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

mail to: [email protected]

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


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


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

�1GcÞ�1GTc S

�1iGuF1; ð60Þ

if F1 is a deterministic vector. On the other hand,

E½uþ1 � ¼ GTuE S�1h

� S�1GcðGTc S

�1GcÞ�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

�1GcÞ�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


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.


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


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


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



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.


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


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


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


whereas the mean-square values are

E½w21ðzÞ� ¼ q2a8

2809z4

576

#� 265z5

144þ 25z6

144

$E

1

D21

# $;


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

# $�;


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.


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

(%)



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


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


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


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


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


2153

288

#�� 4225z

288þ 5915z2

576� 845z3

288þ 169z4

576

$E

1

D21

# $þ#� 65

32þ 65z

32� 13z2

32

$E

1

D1D2

# $�ð116bÞ


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.


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


axial coordinate z

rel

ativ

e er

ror

(%)


Fig. 11. Mean-square values of the displacement in the simply-supported beam.


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


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


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


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



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


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


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


#�� 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


(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


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



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


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

(%)



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


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


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


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


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


#�� 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.


Fig. 19. Mean-square displacement in the clamped–clamped beam.

axial coordinate z

rel

ativ

e er

ror

(%)


axial coordinate z

rel

ativ

e er

ror

(%)



The mean-square values read


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


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


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.


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

# $�


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

# $�


36E

1

D21

# $�þ 289

36E

1

D1D2

# $�

Eðw3w4Þ ¼ q2a82703

64E

1

D21

# $�þ 969

64E

1

D1D2

# $�


48E

1

D21

# $�þ 391

48E

1

D1D2

# $�

Eðh23Þ ¼ q2a8

205

18E

1

D21

# $�þ 133

18E

1

D1D2

# $�



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

# $�


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

# $�


576E

1

D21

# $�þ 299

576E

1

D1D2

# $�

Eðh1h2Þ ¼ q2a8169

576E

1

D21

# $�þ 91

288E

1

D1D2

# $�


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

# $�


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

# $


576E

1

D21

# $�þ 13

46E

1

D1D2

# $�

Eðh2h3Þ ¼ � 169

576q2a8E

1

D21

# $


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

� �)


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

� �)


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

� �)


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

� �)


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

� �)


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

" #

References

[1] Elishakoff I, Ren YJ, Shinozuka M. Some critical observations and attendant new results on the FEM stochastic problems. Chaos,

Solitons & Fractals 1996;7:597–609.

[2] Ghanem RG, Spanos PD. Spectral techniques for stochastic finite-elements. Arch Comput Meth Eng 1997;4:63–100.

[3] Schu€eeller GI, Brenner CE. Stochastic finite elements––A challenge for material science. In: Gerdes A, editor. Advances in Building

Material Science. Freiburg: Aedificatio Publishers; 1996. p. 259–73.

[4] Matthies HG, Brenner ChE, Bucher ChG, Guedes Soares C. Uncertainties in probabilistic numerical analysis of structures and

solids––stochastic finite elements. Struct Safe 1997;19:283–336.

[5] Shinozuka M, Yamazaki F. Stochastic finite element method: an introduction. In: Ariaratnam ST, Schu€eeller GI, Elishakoff I,

editors. Stochastic Structural Dynamics––Progress in Theory and Application. London: Elsevier Applied Science; 1998. p. 291.

[6] Elishakoff I, Ren YJ, Shinozuka M. Some exact solutions for bending of beams with spatially stochastic stiffness. Int J Solids

Struct 1995;32:2315–27 (corrigendum: 33;1996:3491).

[7] Elishakoff I, Ren YJ, Shinozuka M. Finite element analysis of stochastic bars based on exact inverse of stiffness matrix. In: Haldar

A, Guran A, Ayyub B, editors. Uncertainity Modeling in Finite Element, Fatigue and Stability of System. Singapore: World

Scientific; 1997. p. 51–70.

[8] Ren YJ, Elishakoff I. New results in finite element method for stochastic structures. Comput Struct 1998;67:125–35.

[9] Elishakoff I, Ren YJ, Shinozuka M. New formulation of FEM for deterministic and stochastic beams though generalization of

Fuchs� approach. Comput Meth Appl Mech Eng 1997;144:235–43.

[10] Yang TY. Finite element structural analysis. Englewood Cliffs: Prentice-Hall, Inc.; 1986.

[11] Elishakoff I, Lin YK, Zhu LP. Probabilistic and convex modelling of acoustically excited structures. Amsterdam: Elsevier; 1994.

[12] Johnson ME. Multivariate statistical simulation. New York: Wiley & Sons; 1987.

[13] Elishakoff I. Probabilistic methods in the theory of structures. New York: John Wiley & Sons; 1983 (2nd ed. New York: Dover

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Large variation finite element method for beams with stochastic stiffness

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