Analysis of uncertain structural systems under white noise excitation by a response surface method

N. Impollonia & G. Ricciardi Dipartimento di Costruzioni e Tecnologie Avanzate, University of Messina, Italy

Keywords: Stochastic dynamics, uncertain structures, white noise excitation, response surface method

ABSTRACT: A method for the analysis of linear structural systems with uncertain parameters subjected to

stochastic excitation is presented. The excitation is modeled as a Gaussian white noise process and the sta-

tionary stochastic response is characterized in terms of its second order moments. The variability of the re-

sponse due to the uncertainty in the dynamic properties of the structure is investigated. A novel response sur-

face method is developed, expressing an accurate explicit relationship between the derived and basic random

variables. The method is based on the solution of complex eigenvalue problems, one for each basic random

variable, with few eigenvalues different from zero. A complete probabilistic characterization of the response

process can be made. The efficiency and the accuracy of the proposed procedure are evidenced by a numeri-

cal application.

1 INTRODUCTION

The inclusion of the parameter uncertainty in the

analysis of structures is essential for a more complete

understanding of the structural behavior. Such uncer-

tainty can arise because of the numerous assumptions

made when modeling the geometry, material, proper-

ties, constitutive laws, and boundary conditions of the

structural members. Although many non-statistical

methods have been proposed to evaluate the response

of structures with uncertain parameters (or stochastic

structures) to deterministic excitation, the case of sto-

chastic excitation has not been extensively investi-

gated. An efficient method to evaluate the response of

stochastic structure to stochastic excitation is of par-

ticular interest as in this setting the statistical tech-

niques based on Monte Carlo simulation are very

time consuming due to the large number of samples

needed to get satisfactory results.

The orthogonal polynomial expansion proposed

by Ghanem & Spanos (1991) has been successfully

extended to the case of stochastic excitation by Jen-

sen & Iwan (1992). Such method produces accurate

solutions for a second order analysis although it may

become prohibitive for large systems. Furthermore,

the probability density function of uncertain parame-

ters can not be arbitrarily chosen, as only for few of

them suitable polynomials have been presented.

The response surface method can be resorted to

solve the problem (Khuri & Cornell 1987). This is a

very general and conceptually simple technique,

broadly used in several scientific areas for a long

time. For our purpose a response surface is an ex-

plicit function approximating the implicit unknown

limit surface, i.e. the function expressing the de-

pendence of a response quantity on the uncertain pa-

rameters.

The form of the response surface is usually arbi-

trarily chosen to be of polynomial type (linear or

quadratic) and its coefficients are obtained by fitting

the limit surface at a number of points, named sam-

pling points, which constitute realizations of the un-

certain structural parameters (Wong 1985, Bucher &

Bourgund 1990). Then, a deterministic analysis is

required for each sampling point, in order to get the

corresponding value of the limit surface.

The key step of the response surface method is

the choice of the response surface form. Neverthe-

less, no much attention has been devoted in the lit-

erature to this issue, as the use of linear or quadratic

(with or without cross terms) polynomials seems un-

questionable. However, such choice may be inap-

propriate as the limit surface shape is usually far

away from a hyperplane or a quadratic surface,

which consequently furnish only a poor fit of the ac-

tual surface.

69

Applications of Statistics and Probability in Civil Engineering, Der Kiureghian, Madanat & Pestana (eds) © 2003 Millpress, Rotterdam, ISBN 90 5966 004 8

The drawbacks related to the use of inadequate

response surfaces have been already evidenced

(Nelder 1966, Guan & Melchers 2001). The main

trouble is the high sensitivity of response surface to

the sampling point positions so that, remarkably dif-

ferent approximations are created varying a little the

sampling points. Which is the correct choice of the

sampling points to obtain a better fit is still an open

question. Moreover, the use of low order polynomi-

als as response surface is accompanied by some

other disadvantages and neither it is convenient

from a computational point of view. For example, in

some cases, the fitting with inverse polynomials has

been shown to involve no more labor than ordinary

polynomials and to be preferable for several reasons

(Nelder 1966).

A novel response surface technique was recently

presented for the static analysis of stochastic struc-

tures (Falsone & Impollonia 2002). The performance

function adopted is built up by a sum of ratio of

polynomials and turns out to be very effective to fit,

in the spirit of response surface methods, the implicit

response surface. Differently from the classical re-

sponse surface method neither sampling points nor

deterministic analyses are needed. In fact, the coeffi-

cients of the response surface can be retrieved by

solving low dimension real eigenvalue problems.

In the present paper a response surface method

based on rational polynomials is developed to get

the stationary response of stochastic structures under

white noise processes. For the present case, the ei-

gen-problems to be solved, one for each basic ran-

dom variable (i.e. each parameter assumed uncertain

with a given probabilistic description), are complex.

Moreover, the eigen-problems possess only few ei-

genvalues different from zero, so that a very simple

explicit relationship between the basic random vari-

ables and the response is derived, regardless of the

number of degree of freedoms. Such a relationship is

the exact one if a single uncertain parameter is pre-

sent; it is very accurate if several uncertain parame-

ters are present. In this case, however, in order to get

a better approximation, the inclusion of cross terms

is also considered.

The results are given in terms of the second order

statistical moments of the response, although higher

order moments can be straightforwardly derived.

In the applications, the results obtained by the

proposed method are compared with those deriving

from Monte Carlo simulations.

2 FORMULATION

Consider an n-DOF structural system whose dy-

namic properties are predicted by an analytic model

with linear behavior, characterized by the mass,

damping and stiffness ( )n n× matrices M , C and

K , respectively. Moreover, let us assume that the

structure is subjected to a stationary Gaussian white

noise excitation ( )W t with zero mean and intensity

q = 2πS0, S0 being the constant power spectral den-

sity.

The motion differential equations of the system

can be written as follows

( ) ( ) ( ) ( )t t t W t+ + =Mu Cu Ku p (1)

where ( )tu , ( )tu and ( )tu are the n-vectors of nodal

displacements, velocities and accelerations, respec-

tively, and p is the influence forcing n-vector.

By using the 2n-dimensional vector state ap-

proach, Eq.(1) can be written in the following matrix

form:

( ) ( ) ( )t t W t+ =AZ BZ g (2)

where

( )( ) ,

( )

tt

t

= =

u pZ g

u 0 (3)

and

,

= = −

C M K 0A B

M 0 0 M (4)

are symmetric matrices. Eq.(2) can be written in al-

ternative form as follows

( ) ( ) ( )t t W t+ =Z DZ v (5)

where

1

1

1 11 1 1

1

1

11 1 1

,n

−

−

− −− − −

−

−

−− − −

=

− = = −− =

= = −

D A B

K 0 0 I0 M

0 M M K M CM M CM

v A g

p 00 M

0 M pM M CM

(6)

being nI the identity matrix of order n .

If the physical properties of the system, such as

mass, damping and stiffness, are known determinis-

tically, the response process is Gaussian and its

probabilistic characterization is known if the station-

ary second order moments are evaluated. The differ-

ential equations describing the time evolution of

these quantities can be written as follows (Falsone et

al. 1991)

2 2 2 2( ) ( )t t q+ =m D m v (7)

70 Impollonia, N. & Ricciardi, G.

where

2 2 2

1 1

2 2

2

1 1 1 [2] [2]

( ) ( )

( ) ( ) ( )

n n

n n

− −

− − −

= ⊗ + ⊗

= ⊗ + ⊗

= ⊗

= ⊗ =

D D I I D

A B I I A B

v v v

A g A g A g

(8)

In Eqs.(8), the symbol ⊗ means kronecker product

and the exponent in square brackets means kro-

necker power (Bellman 1974, Brewer 1978). In

Eq.(7), the 4n2-vector 2 ( )tm , given as

[2]

2 2( ) [ ( )] [ ( )]t t E t= =m m Z Z (9)

collects all the second order moments between the

displacement and velocities of the nodal coordinates.

Note that all the cross moments appear twice in the

vector 2 ( )tm . Then a condensation can be employed

in order to reduce the size of the problem by clearing

away the repetitions. So operating the problem di-

mension reduces from 24n to 2(2 )N n n .

The stationary response is solution of the follow-

ing system of algebraic equations:

2 2 2q=D m v (10)

From Eq.(10), it appears that the evaluation of the

stationary second order moments of the nodal coor-

dinates requires the inversion of the matrix 2D .

3 PARAMETER UNCERTAINTY

In agreement with realistic models, the mass pa-

rameter fluctuations are negligible, whereas stiffness

and damping parameters commonly suffer from sig-

nificant uncertainty. Then, let us assume that the ma-

trices K and C can be written in the following

form

(0) ( )

1

(0) ( )

1

( )

( )

Ri

ii

Ri

ii

=

=

= + α

+ α

∑

∑

K K K

C C C

αααα

α =α =α =α = (11)

where (0)K , ( )iK , (0)C and ( )i

C are deterministic

matrices and =αααα [α1 α2 … αR]T is the vector col-

lecting the basic random variables. It is worth not-

ing that Eq.(11) does not involve any approximation,

since, whatever the uncertain physical properties are,

the stochastic stiffness matrix can always be ex-

pressed as a linear function of appropriate random

variables iα (eventually by applying a variable

transformation). The basic random variables repre-

sent the fluctuations of the random parameters with

respect to their mean value. (0)K and (0)

C represent

the expected values of the stiffness and damping ma-

trices, respectively, and the summations represent

their fluctuating components. It follows that

(0) ( )

1

(0) ( )

1

( )

( )

Ri

ii

Ri

ii

=

=

= + α

= + α

∑

∑

A A A

B B B

αααα

αααα (12)

where

(0) ( )

(0) ( )

(0) ( )

(0) ( )

,

,

i

i

i

i

= =

= = −

C M C 0A A

M 0 0 0

K 0 K 0B B

0 M 0 0

(13)

and the dynamic matrix D can be written in the

form

(0) ( )

1

( )R

i

ii=

= + α∑D D Dαααα (14)

where

(0)

1 (0) 1 (0)

( )

1 ( ) 1 ( )

,n

i

i i

− −

− −

− =

=

0 ID

M K M C

0 0D

M K M C

(15)

The stationary second order moment of the nodal re-

sponse is solution of the following system of alge-

braic equations

2 2 2( ) ( ) q=D m vα αα αα αα α (16)

where

(0) ( )

2 2 21

( )R

i

ii=

= + α∑D D Dαααα (17)

being

(0) (0) (0)

2 2 2

( ) ( ) ( )

2 2 2

n n

i i i

n n

= ⊗ + ⊗

= ⊗ + ⊗

D D I I D

D D I I D (18)

In Eq.(16), the vector of the second order moments

of the nodal response 2 ( )m αααα depends on the uncer-

tain parameters, then it is a vector of random vari-

ables (derived random variables). Solving the system

governed by Eq.(16) from a stochastic point of view

means to find probabilistic information on the de-

rived random variables 2 ( )m αααα , once that the prob-

abilistic characteristics of the basic random variables

αi are given. In the literature many approaches solv-

ing this problem in an approximate way have been

proposed. In the next section a novel approach will

be presented.

71Analysis of uncertain structural systems under white noise excitation

4 PROPOSED METHOD

Let us first consider the case in which only one de-

sign variable αi is present. In order to operate with

symmetric matrices, by pre-multiplying Eq.(10) by [2] = ⊗A A A and by taking into account Eq.(8), we

obtain

2 2 2q=C m g (19)

where

[2]

2 2

2

= = ⊗ + ⊗= ⊗

C A D A B B A

g g g (20)

By taking into account the presence of the uncertain

parameter αi, the stochastic version of Eq.(19) is

2 2 2( ) ( ) q=C m gα αα αα αα α (21)

The vector of the second order moments of the nodal

response can be written as follows

(0) ( )

2 2 2( ) ( )i

i= + αm m mαααα (22)

where (0)

2m is the vector of the stationary second or-

der moments of the nominal structure, solution of

the following matrix equation

(0) (0)

2 2 2q=C m g (23)

Moreover, the matrix 2 ( )C αααα takes on the following

simpler form:

(0) ( )

2 2 2( ) i

i= + αC C Cαααα (24)

By substituting Eq.(22) and Eq.(24) into Eq.(21), we

obtain

2 2

(0) ( ) (0) ( )

2 2 2 2 2

( ) ( )

( )[ ( )]i i

i i q

≡

+ α + α =

C m

C C m m g

α αα αα αα α (25)

Taking into account of Eq.(23), the following equa-

tion for the unknown vector ( )

2 ( )i

iαm is obtained:

(0) ( ) ( ) ( ) (0)

2 2 2 2 2( ) ( )i i i

i i i+ α α = −αC C m C m (26)

The exact solution of Eq.(26) appears not

straightforward at a first glance, as the inverse of the

parametric matrix (0) ( ) 1

2 2( )ii

−+ αC C is required. How-

ever, a simple expression of the vector ( )

2 ( )i

iαm as a

function of the design parameter αi can be obtained,

as shown in the next.

Let us consider the following eigenproblem rela-

tive to the ( )N N× symmetric matrices (0)

2C and ( )

2

iC :

( ) ( ) ( ) (0) ( )

2 2

i i i i= λC CΨ ΨΨ ΨΨ ΨΨ Ψ (27)

which is characterized by a small number p of

complex eigenvalues different from zero, with

p N , being 2(2 )N n n .

By imposing the following ortonormalization condi-

tion

( ) (0) ( )

2

i T i

p=C IΨ ΨΨ ΨΨ ΨΨ Ψ (28)

( ) ( ) ( ) ( )

1 2[ ]i i i i

p= being the N p× matrix

listing the p eigenvectors ( )i

j (with 1, 2, ,j p= ),

we have

( ) ( ) ( ) ( )

2

i T i i i=CΨ ΨΨ ΨΨ ΨΨ Ψ (29)

( ) ( ) ( ) ( )

1 2diag[ ]i i i i

p= λ λ λ being the p p× diagonal

matrix listing the p eigenvalues ( )i

jλ (with

1, 2, ,j p= ). By introducing the following coor-

dinate transformation

( ) ( ) ( ) (0) ( ) (0)

2 2 2 2( ) ( ), i i i i

i iα = α =m n m n (30a,b)

in Eq.(26), by pre-multiplying by ( )i TΨΨΨΨ and taking

into account of Eqs.(28) and (29), Eq.(26) becomes

( ) ( ) ( ) (0)

2 2( ) ( )i i i

p i i i+ α α = −αI n n (31)

From the previous equation, it is easy to obtain

( ) ( ) (0)

2 2( ) ( )i i

i iα = αn nΞΞΞΞ (32)

where

(0) ( ) (0) (0) ( )

2 2 2 2

i T i T q= =n C m gΨ ΨΨ ΨΨ ΨΨ Ψ (33)

and

( ) ( ) 1 ( )( ) ( )i i i

i i p i

−α = −α + αIΞΞΞΞ (34)

is a diagonal matrix, whose generic j-th element is

given as

( )

( )

( )( ) , 1, 2, ,

1

i

i ji

j i i

i j

j pα λ

Ξ α = − =+ α λ

(35)

By replacing Eq.(32) and Eq.(33) into Eq.(30a), the

exact relationship between the vector ( )

2 ( )i

iαm and

the uncertain parameter iα is determined as follows

( ) ( ) ( ) ( )

2 2( ) ( )i i i i T

i i qα = αm gΨ Ξ ΨΨ Ξ ΨΨ Ξ ΨΨ Ξ Ψ (36)

If more than one uncertain parameter is present, as

a first approximation we assume that the cross ef-

fects of the uncertainties are negligible and the su-

perposition principle is applied. Therefore the solu-

tion is the sum of the single contributions, that is

(0) ( )

2 2 21

( ) ( )R

i

ii=

= + α∑m m mαααα (37)

72 Impollonia, N. & Ricciardi, G.

where R is the number of the uncertain parameters

and the generic ( )

2 ( )i

iαm is given by Eq.(36). It is

obvious that Eq.(37) is an approximation if 2R ≥ .

However, it is possible to find the expression of the

residue associated with the use of Eq.(37) by replac-

ing this one and Eq.(17) into Eq.(16):

( )(0) (0)

2 2 2

(0) ( ) ( ) ( ) (0)

2 2 2 2 21

( ) ( )

2 21 1

Residue

( ) ( )

( )

Ri i i

i i ii

R Ri j

i ji j

i j

q

=

= =≠

≡ −

+ + α α + α

= − α α

∑

∑ ∑

D m g

D D m D m

D m

(38)

As it will be seen in the applications, this residue

implies an acceptable error in the stochastic charac-

terization of the structural response, even for high

uncertainty level in the parameters. Eq.(38) leads to

the consideration that it is possible to further im-

prove the explicit response surface given by Eq.(37)

by adding to this expression proper crossing terms

so to read

(0) ( )

2 2 21

(0) 1 ( ) ( )

2 2 21 1

( ) ( )

( ) ( )

Ri

ii

R Ri j

i ji j

i j

=

−

= =≠

= + α

− α α

∑

∑ ∑

m m m

D D m

αααα

(39)

In this case the residue is reduced, being quadratic

with respect to parameter fluctuations (assumed

smaller than one), namely

( ) (0) 1 ( )

2 2 21 1

Residue ( ) ( )R R R

k j

i k ji j k

i j

−

= =≠

= α α α∑ ∑ ∑ D D m (40)

In the case of significant uncertainties, this correc-

tion procedure can be easily extended in order to re-

duce the residue and, then, to improve the response

surface approximation.

The stochastic characterization of the response

can be made evaluating its statistical moments, eas-

ily available starting from Eqs. (36) and (37). For

example, for the first and second order moment of

the r-th component of the second order vector mo-

ment of the response we can write:

1 (0) ( )

2, 2, 2, 2,1

2 2

2, 2,

(0) 2 (0) ( )

2, 2, 2,1

( ) ( )

2, 2,1 1

[ ( )] [ ( )]

[( ( )) ]

( ) 2 [ ( )]

[ ( ) ( )]

Ri

r r r r ii

r r

Ri

r r r ii

R Ri j

r i r ji j

E m m E m

E m

m m E m

E m m

=

=

= =

µ = = + α

µ =

= + α

+ α α

∑

∑

∑ ∑

αααα

αααα (41)

where ,

t

s rµ means the moment of order t of the r-th

component of the vector collecting the moments of

s-th order of the response.

Alternatively, one can apply the Monte Carlo

simulation on Eq.(37) or Eq.(39) to get the statistical

moments and the probability density functions of the

second order moments of the response. Clearly such

a simulation is enormously more efficient than a

simulation applied directly to Eq.(16), being very

small the time required to get a response sample.

Note that Eq.(37) and Eq.(39) have been derived

without any assumptions on the probabilistic struc-

ture of the basic random variables.

5 APPLICATION

As an application of the proposed method, the char-

acterization of the second order moment of the re-

sponse of a shear-type frame structure under white

noise base excitation ( )W t is presented. In particu-

lar, a three-degree-of-freedom system ( 3n ) with

uniform mass distribution along the height of the

structure is considered, with 1jm m= = ( 1,2,3j = );

on the contrary, a nonuniform nominal stiffness dis-

tribution is assumed, as shown in fig. 1, with (0)

1 2 2k k= = and (0) (0)

2 3 1k k k= = = (see Fig.1). The

structure is assumed classically damped, with (0)

jζ( 1,2,3j = ) equals to 5% of nominal critical damp-

ing in all the three modes. The intensity of the white

noise is 1q = .

Figure 1. Model of the structural system

u1(t)

u2(t)

u3(t)

W(t)

k1

k2

k3

m

m

m

73Analysis of uncertain structural systems under white noise excitation

The influence of the uncertainties in the system

parameters, in conjunction with that inherent to the

applied dynamic load on the response variability, is

investigated. In particular, the nodal displacements

( )ju t and velocities ( )ju t (for 1,2,3j = ) are con-

sidered as response quantities and the second order

moment of the nodal response of the type

[ ( ) ( )]r s

j kE u t u t (with , 1, 2,3j k = and 2r s+ = ) are

investigated. Two types of structural uncertainties

are considered: uncertain structural stiffness and un-

certain modal damping ratios. The uncertain parame-

ters are modeled as random variables in terms of

their mean value and deviatoric components in the

following form

(0)

(0)

3

(1 )

1, 2,3

(1 )

j j j

j j j

k k

j

+

= + α

=ζ = ζ + α

(42)

where 1 2 6, , , α α α (being 6R ) are the basic

random variables.

For the present case, the vector 2 ( )m , has 24 36n components before the condensation and

2(4 ) / 2 21n n after the condensation. All the vec-

tors and matrix are condensed so to reduce the di-

mension of the problem to 21. The eigen-problems

in Eq.(27) have just 5 eigenvalues different from

zero for 1,2 ,6i , so that just 5 eigenvelues and

eigenvectors are needed to obtain each of the vectors ( )

2 ( )i

iαm . It turns that for all the six cases 4 eigen-

values are complex and one is real. For example the

component of ( )

2 ( )i

iαm relative to the second order

moment of the top displacement as a function of the

uncertain parameter 3α reads

2

3

5.684[ ] 36.273

1

0.05925 0.02094

0.8325 0.6153

14.444 13.801

1.0386 1.9116

E u = ++−+

− +++

+ +

(43)

Eq.(43) expresses the exact relationship between a

derived and a basic random variable, which is inde-

pendent from the probabilistic information on the

random variable 3α .

The accuracy of the proposed method in the case

more the one basic variable are present can be as-

sessed from Figure 2.

Figure 2. Percentage error ( ) / 100exact approx exact of

the approximate relationship between the second order moment

of the top floor displacement and the basic variable 2α (a) and

3α (b) for given values of the basic variable 1α .

The figure shows that even neglecting the cross

terms by using Eq.(37) the error is very small de-

spite of large fluctuations of the basic random vari-

ables. Furthermore, the inclusion of cross terms by

Eq.(39) produces a better approximation with negli-

gible errors.

Once the probabilistic structure of the basic ran-

dom variables is assigned, its counterpart for the de-

rived random variables can be easily obtained.

Figures 3 and 4 portray, respectively, the mean

value and the mean square value of the conditional

second order moment of top floor displacement. In

the figures some of or all the variables

1 2 6, , , α α α are simultaneously considered zero

mean independent random variables, uniformly dis-

tributed in the intervals ( , )a a− .

(a)

(b)

74 Impollonia, N. & Ricciardi, G.

Figure 3. Mean value of the conditional second order moment of

the top floor displacement. Dotted line: 1 2 3, ,α α α are uncertain

and 4 5 6 0α = α = α = . Dashed line: 1 2 3 0α = α = α = and

4 5 6, ,α α α are uncertain. Solid line: 1 2 6, , ,α α α are uncertain.

Symbols: Monte Carlo simulation.

Figure 4. Mean square value of the conditional second order

moment of the top floor displacement. Dotted line: 1 2 3, ,α α αare uncertain and

4 5 6 0α = α = α = . Dashed line: 1 2 3 0α =α = α =

and 4 5 6, ,α α α are uncertain. Solid line: 1 2 6, , ,α α α are

uncertain. Symbols: Monte Carlo simulation.

The approximate statistical moments are derived,

neglecting the cross terms, by integrating Eq.(37)

and its square times the joint probability density

function. The accuracy is evidenced by comparison

with the results pertaining to Monte Carlo simula-

tions. Obviously, such performance is preserved

with any other probability density function of the

basic random variables.

The pdf of the conditional second order moment

of top floor displacement is reported in Figures 5-7.

Figure 5 refers to the case of uncertain stiffness

(the basic random variables 1 2 3, ,α α α are inde-

pendent and uniformly distributed in the

range ( 0.4, 0.4) ) and deterministic critical damping

ratios ( 4 5 6 0α = α = α = ).

Figure 6 refers to the case of uncertain critical

damping ratios ( 4 5 6, ,α α α are independent and uni-

formly distributed in the range ( 0.4, 0.4) ) and de-

terministic stiffness ( 1 2 3α = α = α =0).

Figure 7 refers to the case of uncertain stiffness

and critical damping ratios ( 1 2 6, , ,α α α are inde-

pendent and uniformly distributed in the

range ( 0.4, 0.4) ).

The probability density function is obtained by sta-

tistically combining the samples given by Eq.(37)

(which does not include cross terms) or by Eq.(39)

(which includes cross terms). Such a simulation is

enormously more efficient than a simulation applied

directly to Eq.(21), whose results are reported by

symbols. In fact, the time required to get a response

sample by Eq.(37) or Eq.(39) is much smaller than

that required by Eq.(21).

The solution given by the proposed formulation is

practically exact in the case only the damping is un-

certain (Fig.6), even if the cross terms are neglected.

If uncertain stiffness is present (Figs.5 and 7), then,

the solution is still very accurate especially if the

cross terms are taken into account.

6 CONCLUSIONS

An approach to the stochastic analysis of structures

with uncertain parameters excited by white noise

processes has been introduced. It can be considered

as a particular response surface method which pro-

duces an explicit relationship between the derived

and the basic random variables by solving complex

eigen-problems.

The numerical results demonstrate that the pro-

posed approach provides very accurate estimates of

both the statistical moments and probability density

function of the system response, even when high

levels of uncertainty are present. The response con-

sidered in the paper is the conditional second order

statistical moments, although also higher order sta-

tistics can be derived.

The inclusion of cross terms in the response sur-

face is formulated so to reduce the error of the ap-

proximate explicit relationship.

75Analysis of uncertain structural systems under white noise excitation

Figure 5. Pdf of the conditional second order moment of the top

floor displacement for independent 1 2 3, ,α α α uniformly distrib-

uted in the range ( 0.4,0.4) and 4 5 6 0α = α = α = . Dashed line:

proposed method without crossing terms. Solid line: proposed

method with crossing terms. Symbols: Monte Carlo simulation.

REFERENCES

Bellman, R., 1974. Introduction to Matrix Analysis. New York: McGraw-Hill.

Brewer, J.W., 1978. Kronecker Products and Matrix Calculus in System Theory. Transactions on Circuits and Systems(IEEE) Vol. cas-25(9): 772-781.

Bucher, C.G. & Bourgund, U. 1990. A Fast and Efficient Re-sponse Surface Approach for Structural Reliability Prob-lems. Structural Safety Vol. 7: 57-66. Falsone, G., Muscoli-no, G. & Ricciardi, G. 1991. Combined Dynamic Response of Primary and Multiply Connected Cascated Secondary Subsystems. Earthquake Engineering and Structural Sy-namics Vol. 20: 749-767.

Falsone, G. & Impollonia, N. 2002. A New Approach for the Stochastic Analysis of Finite Element Modelled Structures with Uncertain Parameters. Computer Methods in Applied Mechanics and Engineering Vol. 191(44): 5067-5085.

Ghanem, R. G. & Spanos, P. D., 1991. Stochastic Finite Ele-ments: A Spectral Approach, Springer Verlag, Berlin.

Guan, X.L. & Melchers, R.E. 2001. Effect of Response Surface Parameter Variation on Structural Reliability Estimates. Structural Safety Vol.23: 429-444.

Jensen, H. & Iwan, W. D. 1992. Response of Systems with Uncertain Parameters to Stochastic Excitation, Journal of Engineering Mechanics (ASCE), Vol. 118(5), 1012-1025.

Khuri, A. & Cornell, J.A. 1987. Response Surface: Designs and Analyses. New York: Dekker.

Nelder, J.A. 1966. Inverse Polynomials, a Useful Group of Multifactor Response Function. Biometrics Vol.22: 128-141.

Wong, F.S. 1985. Slope Reliability and Response Surface Method. Journal of Geotechnical Engineering ASCE Vol.111(1): 32-53.

Figure 6. Pdf of the conditional second order moment of the top

floor displacement for independent 4 5 6, ,α α α uniformly distrib-

uted in the range ( 0.4,0.4) and 1 2 3 0α = α = α = . Dashed line:

proposed method without crossing terms. Solid line: proposed

method with crossing terms. Symbols: Monte Carlo simulation.

Figure 7. Pdf of the conditional second order moment of the top

floor displacement for independent 1 2 6, , ,α α α uniformly dis-

tributed in the range ( 0.4,0.4) . Dashed line: proposed method

without crossing terms. Solid line: proposed method with cross-

ing terms. Symbols: Monte Carlo simulation.

76 Impollonia, N. & Ricciardi, G.