þ ¼ D2w000
2 ð1Þ ð92dÞ
w2ð2Þ ¼ w3ð2Þ ð92eÞ
w01ð1Þ ¼ w0
2ð1Þ ð92fÞ
D2w002ð2Þ ¼ D1w00
3ð2Þ ð92gÞ
D2w0002 ð2Þ ¼ D1w000
3 ð2Þ ð92hÞ
Therefore, the constants are expressed as follows:
a1 ¼ �3; b1 ¼9
2; c1 ¼ 0; d1 ¼ 0;
a2 ¼ �3; b2 ¼9
20; c2 ¼ � 19 D1 � D2ð Þ
6D1
; d2 ¼11 D2 � D1ð Þ
8D1
;
a3 ¼ �3; b3 ¼ 0; c3 ¼7 D1 � D2ð Þ
6D2
; d3 ¼ � 13 D1 � D2ð Þ8D2
ð93Þ
Finally, the displacement functions become
w1ðzÞ ¼qa4z2
2D1
9
2
#� z þ z2
12
$ð94aÞ
w2ðzÞ ¼ qa4#�� 11
8D1
þ 11
8D2
$þ 19
6D1
#� 19
6D2
$z þ 9z2
4D2
� z3
2D2
þ z4
24D2
�ð94bÞ
w3ðzÞ ¼ qa413
8D1
#�� 13
8D2
$þ#� 7
6D1
þ 7
6D2
$z þ 9z2
4D1
� z3
2D2
þ z4
24D1
�ð94cÞ
The final expression for the mean functions becomes
E½w1ðzÞ� ¼ E½w2ðzÞ� ¼ E½w3ðzÞ�; E½wðzÞ� ¼ E1
D1
# $qa4z2
2
z2
12
#� z þ 9
2
$ð95Þ
The mean-square values of the displacement E½w2i ðzÞ� are given by
E½w21ðzÞ� ¼ q2a8E
1
D21
# $81
16z4
#� 9
4z5 þ 7
16z6 � z7
24þ z8
576
$ð96aÞ
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 759
E½w22ðzÞ� ¼ q2a8 E
1
D21
# $121
32
#�� 209
12z þ 3779
144z2 þ 125
8z3 þ 267
32z4 � 181
72z5 þ 7
16z6 � 1
24z7 þ 1
576z8$
þ E1
D1D2
# $#� 121
32þ 209
32z � 3779
144z2 þ 125
8z3 � 105
32z4 þ 19
75z5$�
ð96bÞ
E½w23ðzÞ� ¼ q2a8 E
1
D21
# $169
32
#�� 91
12z þ 1445
144z2 � 55
8z3 þ 125
96z4 � 7
72z5 þ 7
16z6 � 1
24z7 þ 1
576z8$
þ E1
D1D2
# $#� 169
32þ 91
12z � 1445
144z2 þ 55
8z3 � 125
96z4 � 7
72z5$�
ð96cÞ
Let us proceed now with the estimates of the probabilistic response derivable via the finite element method. The mean
displacement functions are given in Eq. (41). In the present example, w1 and h1 vanish identically. The expectation of the
inverse of the stiffness matrix equals
EðS�1Þ ¼
14E 1
D1
� �0 0 0 0 0
0 1300
E 1D2
� �0 0 0 0
0 0 14E 1
D1
� �0 0 0
0 0 0 1300
E 1D2
� �0 0
0 0 0 0 14E 1
D1
� �0
0 0 0 0 0 1300
E 1D2
� �
2666666666666664
3777777777777775
ð97Þ
The numeric matrix Q1 reads
Q1 ¼1
5
0 �1 0 1 0 052
12
�52
12
0 0
0 0 0 �1 0 1
0 0 52
12
�52
12
0 0 0 0 0 �10 0 0 0 5
212
26666664
37777775
ð98Þ
The equivalent load is represented by
F T1 ¼ qa 1 0 1 0 1
2� 1
12
� ð99Þ
Therefore, taking into account Eq. (73), the mean vector becomes
Eðu1Þ ¼ Q�T1 EðS�1ÞQ�1
1 F1;
Eðw2Þ Eðh2Þ Eðw3Þ Eðh3Þ Eðw4Þ Eðh4Þ½ �T ¼ qa4E1
D1
� �4312
193
343
263
814
9� T ð100Þ
The auto-correlation matrix is given in Appendix A. Eqs. (100) yields the mean displacements at the nodes. One in-
terpolates the mean displacement via the shape functions:
E½wiðzÞ� ¼ N1ðzÞE½wi1� þ N2ðzÞaE½hi1� þ N3ðzÞE½wi2� þ N4ðzÞaE½hi2�;E½w2
i ðzÞ� ¼ E½ðN1ðzÞwi1 þ N2ðzÞ þ N3ðzÞwi2 þ N4ðzÞahi2Þ2�ð101Þ
Thus we obtain for the mean displacement over the three segments:
E½w1ðzÞ� ¼ qa453z2
24
#� 5z3
12
$E
1
D1
# $;
E½w2ðzÞ� ¼ qa4#� 1
6þ z2þ 41z2
24� z3
4
$E
1
D1
# $;
E½w3ðzÞ� ¼ qa4#� 3
2þ 5z
2þ 17z2
24� z3
12
$E
1
D1
# $ ð102Þ
760 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
whereas the mean-square values are
E½w21ðzÞ� ¼ q2a8
2809z4
576
#� 265z5
144þ 25z6
144
$E
1
D21
# $;
E½w22ðzÞ� ¼ q2a8
965
288
#�� 1091z
72þ 6125z2
288� 1399z3
144þ 2449z4
576� 41z5
48þ z6
16
$E
1
D21
# $
þ#� 319
96þ 1079z
72� 6217z2
288þ 1657z3
144� 19z4
12
$E
1
D1D2
# $�;
E½w23ðzÞ� ¼ q2a8
85
32
#�� 83z
24þ 955z2
288þ 269z3
144þ 161z4
576� 17z5
144þ z6
144
$E
1
D21
# $
þ#� 13
32� 97z
24þ 233z2
288þ 277z3
144� 7z4
36
$E
1
D1D2
# $�
ð103Þ
The mean function and the mean-square value of the displacement by both methods are plotted in Figs. 6 and 7. The
relative error between the exact solution and the finite element method is plotted in Figs. 8 and 9. The calculations are
performed for a correlation q ¼ 0:5 and k ¼ 0:6 implying the coefficient of variation of the stiffness of 30%. This co-
efficient of variation is beyond the scope of applicability of the conventional perturbation methods. In our present
scheme, the relative error in comparison with the exact solution is smaller than 0.02%. We notice a remarkable phe-
nomenon that the finite element method yields the exact results in the nodes.
3.2. Beam simply-supported at both ends
The method used in this case coincides with that for the clamped-free beam. The exact solution is based on the Eq.
(87). To perform the direct integration, we have to separate the beam into three segments of uniform stiffness. The
integration result is
w1ðzÞ ¼qa4
D1
1
24z4
#þ a1
6z3 þ b1
2z2 þ c1z þ d1
$ð104aÞ
w2ðzÞ ¼qa4
D2
1
24z4
#þ a2
6z3 þ b2
2z2 þ c2z þ d2
$ð104bÞ
w3ðzÞ ¼qa4
D1
1
24z4
#þ a3
6z3 þ b3
2z2 þ c3z þ d3
$ð104cÞ
0
2
4
6
8
1 0
1 2
0 1 2 3axial coordinate z
analytical result
present method
E[w
(z)]
Fig. 6. Mean displacement in the clamped-free beam.
Fig. 7. Mean-square values of the displacement in the clamped-free beam.
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 761
The boundary conditions of the case read
w1ð0Þ ¼ 0; at z ¼ 0 ð105aÞ
D1w001ð0Þ ¼ 0; at z ¼ 0 ð105bÞ
w3ð3Þ ¼ 0; at z ¼ 0 ð105cÞ
D1w003ð0Þ ¼ 0; at z ¼ 0 ð105dÞ
The continuity conditions coincide with Eqs. (92). The integration constants are expressed as
a1 ¼ � 3
2; b1 ¼ 0; c1 ¼
13D1 þ 14D2
24D2
; d1 ¼ 0;
a2 ¼ � 3
2; b2 ¼ 0; c2 ¼
9
8; d2 ¼ � 3 D1 � D2ð Þ
8D1
;
a3 ¼ � 3
2; b3 ¼ 0; c3 ¼
�13D1 þ 14D2
24D2
; d3 ¼13 D1 � D2ð Þ
8D2
ð106Þ
0 . 0 0
0 . 0 0
0 . 0 0
0 . 0 0
0 . 0 1
0 . 0 1
0 . 0 1
0 . 0 1
0.01
0.02
0.02
0 1 2 3
axial coordinate z
rel
ativ
e er
ror
(%)
Fig. 8. Relative error.
- 0 . 0 1
0 . 0 0
0 . 0 1
0 . 0 1
0 . 0 2
0.02
0.03
0.03
0.04
0 1 2 3
axial coordinate z
rel
ativ
e er
ror
(%)
Fig. 9. Relative error.
762 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
Finally, the displacement functions are obtained as
w1ðzÞ ¼ qa47z
12D1
#þ 13z24D2
� z3
4D1
þ z4
24D1
$ð107aÞ
w2ðzÞ ¼ qa43
8D1
#� 3
8D2
þ 9z8D2
� z3
4D2
þ z4
24D2
$ð107bÞ
w3ðzÞ ¼ qa4#� 13
16D1
þ 13
16D2
þ 5z3D1
� 13z24D2
� z3
4D1
þ z4
24D1
$ð107cÞ
The mean functions read
E½w1ðzÞ� ¼ E½w2ðzÞ� ¼ E½w3ðzÞ�; E½wðzÞ� ¼ qa4E1
D1
# $z4
24
#� z3
4þ 9
8
$ð108Þ
because Eð1=D1Þ ¼ Eð1=D2Þ. Although, remarkably, the mathematical expressions E½wiðzÞ� coincide, still they differ
since the coordinate z is different in each segment. The mean-square values of the displacement E½w2i ðxÞ� are given by
E½w21ðxÞ� ¼ q2a8 E
1
D21
# $365
576z2
#�� 7
24z4 � 7
144z5 þ 1
16z6 � 1
48z7 þ 1
576z8$
þ E1
D1D2
# $91
144z2
#� 13
48z4 þ 13
288z5$�
ð109aÞ
E½w22ðxÞ� ¼ q2a8 E
1
D21
# $9
32
#�� 27
32z þ 81
64z2 þ 3
16z3 � 19
32z4 � 3
32z5 þ 1
16z6 � 1
48z7 þ 1
576z8$
þ E1
D1D2
# $#� 9
32þ 27
32z � 3
16z3 þ 1
32z4$�
ð109bÞ
E½w23ðxÞ� ¼ q2a8 E
1
D21
# $169
32
#�� 689
96z þ 1769
576z2 þ 13
16z3 � 31
32z4 þ 5
36z5 þ 1
16z6 � 1
48z7 þ 1
576z8$
þ E1
D1D2
# $#� 169
32þ 689
96z � 65
36z2 � 13
16z3 þ 13
32z4 � 13
288z5$�
ð109cÞ
As far as the finite element method is concerned, the mean displacement is derivable from Eq. (41). In this example, w1
and w4 are vanishing identically. The expectation of the inverse of the stiffness matrix equals
EðS�1Þ ¼
14E 1
D1
� �0 0 0 0 0
0 1300
E 1D2
� �0 0 0 0
0 0 14E 1
D1
� �0 0 0
0 0 0 1300
E 1D2
� �0 0
0 0 0 0 14E 1
D1
� �0
0 0 0 0 0 1300
E 1D2
� �
266666666666664
377777777777775
ð110Þ
The numeric matrix Q1 reads in this case
Q1 ¼1
5
� 52
12
0 0 0 0
0 �1 0 1 0 052
12
� 52
12
0 00 0 0 �1 0 1
0 0 52
12
�52
12
0 0 0 0 52
12
26666664
37777775
ð111Þ
The equivalent load is
F T1 ¼ qa 1
121 0 1 0 � 1
12
� T ð112Þ
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 763
Therefore, the mean vector becomes
Eðu1Þ ¼ Q�T1 EðS�1ÞQ�1
1 F1 ð113Þ
Eðw2Þ Eðh2Þ Eðw3Þ Eðh3Þ Eðw4Þ Eðh4Þ½ �T ¼ qa4E1
D1
� �98
1112
1324
1112
� 1324
� 98
� T ð114Þ
The auto-correlation matrix is given in Appendix B. Eqs. (113) and (114) presents the mean displacement at the nodes.
The mean displacements over the three segments read
E½w1ðzÞ� ¼ qa4#� 13
12� 17
24z þ 23z2
24� z3
6
$E
1
D1
� �;
E½w2ðzÞ� ¼ qa4#� 7
3þ 65z
24� 13z2
24
$E
1
D1
� �;
E½w3ðzÞ� ¼ qa4#� 3
2þ 29z
8� 37z2
24þ z3
6
$E
1
D1
� �ð115Þ
The mean-square values are
E½w21ðzÞ� ¼ q2a8
3887
72
#�� 7943
72z þ 2923
32z2 � 349
9z3 þ 2497z4
288� 29z5
32þ z6
36
$E
1
D21
# $
þ 65
24z
#� 247
72z2 þ 403
288z3 � 13z4
72
$E
1
D1D2
# $�ð116aÞ
E½w22ðzÞ� ¼ q2a8
2153
288
#�� 4225z
288þ 5915z2
576� 845z3
288þ 169z4
576
$E
1
D21
# $þ#� 65
32þ 65z
32� 13z2
32
$E
1
D1D2
# $�ð116bÞ
E½w23ðzÞ� ¼ q2a8
397
32
#�� 2669z
96þ 15719z2
576� 125z3
9þ 241z4
64� 37z5
72þ z6
36
$E
1
D21
# $
þ#� 325
32þ 1625z
96� 2743z2
288þ 637z3
288� 13z4
72
$E
1
D1D2
# $�ð116cÞ
The mean function and the second moment of the displacement by both methods are portrayed in Figs. 10 and 11. The
relative error is shown in Figs. 12 and 13. The calculations are performed for a correlation coefficient q ¼ 0:5 and
coefficient k is fixed at 0:6, implying a coefficient of variation of the stiffness of 30%. The relative error of the maximum
displacement is about 0.2% for the mean function and less than 0.01% for the mean-square, again illustrating the
excellent accuracy achieved by the present scheme of the finite element method.
Fig. 10. Mean displacement in the simply-supported beam.
764 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
4. Statically indeterminate beams
4.1. Clamped/simply-supported beam
The third example presents a beam clamped at one end and simply-supported at the other end. We first obtain the
exact solution. To perform the integration in Eq. (87), we follow the previous procedure of separation
w1ðzÞ ¼qa4
D1
1
24z4
#þ a1
6z3 þ b1
2z2 þ c1z þ d1
$ð117aÞ
w2ðzÞ ¼qa4
D2
1
24z4
#þ a2
6z3 þ b2
2z2 þ c2z þ d2
$ð117bÞ
Fig. 12. Relative error.
axial coordinate z
rel
ativ
e er
ror
(%)
Fig. 13. Relative error.
Fig. 11. Mean-square values of the displacement in the simply-supported beam.
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 765
w3ðzÞ ¼qa4
D1
1
24z4
#þ a3
6z3 þ b3
2z2 þ c3z þ d3
$ð117cÞ
The boundary conditions of the case are:
w1ð0Þ ¼ 0; at z ¼ 0 ð118aÞ
w01ð0Þ ¼ 0; at z ¼ 0 ð118bÞ
w3ð3Þ ¼ 0; at z ¼ 0 ð118cÞ
D1w001ð0Þ ¼ 0; at z ¼ 0 ð118dÞ
Satisfaction the continuity conditions in the Eqs. (92) leads to the following expression for the integration constants
a1 ¼ a2 ¼ a3 ¼ � 3ð41D1 þ 94D2Þ8ð7D1 þ 20D2Þ
; b1 ¼ b2 ¼ b3 ¼9ð13D1 þ 14D2Þ8ð7D1 þ 20D2Þ
;
c1 ¼ 0; c2 ¼ � 389D21 � 319D1D2 � 70D2
2
48D1ð7D1 þ 20D2Þ; c3 ¼ � 13D2
1 þ 649D1D2 � 662D22
48D2ð7D1 þ 20D2Þ;
d1 ¼ 0; d2 ¼49D2
1 � 71D1D2 þ 22D22
16D1ð7D1 þ 20D2Þ; d3 ¼
13ðD21 þ 25D1D2 � 26D2
2Þ16D2ð7D1 þ 20D2Þ
ð119Þ
The displacement functions are
w1ðzÞ ¼ qa4 63
160D1
z2#
þ 729
160ð7D1 þ 20D2Þz2 � 47
160D1
z3 � 81
160ð7D1 þ 20D2Þz3 þ z4
24D1
$ð120aÞ
w2ðzÞ ¼ qa4 11
160D1
#þ 7
16D2
� 2187
160ð7D1 þ 20D2Þþ 7
96D1
z � 389
336D2
z þ 6561
224ð7D1 þ 20D2Þz
þ 63
160D1
z2 þ 729
160ð7D1 þ 20D2Þz2 � 41
112D2
z3 � 81
56ð7D1 þ 20D2Þz3 þ z4
24D2
$ð120bÞ
w3ðzÞ ¼ qa4
#� 169
160D1
þ 13
112D2
þ 28431
1120ð7D1 þ 20D2Þþ 331
480D1
z � 13
336D2
z � 19683
1120ð7D1 þ 20D2Þz
þ 63
160D1
z2 þ 729
160ð7D1 þ 20D2Þz2 � 47
160D1
z3 � 81
160ð7D1 þ 20D2Þz3 þ z4
24D1
$ð120cÞ
Final expressions for the mean functions are
E½w1ðzÞ� ¼ qa4 E1
D1
# $63z2
160
#�� 47z3
160þ z4
24
$þ E
1
7D1 þ 20D2
# $729z2
160
#� 81z3
160
$�ð121aÞ
E½w2ðzÞ� ¼ qa4 E1
D1
# $81
160
#�þ 243z
224þ 117z2
112� 41z3
160þ z4
24
$
þ E1
7D1 þ 20D2
# $#� 2187
160þ 6561z
224� 729z2
160þ 81z3
56
$�ð121bÞ
E½w3ðzÞ� ¼ qa4 E1
D1
# $#�� 1053
1120þ 729z1120
þ 63z2
160� 41z3
160þ z4
24
$
þ E1
7D1 þ 20D2
# $28431
1120
#� 19683z
1120þ 729z2
160� 81z3
160
$�ð121cÞ
The mean-square values of the displacement E½w2i ðxÞ� are given by
E½w21ðzÞ� ¼ q2a8 E
1
D21
# $3969z4
25600
(� 2961z5
12800þ 2629z6
25600� 47z7
1920þ z8
576
!
þ E1
ð7D1 þ 20D2Þ2
" #531441z4
25600
#� 59049z5
12800þ 6561z6
25600
$
þ E1
D1ð7D1 þ 20D2Þ
" #45927z4
12800
#� 19683z5
6400þ 4293z6
12800� 27z7
640
$)ð122aÞ
766 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
E½w22ðzÞ� ¼ q2a8 E
1
D21
# $5021
25600
(� 7703z
7680þ 1020461z2
451584� 4295z3
1568þ 24779z4
12544� 48619z5
56448þ 2773z6
12544
� 47z7
1344þ z8
576
!þ E
1
D1D2
# $77
1280
#� 641z6720
� 127z2
5040þ 457z3
4480� 61z4
1280þ 7z5
1152
$
þ E1
ð7D1 þ 20D2Þ2
" #4782969
25600
#� 14348907z
17920þ 304515693z2
250880� 12577437z3
15680þ 1594323z4
6272
� 59049z5
1568þ 6561z6
3136
$þ E
1
D1ð7D1 þ 20D2Þ
" ##� 24057
12800þ 3645z
1792þ 444469z2
17920� 3807z3
2240
þ 27z4
128
$þ E
1
D2ð7D1 þ 20D2Þ
� �#� 15309
1280þ 8019z
140� 844911z2
7840þ 3217887z3
31360� 3332367z4
62720
þ 94041z5
6272� 6723z6
3136þ 27z7
224
$)ð122bÞ
E½w23ðzÞ� ¼ q2a8 E
1
D21
# $1416389
1254400
(� 2757911z
1881600� 801053z2
2257920þ 7447z3
6400� 25967z4
76800� 20029z5
115200þ 3049z6
25600
� 47z7
1920þ z8
576
!þ E
1
D1D2
# $#� 2197
8960þ 325z1344
þ 767z2
20160� 221z3
2240þ 871z4
26880� 13z5
4032
$
þ E1
ð7D1 þ 20D2Þ2
" #808321761
1254400
#� 559607373z
627200þ 27103491z2
50176� 8325909z3
44800þ 6908733z4
179200
� 59049z5
12800þ 6561z6
25600
$þ E
1
D1ð7D1 þ 20D2Þ
" ##� 4804839
89600þ 3231657z
44800� 248589z2
17920
� 239679z3
11200þ 1373571z4
89600� 203391z5
44800þ 8667z6
12800� 27z7
640
$þ E
1
D2ð7D1 þ 20D2Þ
� �369603
62720
#
� 9477z1568
þ 9477z2
3920� 1053z3
2240þ 351z4
8960
$)ð122cÞ
For statically indeterminate beams, the implementation of the present method is more elaborate since we have to
separate the numerical matrix Q into two sub-matrices corresponding to the constrained and the unconstrained nodes.
The compliance matrix is defined by
S�1 ¼
14
1D1
0 0 0 0 0
0 1300
1D1
0 0 0 0
0 0 14
1D2
0 0 0
0 0 0 1300
1D2
0 0
0 0 0 0 14
1D1
0
0 0 0 0 0 1300
1D1
2666666664
3777777775
ð123Þ
To define the sub-matrices Gc and Gu, we start from the numerical matrix Q1 of the statically determinate case of a
clamped-free beam (see Eq. (97)). We first deduce G, the inverse matrix of Q1
G ¼
1 2 3 2 5 2
�5 0 �5 0 �5 00 0 1 2 3 2
0 0 �5 0 �5 0
0 0 0 0 1 2
0 0 0 0 �5 0
26666664
37777775
ð124Þ
From this matrix, we get the sub-matrix Gc of the constrained nodes, namely the fifth column of the matrix G, the one
which corresponds to the translation of the fourth node:
GTc ¼ ½ 5 �5 3 �5 1 �5 �T ð125Þ
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 767
The sub-matrix Gu corresponds to the unconstraint nodes. It reads
Gu ¼
1 2 3 2 2�5 0 �5 0 00 0 1 2 20 0 �5 0 00 0 0 0 20 0 0 0 0
2666664
3777775 ð126Þ
The equivalent load of this case is
F T1 ¼ qa 1 0 1 0 � 1
12
� T ð127Þ
Therefore, according to Eq. (35), the displacement vector is
w2
h2
w3
h3
h4
266664
377775 ¼ qa4
1120
17D1þ 486
7D1þ20D2
� �196
7D1þ 729
7D1þ20D2
� �148
10310D1
þ 137D2
þ 14 82370ð7D1þ20D2Þ
� �� 1
48� 7
2D1þ 13
7D2þ 3645
14ð7D1þ20D2Þ
� �� 1
48915D1
þ 137D2
þ 656135ð7D1þ20D2Þ
� �
26666666664
37777777775
ð128Þ
The correlation matrix is given in the Appendix C. As in the statically determinate case, the mean displacement and the
mean-square value of the displacement are expressible via the shape function. The equations of themean displacement are
E½w1ðzÞ� ¼ qa4 169
480z2
#�� 101
480z3$E
1
D1
# $þ 729
160z2
#� 81
160z3$E
1
7D1 þ 20D2
# $�ð129aÞ
E½w2ðzÞ� ¼ qa4163
480
#�� 131
224zþ 169
336z2 � 13
112z3$E
1
D1
# $þ#� 2187
160þ 6561
224z� 729
56z2 þ 81
56z3$E
1
7D1 þ 20D2
# $�ð129bÞ
E½w3ðzÞ� ¼ qa4
#�� 2733
1120þ 3529
1120z � 551
480z2 þ 59
480z3$E
1
D1
# $
þ 28431
1120
#� 19683
1120z þ 729
160z2 � 81
160z3$E
1
7D1 þ 20D2
# $�ð129cÞ
The second moment read
E½w21ðzÞ� ¼ q2a8
(28561
230400z4
#� 17069
115200z5 þ 10201
230400z6$E
1
D21
!þ 531441
25600z4
#� 59049
12800z5
þ 6561
25600z6$E
1
ð7D1 þ 20D2Þ2
" #þ 41067
12800z4
#� 14553
6400z5 þ 2727
12800z6$E
1
D1ð7D1 þ 20D2Þ
" #)ð130aÞ
E½w22ðzÞ� ¼ q2a8
17989
230400
#�� 55843
161280z þ 320797
451584z2 � 20449
28224z3 þ 45799
112896z4 � 2197
18816z5
þ 169
12544z6$E
1
D21
!þ 143
3840
#� 1027
20160z � 1079
40320z2 þ 1157
20160z3 � 13
768z4$E
1
D1D2
# $
þ 4782969
25600
#� 14348907
17920z þ 304515693
250880z2 � 12577437
15680z3 þ 1594323
6272z4 � 59049
1568z5
þ 6561
3136z6$E
1
7D1 þ 20D2
� �2" #
þ
� 24057
12800þ 3645
1792z � 44469
17920z2 � 3807
2240z3
þ 27
128z4!
E1
D1ð7D1 þ 20D2Þ
� �þ#� 9477
1280þ 9477
280z � 1860651
31360z2 þ 396279
7840z3 � 273429
12544z4
þ 1755
392z5 � 1053
3136z6$E
1
D2ð7D1 þ 20D2Þ
� ��ð130bÞ
768 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
E½w23ðzÞ� ¼ q2a8
8213669
1254400
#�� 30699671
1881600z þ 36225307
2257920z2 � 457987
57600z3 þ 28561
230400z4 � 17069
115200z5
þ 10201
230400z6$E
1
D21
!þ#� 5317
8960þ 1261
1344z � 2587
5040z2 þ 169
1440z3 � 767
80640z4$E
1
D1D2
# $
þ 808321761
1254400
#� 559607373
6272600z þ 27103491
50176z2 � 8325909
44800z3 þ 6908733
179200z4 � 59049
12800z5
þ 6561
25600z6$E
1
ð7D1 þ 20D2Þ2
" #þ#� 11628279
89600þ 11279817
44800z � 3470769
17920z2 þ 250371
3200z3
� 1613709
89600z4 þ 14607
6400z5 � 1053
3136z6$E
1
D1ð7D1 þ 20D2Þ
" #þ 369603
62720
#� 9477
1568z þ 9477
3920z2
� 1053
2240z3 þ 351
8960z4$E
1
D2ð7D1 þ 20D2Þ
� ��ð130cÞ
The results obtained by both methods are the functions of the following mathematical expectations E½ð7D1 þ 20D2Þ�1�,E½ð7D1 þ 20D2Þ�2�, E½D�1
1 ð7D1 þ 20D2Þ�1�, E½D�12 ð7D1 þ 20D2Þ�1�. These expectations must be calculated using the
definition
E½f ðx; yÞ� ¼Z þ1
�1
Z þ1
�1f ðx; yÞpab dxdy ð131Þ
but these integrations are too cumbersome because one has to integrate a rational function over the elliptic domain of
definition of the random variables. Therefore, we use the same approach as described in Eqs. (79)–(86). This approach
aims at modification in two steps. First, we approximate the rational function by a Taylor expansion. Then, we change
the original variables to the elliptic ones in order to achieve an additional simplification the integration domain. This
method yields these desired expectations as polynomial expansion in terms of k and q:
E1
7D1 þ 20D2
# $¼ a3
D0
1
27
"þ k2ð449þ 280qÞ
78732þ k4ð449þ 280qÞ2
114791256þ 5k6ð449þ 280qÞ3
669462604992
þ 7k8ð449þ 280qÞ4
976076478078336þ 7k10ð449þ 280qÞ5
948746336692142592þ � � �
#ð132Þ
E1
ð7D1 þ 20D2Þ2
" #¼ a6
D20
1
729
"þ k2ð449þ 280qÞ
708588þ 5k4ð449þ 280qÞ2
3099363912þ 35k6ð449þ 280qÞ3
18075490334784
þ 7k8ð449þ 280qÞ4
2928229434235008þ 77k10ð449þ 280qÞ5
25616151090687849984þ � � �
#ð133Þ
E1
D1ð7D1 þ 20D2Þ
� �¼ a6
D20
1
27
�þ k2
1367þ 820q78732
þ 5k4217721þ 228920q þ 84800q2
114791256
þ 5k6767083283þ 1031840940q þ 720427200q2 þ 192592000q3
669462604992þ � � �
�ð134Þ
E1
D2ð7D1 þ 20D2Þ
� �¼ a6
D20
1
27
�þ k2
1718þ 469q78732
þ k41672669þ 829402q þ 155134q2
114791256
þ k6996243064þ 669397911q þ 237553176q2 þ 33908294q3
669462604992þ � � �
�ð135Þ
The results are plotted on Figs. 14–16. The relative error in contrast with the numerical evaluation of the integrals is
order of 1%. This error is still extremely small for a coefficient of variation k as large as 30%. One may argue that the
expression (132)–(135) amount de facto to the use of the perturbation technique. Yet, in this scheme arbitrary number of
terms can be included, whereas in the conventional perturbation or polynomial chaos expansions only very few terms
are taken into account. Also, the expressions (132)–(135) are derived here for convenience. For larger coefficients of
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 769
variation one can utilize a numerical method to arrive at the desired mathematical expectations of the function
E½D�a1 D�b
2 ðxD1 þ xD2Þ�c� where a, b, x, x and c are positive coefficients.
4.2. Beam clamped at both ends
To obtain an exact solution, we integrate the differential equation (87). The boundary conditions of the case are
w1ð0Þ ¼ 0; at z ¼ 0 ð136aÞ
w01ð0Þ ¼ 0; at z ¼ 0 ð136bÞ
Fig. 14. Mean displacement in the clamped/simply-supported beam.
Fig. 15. Mean-square values of the displacement in the simply-supported beam.
-0.010.000.010.020.030.040.050.060.070.080.090.10
0 1 2 3
axial coordinate z
rel
ativ
e er
ror
(%)
Fig. 16. Relative error.
770 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
w3ð3Þ ¼ 0; at z ¼ 0 ð136cÞ
w03ð0Þ ¼ 0; at z ¼ 0 ð136dÞ
The satisfaction of the boundary condition (Eqs. (136)) and the continuity conditions in (Eqs. (92)) result in the
mathematical expectations are
E½w1ðzÞ� ¼qa4
4E
1
D1
# $7z2
6
#�� z3 þ z4
6
$þ z2
6E
1
D1 þ 2D2
# $�ð137aÞ
E½w2ðzÞ� ¼qa4
4E
1
D1
# $1
#�� 2z þ 13z2
6� z3 þ z4
6
$þ E
1
D1 þ 2D2
# $ð � 3þ 6z � 2z2Þ
�ð137bÞ
E½w3ðzÞ� ¼qa4
4E
1
D1
# $#�� 3þ 2z þ 7z2
6� z3 þ z4
6
$þ E
1
D1 þ 2D2
# $ð9� 6z þ z2Þ
�ð137cÞ
The mean-square values equal
E½w21ðzÞ� ¼ q2a8 E
1
D21
# $49z4
576
(� 7z5
48þ 25z6
288� z7
48þ z8
576
!þ z4
16E
1
ðD1 þ 2D2Þ2
" #
þ E1
D1ðD1 þ 2D2Þ
� �7z4
48
� z5
8þ z6
48
!)ð138aÞ
E½w22ðzÞ� ¼ q2a8 E
1
D21
# $5
144
(� z6þ 31z2
72� 5z3
8þ 107z4
192� 5z5
16þ 31z6
288� z7
48þ z8
576
!
þ E1
D1D2
# $1
36
#� z12
þ 13z2
144� z3
24þ z4
144
$þ E
1
ðD1 þ 2D2Þ2
" #9
16
#� 9z
4þ 3z2 � 3z3
4þ z4
16
$
þ E1
D1ðD1 þ 2D2Þ
" ##� 1
8þ z4� z2
12
$þ E
1
D2ðD1 þ 2D2Þ
� �#� 1
4þ 5z
4� 119z2
48þ 5z3
2
� 65z4
48þ 3z5
8� z6
24
$)ð138bÞ
E½w23ðzÞ� ¼ q2a8 E
1
D21
# $9
16
(� 3z
4� 3z2
16þ 2z3
3� 131z4
576� 5z5
48þ 25z6
288� z7
48þ z8
576
!
þ E1
ðD1 þ 2D2Þ2
" #81
16
#� 27z
4þ 27z2
8� 3z3
4þ z4
16
$
þ E1
D1ðD1 þ 2D2Þ
" ##� 27
8þ 9z
2� 9z2
16� 7z3
4þ 13z4
12� z5
4þ z6
48
$)ð138cÞ
We separate the numerical matrix Q into two sub-matrices corresponding to the constrainted and the unconstrainted
nodes, respectively. The compliance matrix is defined as follows:
S�1 ¼
14
1D1
0 0 0 0 0
0 1300
1D1
0 0 0 0
0 0 14
1D2
0 0 0
0 0 0 1300
1D2
0 0
0 0 0 0 14
1D1
0
0 0 0 0 0 1300
1D1
2666666664
3777777775
ð139Þ
To introduce the sub-matrices Gc and Gu, we start from the numerical matrix Q1 occurring in statically determinate case
of a clamped-free beam (see Eq. (97)). We first deduce G, the inverse matrix of Q1 Eq. (124). From this matrix, we get
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 771
the sub-matrix Gc of the constrained nodes, namely the two last columns of the matrix G, the columns which correspond
to the translation and the rotation of the fourth node:
GTc ¼ 5 �5 3 �5 1 �5
2 0 2 0 2 0
� �Tð140Þ
The sub-matrix Gu corresponds to the unconstrainted nodes. It reads
Gu ¼
1 2 3 2
�5 0 �5 00 0 1 2
0 0 �5 0
0 0 0 0
0 0 0 0
26666664
37777775
ð141Þ
The equivalent load of this case is
F T1 ¼ qa 1 0 1 0½ �T ð142Þ
Therefore, according to Eq. (35), the displacement vector becomes
w2
h2
w3
h3
2664
3775 ¼ qa4
112
1D1þ 3
D1þ2D2
� �1
2ðD1þ2D2Þ112
1D1þ 3
D1þ2D2
� �� 1
2ðD1þ2D2Þ
2666664
3777775 ð143Þ
We can immediately notice a strict symmetry of the result on the sense that w2 ¼ w3 and h2 ¼ h3. The correlation matrix
is reported in the Appendix D. The mean displacement and the mean-square value of the displacement are
E½w1ðzÞ� ¼ qa4 1
4z2
#�� 1
6z3$E
1
D1
# $þ 1
4z2E
1
D1 þ 2D2
# $�ð144aÞ
E½w2ðzÞ� ¼ qa4 1
12E
1
D1
# $�þ#� 3
4þ 3
2z � 1
2z2$E
1
D1 þ 2D2
# $�ð144bÞ
E½w3ðzÞ� ¼ qa4
#�� 9
4þ 3z � 5
4z2 þ 1
6z3$E
1
D1
# $þ 1
49�
� 6z þ z2�E
1
D1 þ 2D2
# $�ð144cÞ
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0 1 2 3
E[w
(z)]
analytical resultpresent method
axial coordinate z
Fig. 17. Mean displacement in the clamped–clamped beam.
772 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
Fig. 19. Mean-square displacement in the clamped–clamped beam.
axial coordinate z
rel
ativ
e er
ror
(%)
Fig. 18. Relative error.
axial coordinate z
rel
ativ
e er
ror
(%)
Fig. 20. Relative error.
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 773
The mean-square values read
E½w21ðzÞ� ¼ q2a8
1
16z4
#(� 1
12z5 þ 1
36z6$E
1
D21
# $þ 1
16z4E
1
ðD1 þ 2D2Þ2
" #þ 1
8z4
#� 1
12z5$E
1
D1ðD1 þ 2D2Þ
� �)
ð145aÞ
E½w22ðzÞ� ¼ q2a8
1
144E
1
D21
# $(þ 9
16
� 9
4z þ 3z2 � 3
2z3 þ 1
4z4!
E1
ðD1 þ 2D2Þ2
" #
þ#� 1
8þ 1
4z � 1
12z2$E
1
D1ðD1 þ 2D2Þ
" #)ð145bÞ
E½w23ðzÞ� ¼ q2a8
81
16
#(� 27
2z þ 117
8z2 � 33
4z3 þ 41
16z4 � 5
12z5 þ 1
36z6$E
1
D21
# $
þ#� 81
8þ 81
4z � 63
4z2 � 6z3 � 9
8z4 � 5
12z5 þ 1
36z6$E
1
D1ðD1 þ 2D2Þ
� �
þ 81
16
#� 27
4z þ 27
2z2 � 3
4z3 þ 1
16z4$E
1
ðD1 þ 2D2Þ2
" #)ð145cÞ
The obtained results are functions of the mathematical expectations of the following functions of D1 and D2:
E½ðD1 þ 2D2Þ�1�, E½ðD1 þ 2D2Þ�2� and E½D�11 ðD1 þ 2D2Þ�1�. These expectations are calculated using the same approach
describe from Eqs. (79)–(86). The final results are
E1
D1 þ 2D2
# $¼ a3
D0
1
3
"þ k2ð5þ 4qÞ
108þ k4ð5þ 4qÞ2
1944þ 5k6ð5þ 4qÞ3
139968þ 7k8ð5þ 4qÞ4
2519424þ 7k10ð5þ 4qÞ5
30233088
#ð146aÞ
E1
ðD1 þ 2D2Þ2
" #¼ a6
D20
1
9
"þ k2ð5þ 4qÞ
108þ 5k4ð5þ 4qÞ2
5832þ 35k6ð5þ 4qÞ3
419904þ 7k8ð5þ 4qÞ4
839808þ 77k10ð5þ 4qÞ5
90699264
#
ð146bÞ
E1
D1ðD1 þ 2D2Þ
� �¼ a6
D20
1
3
�þ k2
17þ 10q108
þ k4169þ 172q þ 64q2
1944þ k6
7453þ 9582q þ 6672q2 þ 1808q3
139968
þ k8446003þ 630760q þ 606144q2 þ 316480q3 þ 67328q4
12597120
�ð146cÞ
The results are plotted on Figs. 17–20. The relative error is again about 1%.
5. Conclusion
This study is a generalization of the method developed in Ref. [9] applied to beams with stochastic stiffness or
loading under different boundary conditions. The originality of this method consists in the representation of the
stiffness in three matrices, two of which are numerical ones, whereas one of them contains a stochastic stiffness.
Therefore, the inversion of the stiffness matrix is direct and without approximations contrary to the conventional
perturbation methods. Moreover, the present formulation is free of the crucial assumption needed in the perturbation
approach, namely that the coefficient of variation is small. Indeed, in the presented examples, the obtained relative
error between both approaches is extremely small despite the adopted value of 30% coefficient of variation of
the stiffness. The exact analytical solution and the FEM results match perfectly at each node. The present results are
obtained for both statically determinate and statically indeterminate beams. They deal with only a stochastic
stiffness. The general case of both random stiffness and the random loads are under study and will be reported else-
where.
774 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
Acknowledgements
This study was conducted when Mr. O. Rollot was at the Florida Atlantic University�s Department of Mechanical
Engineering during August 1998–December 1998 as a Visiting Research Fellow on a special training program from
IFMA, France. The work of I. Elishakoff was supported by the National Science Foundation SGER grant (Program
Director: Dr. K.P. Chong). The opinions and recommendations in this paper are of the authors only and do not reflect
the views of the National Science Foundation.
Appendix A. Elements of the nodal displacement correlation matrix for a clamped-free beam
Eðw22Þ ¼
1849
576q2a8E
1
D21
# $
Eðw2h2Þ ¼817
144q2a8E
1
D21
# $
Eðw2w3Þ ¼ q2a85117
576E
1
D21
# $�þ 731
576E
1
D1D2
# $�
Eðw2h3Þ ¼ q2a8817
144E
1
D21
# $�þ 301
144E
1
D1D2
# $�
Eðw2w4Þ ¼ q2a8473
32E
1
D21
# $�þ 215
64E
1
D1D2
# $�
Eðw2h4Þ ¼ q2a8215
36E
1
D21
# $�þ 301
144E
1
D1D2
# $�
Eðh22Þ ¼
361
36q2a8E
1
D21
# $
Eðh2w3Þ ¼ q2a82261
144E
1
D21
# $�þ 323
144E
1
D1D2
# $�
Eðh2h3Þ ¼ q2a8 361
36E
1
D21
# $�þ 133
36E
1
D1D2
# $�
Eðh2w4Þ ¼ q2a8209
8E
1
D21
# $�þ 95
16E
1
D1D2
# $�
Eðh2h4Þ ¼ q2a8 95
9E
1
D21
# $�þ 133
36E
1
D1D2
# $�
Eðw23Þ ¼ q2a8
2261
144E
1
D21
# $�þ 323
144E
1
D1D2
# $�
Eðw3h3Þ ¼ q2a8595
36E
1
D21
# $�þ 289
36E
1
D1D2
# $�
Eðw3w4Þ ¼ q2a82703
64E
1
D21
# $�þ 969
64E
1
D1D2
# $�
Eðw3h4Þ ¼ q2a8833
48E
1
D21
# $�þ 391
48E
1
D1D2
# $�
Eðh23Þ ¼ q2a8
205
18E
1
D21
# $�þ 133
18E
1
D1D2
# $�
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 775
Eðh3w4Þ ¼ q2a8453
16E
1
D21
# $�þ 249
16E
1
D1D2
# $�
Eðh3h4Þ ¼ q2a8143
12E
1
D21
# $�þ 91
12E
1
D1D2
# $�
Eðw24Þ ¼ q2a8
4581
64E
1
D21
# $�þ 495
16E
1
D1D2
# $�
Eðw4h4Þ ¼ q2a8475
16E
1
D21
# $�þ 127
8E
1
D1D2
# $�
Eðh24Þ ¼ q2a8
449
36E
1
D21
# $�þ 70
9E
1
D1D2
# $�
Appendix B. Elements of the nodal displacement correlation matrix for a simply-supported beam
Eðh21Þ ¼ q2a8
365
576E
1
D21
# $�þ 91
144E
1
D1D2
# $�
Eðh1w2Þ ¼ q2a8295
576E
1
D21
# $�þ 299
576E
1
D1D2
# $�
Eðh1h2Þ ¼ q2a8169
576E
1
D21
# $�þ 91
288E
1
D1D2
# $�
Eðh1w3Þ ¼ q2a8295
576E
1
D21
# $�þ 299
576E
1
D1D2
# $�
Eðh1h3Þ ¼ q2a8�� 169
576E
1
D21
# $� 91
288E
1
D1D2
# $�
Eðh1h4Þ ¼ q2a8�� 365
576E
1
D21
# $� 91
144E
1
D1D2
# $�
Eðw22Þ ¼ q2a8
125
288E
1
D21
# $�þ 13
32E
1
D1D2
# $�
Eðw2h2Þ ¼ q2a8169
576E
1
D21
# $�þ 13
64E
1
D1D2
# $�
Eðw2w3Þ ¼ q2a8125
288E
1
D21
# $�þ 13
32E
1
D1D2
# $�
Eðw2h3Þ ¼ q2a8�� 169
576E
1
D21
# $� 13
64E
1
D1D2
# $�
Eðw2h4Þ ¼ q2a8�� 295
576E
1
D21
# $� 299
576E
1
D1D2
# $�
Eðh22Þ ¼
169
576q2a8E
1
D21
# $
Eðh2w3Þ ¼ q2a8169
576E
1
D21
# $�þ 13
46E
1
D1D2
# $�
Eðh2h3Þ ¼ � 169
576q2a8E
1
D21
# $
776 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
Eðh2h4Þ ¼ q2a8
�� 169
576E
1
D21
# $� 91
288E
1
D1D2
# $�
Eðw23Þ ¼ q2a8
125
288E
1
D21
# $�þ 13
32E
1
D1D2
# $�
Eðw3h3Þ ¼ q2a8�� 169
576E
1
D21
# $� 13
64E
1
D1D2
# $�
Eðw3h4Þ ¼ q2a8�� 295
576E
1
D21
# $� 299
576E
1
D1D2
# $�
Eðh23Þ ¼
169
576q2a8E
1
D21
# $
Eðh3h4Þ ¼ q2a8 169
576E
1
D21
# $�þ 91
288E
1
D1D2
# $�
Eðh24Þ ¼ q2a8
365
576E
1
D21
# $�þ 91
144E
1
D1D2
# $�
Appendix C. Elements of the nodal displacement correlation matrix for a clamped/simply-supported beam
Eðw22Þ ¼ q2a8
289
14400E
1
D21
# $(þ 6561
400E
1
ð7D1 þ 20D2Þ2
" #þ 459
400E
1
D1ð7D1 þ 20D2Þ
� �)
Eðw2h2Þ ¼ q2a8119
115200E
1
D21
# $(þ 19683
640E
1
ð7D1 þ 20D2Þ2
" #þ 351
256E
1
D1ð7D1 þ 20D2Þ
� �)
Eðw2w3Þ ¼ q2a81751
57600E
1
D21
# $(þ 221
40230E
1
D1D2
# $þ 400221
22400E
1
ð7D1 þ 20D2Þ2
" #
þ 66933
44800E
1
D1ð7D1 þ 20D2Þ
� �þ 351
2240E
1
D2ð7D1 þ 20D2Þ
� �)
Eðw2h3Þ ¼ q2a8119
11520E
1
D21
# $(� 221
40230E
1
D1D2
# $� 19683
896E
1
ð7D1 þ 20D2Þ2
" #
� 4239
8960E
1
D1ð7D1 þ 20D2Þ
� �� 351
2240E
1
D2ð7D1 þ 20D2Þ
� �)
Eðw2h4Þ ¼ q2a8(
� 1547
28800E
1
D21
# $� 221
40230E
1
D1D2
# $� 177147
11200E
1
ð7D1 þ 20D2Þ2
" #
� 46791
22400E
1
D1ð7D1 þ 20D2Þ
� �� 351
2240E
1
D2ð7D1 þ 20D2Þ
� �)
Eðh22Þ ¼ q2a8
49
9216E
1
D21
# $(þ 59049
1024E
1
ð7D1 þ 20D2Þ2
" #þ 567
512E
1
D1ð7D1 þ 20D2Þ
� �)
Eðh2w3Þ ¼ q2a8721
46080E
1
D21
# $(þ 13
4608E
1
D1D2
# $þ 1200663
35840E
1
ð7D1 þ 20D2Þ2
" #
þ 999
512E
1
D1ð7D1 þ 20D2Þ
� �þ 1053
3584E
1
D2ð7D1 þ 20D2Þ
� �)
O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779 777
Eðh2h3Þ ¼ q2a849
9216E
1
D21
# $(� 13
4608E
1
D1D2
# $þ 295245
7168E
1
ð7D1 þ 20D2Þ2
" #
þ 81
512E
1
D1ð7D1 þ 20D2Þ
� �� 1053
3584E
1
D2ð7D1 þ 20D2Þ
� �)
Eðh2h4Þ ¼ q2a8(
� 637
23040E
1
D21
# $� 13
4608E
1
D1D2
# $� 531441
17920E
1
ð7D1 þ 20D2Þ2
" #
� 405
128E
1
D1ð7D1 þ 20D2Þ
� �� 1053
3584E
1
D2ð7D1 þ 20D2Þ
� �)
Eðw23Þ ¼ q2a8
536741
11289600E
1
D21
# $(þ 1339
80640E
1
D1D2
# $þ 2441348
1254400E
1
ð7D1 þ 20D2Þ2
" #
þ 169641
89600E
1
D1ð7D1 þ 20D2Þ
� �þ 21411
62720E
1
D2ð7D1 þ 20D2Þ
� �)
Eðw3h3Þ ¼ q2a831949
2257920E
1
D21
# $(� 22113
40230E
1
D1D2
# $� 1200663
50176E
1
ð7D1 þ 20D2Þ2
" #
� 15093
17920E
1
D1ð7D1 þ 20D2Þ
� �� 5967
15680E
1
D2ð7D1 þ 20D2Þ
� �)
Eðw3h4Þ ¼ q2a8(
� 155909
1881600E
1
D21
# $� 247
10752E
1
D1D2
# $� 10805967
627200E
1
ð7D1 þ 20D2Þ2
" #
� 56241
22400E
1
D1ð7D1 þ 20D2Þ
� �� 8073
25088E
1
D2ð7D1 þ 20D2Þ
� �)
Eðh23Þ ¼ q2a8
3077
451584E
1
D21
# $(� 13
2304E
1
D1D2
# $þ 1476225
50176E
1
ð7D1 þ 20D2Þ2
" #� 405
512E
1
D1ð7D1 þ 20D2Þ
� �
þ 5265
12544E
1
D2ð7D1 þ 20D2Þ
� �)
Eðh3h4Þ ¼ q2a8(
� 9841
376310E
1
D21
# $þ 921
7680E
1
D1D2
# $þ 531441
25088E
1
ð7D1 þ 20D2Þ2
" #
þ 567
320E
1
D1ð7D1 þ 20D2Þ
� �þ 45279
125440E
1
D2ð7D1 þ 20D2Þ
� �)
Eðh24Þ ¼ q2a8
204997
1411200E
1
D21
# $(þ 169
5760E
1
D1D2
# $þ 4782969
313600E
1
ð7D1 þ 20D2Þ2
" #
þ 9477
3200E
1
D1ð7D1 þ 20D2Þ
� �þ 9477
31360E
1
D2ð7D1 þ 20D2Þ
� �)
Appendix D. Correlation matrix of elements of the nodal displacement correlation matrix for a clamped/clamped beam
Eðw22Þ ¼ q2a8
1
144E
1
D21
# $(þ 1
16E
1
ðD1 þ 2D2Þ2
" #þ 1
24E
1
D1ðD1 þ 2D2Þ
� �)
778 O. Rollot, I. Elishakoff / Chaos, Solitons and Fractals 17 (2003) 749–779
Eðw2h2Þ ¼ q2a81
8E
1
ðD1 þ 2D2Þ2
" #(þ 1
24E
1
D1ðD1 þ 2D2Þ
� �)
Eðw2w3Þ ¼ q2a81
144E
1
D21
# $(þ 1
16E
1
ðD1 þ 2D2Þ2
" #þ 1
24E
1
D1ðD1 þ 2D2Þ
� �)
Eðw2h3Þ ¼ q2a8(
� 1
8E
1
ðD1 þ 2D2Þ2
" #� 1
24E
1
D1ðD1 þ 2D2Þ
� �)
Eðh22Þ ¼
1
4q2a8E
1
ðD1 þ 2D2Þ2
( )
Eðh2w3Þ ¼ q2a81
8E
1
ðD1 þ 2D2Þ2
" #(þ 1
24E
1
ðD1 þ 2D2Þ2
" #)
Eðh2h3Þ ¼ � 1
4q2a8E
1
ðD1 þ 2D2Þ2
" #
Eðw23Þ ¼ q2a8
1
144E
1
D21
# $(þ 1
16E
1
ðD1 þ 2D2Þ2
" #þ 1
24E
1
D1ðD1 þ 2D2Þ
� �)
Eðw3h3Þ ¼ q2a8(
� 1
8E
1
ðD1 þ 2D2Þ2
" #� 1
24E
1
D1ðD1 þ 2D2Þ
� �)
Eðh23Þ ¼
1
4q2a8E
1
ðD1 þ 2D2Þ2
" #
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