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description
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.
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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.