Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method
Direct Monte-Carlo simulation method (MCS)
 plays an important role in modern computational
 methods. For a large class of multidimensional
 problems of radiative transport theory,
 the MCS procedure is the only available
 method allowing the consideration of...
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| Date: | 2008 |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2008
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| Cite this: | Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method / M. Labou // Проблемы прочности. — 2008. — № 6. — С. 54-70. — Бібліогр.: 29 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859990717030465536 |
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| author | Labou, M. |
| author_facet | Labou, M. |
| citation_txt | Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method / M. Labou // Проблемы прочности. — 2008. — № 6. — С. 54-70. — Бібліогр.: 29 назв. — англ. |
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| container_title | Проблемы прочности |
| description | Direct Monte-Carlo simulation method (MCS)
plays an important role in modern computational
methods. For a large class of multidimensional
problems of radiative transport theory,
the MCS procedure is the only available
method allowing the consideration of all varieties
of geometric and physical presumptions.
This gives rise to computational investigation
of actual and practical problems such as neutron-
transport problem, reactor shielding, and
impurity diffusion in random fields. MCS
method enables the determination of the
first-passage probability that the system response
exceeds prescribed safe domain. In this
paper, the failure probabilities of a plate subjected
to seismic load are determined using the
MCS simulation method. For the purpose of reliability
analysis, the von Mises criterion is
used to determine the limit state failure. For the
seismic load, the statistical model of earthquake-
induced ground acceleration proposed
by Ruiz and Penzien is used.
Прямий метод моделювання Монте-Карло відіграє важливу роль у сучасних
розрахункових методиках. Для великого класу багатовимірних задач радіаційної
теорії переносу тільки даний метод дозволяє розглядати всі геометричні
та фізичні припущення. За допомогою методу можна розв’язати
такі актуальні і практично важливі проблеми, як нейтронне перенесення,
екранування реактора та дифузія домішок у випадкових областях. Метод
дозволяє оцінити вірогідність першого проходження події, коли відклик
системи виходить за рамки заданої безпечної області. Оцінюється вірогідність
руйнування пластини при сейсмічному навантаженні. Розрахунок
надійності системи проводиться по критерію Мізеса, за допомогою якого
визначається граничний стан руйнування пластини. Для описання сейсмічного
навантаження використовується запропонована Руісом та Пенцьєном
статистична модель прискорення грунту внаслідок землетрусу.
Прямой метод моделирования Монте-Карло играет важную роль в современных расчетных
методиках. Для большого класса многомерных задач радиационной теории переноса только
данный метод позволяет рассмотреть все геометрические и физические допущения. С
помощью метода можно решить такие актуальные и практически важные проблемы, как
нейтронный перенос, экранирование реактора и диффузия примесей в случайных областях.
Метод позволяет оценить вероятность первого прохождения события, при котором
отклик системы выходит за рамки заданной безопасной области. Оценивается вероятность
разрушения пластины, подвергнутой сейсмической нагрузке. Расчет надежности
системы проводится по критерию Мизеса, с помощью которого определяется предельное
состояние разрушения пластины. Для описания сейсмической нагрузки используется предложенная
Руисом и Пенцьеном статистическая модель ускорения грунта, вызванного землетрясением.
|
| first_indexed | 2025-12-07T16:31:15Z |
| format | Article |
| fulltext |
UDC 539.4
Evaluation of Failure Probabilities of Mechanical Systems under
Seismic Action by the Monte-Carlo Simulation Method
M. Labou
Mechanical Engineering Department, University of Quebec, Montreal, Canada
УДК 539.4
Оценка вероятности разрушения механических систем при
сейсмическом воздействии с помощью метода моделирования
Монте-Карло
М. Лабу
Отделение машиностроения при Квебекском университете, Монреаль, Канада
Прямой метод моделирования Монте-Карло играет важную роль в современных расчетных
методиках. Для большого класса многомерных задач радиационной теории переноса только
данный метод позволяет рассмотреть все геометрические и физические допущения. С
помощью метода можно решить такие актуальные и практически важные проблемы, как
нейтронный перенос, экранирование реактора и диффузия примесей в случайных областях.
Метод позволяет оценить вероятность первого прохождения события, при котором
отклик системы выходит за рамки заданной безопасной области. Оценивается вероят
ность разрушения пластины, подвергнутой сейсмической нагрузке. Расчет надежности
системы проводится по критерию Мизеса, с помощью которого определяется предельное
состояние разрушения пластины. Для описания сейсмической нагрузки используется предло
женная Руисом и Пенцьеном статистическая модель ускорения грунта, вызванного земле
трясением.
К л ю ч е в ы е с л о в а : метод моделирования Монте-Карло, вероятность первого
прохождения, оценка надежности, критерий Мизеса, ускорение грунта.
Introduction. The need for a stochastic analysis of engineering systems
stems from the fact that an important class of structural loads, which evolves with
time, exhibits a strong variability in both intensity and frequency content. This
class encompasses environmental loads (e.g., wind loads on tall buildings and
bridges, hydrodynamic forces on offshore and marine structures, seismic loads on
buildings and dams), as well as loads caused by human activity (e.g., traffic loads
on bridges, blasts and explosions, loads imposed by industrial machinery).
During the initial of the design stage, it is a common practice to perform a
structural analysis by treating these loads as pseudo-static and the initial loads at a
later stage. Implementation of the deterministic techniques is feasible in a routine
fashion, provided that both structural properties and excitation vectors can be
precisely described. There are certain cases however, for which the excitation
process and/or certain structural characteristics are either unknown accurately or
are random in nature [1]. Usually these are hazards due to natural phenomena
© M. LA B O U , 2008
54 ISSN 0556-171X. Проблемы прочности, 2008, № 6
Evaluation o f Failure Probabilities o f Mechanical Systems
such as earthquakes, winds, or sea-waves which result in loading processes that
are not only mathematically complex, but also exhibit a strong element of
randomness. Such time-dependent excitation processes are called random (or
stochastic) processes. Thus, it is almost certain that any future event will result in
a loading realization which will differ, to certain extinct, from all previous records
that are currently available. In such cases, an effort to compute a unique
deterministic solution of the dynamic problem, which will probably never be
realized during the lifetime of the structure, is of little practical value.
The response of a structural system is difficult to predict when uncertainty
stems from the probabilistic nature of the loading function. In other cases,
uncertainty refers to the parameters of the structural system itself, (e.g., the exact
nature of the constitutive law for materials used, geometric imperfections due to
accidental errors in the construction process). The complexity in the analysis
procedure increases, when uncertainties in both the loading process and the
structural system are combined. So, the random nature of loads affecting structures
as well as uncertainties associated with their resistances necessitates a probabilistic
approach to the analysis of structural safety.
In the design of structural systems, safety is an important issue to be
considered. Reliability, which is defined as the probability that the system meets
some specified demands for a specified time period under specified environmental
conditions is used as a probabilistic measure to evaluate the reliability of
structural systems. The performance function of a structural system must be
determined to describe the behavior of the system and to identify the relationship
between the basic parameters in the system. The concept of reliability has become
a power tool which enables designers to deal probabilistically with uncertainty,
when sufficient knowledge of the random variables and functions is available
[2-4].
The number of uncertainties involved in a joint environment system behavior
is large. The seismicity is random in recurrence magnitude and location and
generated by a number of independent sources. The performance of the components
is random. Moreover, the system occupies such a large environment that it is
never uniformly affected. However, the effects of earthquakes are transmitted
indirectly, through the flow alterations in the network, to sites far beyond the felt
area of an earthquake. It is readily seen, that the modeling of the stochastic
process, given such a number of probabilistic degrees of freedom, constitutes a
challenge to any earthquake engineering approach to the problem. Then, the
modeling of the system is a separate complex task, if it is to be stochastically
analyzed. Moreover, the risk analysis can not be undertaken without setting
adequate performance criteria [5].
Performance criteria that are usable from the macroscale point of view have
undergone research by Duke and Moran [6]. In the field of system modeling, for
instance, there has been research of analytical solutions to spatially distributed
networks subjected to seismic hazards [7]. However, it is felt that a more complete
development in the network seismic performance field has been precluded by the
limitations of purely analytic methodologies. Therefore, different approaches to
the problem have to be devised, and one promising methodology is computer
simulation.
ISSN 0556-171X. npoôëeMbi npounocmu, 2008, N 6 55
M. Labou
Computer simulation has been defined as an experimental arm to operations
research, and as a method that abstracts the essence of problems revealing their
underlying structures [8]. Analytical models convolve one fundamental relationship
with the next to build up an overall final model. Simulation, on the other hand,
structures the overall model chaining the elemental submodels that much resembles
the actual concatenation of events in the real system. Flexibility is gained to
interactively improve the model, and physical insight to the problem is retained
for the iconic nature of the procedure. More important, simulation is frequently
the only available solution to problems that cannot be approached analytically.
Computer simulation takes samples of the probability distribution functions found
along the process by means of random number generators [9]. This sampling
procedure is called Monte-Carlo simulation technique.
1. G eneral C haracteristics of the M onte-C arlo Sim ulation M ethod.
Numerical simulation of random fields is one of the cornerstone problems of
stochastic dynamics. In fact, the Monte-Carlo method and the process of stochastic
field realizations are essentially two sides of the same coin; thus, the first method
is used for computing statistical measures of random field from its realizations,
while through the second method sample values of the field are produced from its
statistical measures. We note here that the Monte-Carlo simulation method is
completely general and its range of applicability is very wide. It is used for
solving problems in mathematics, physics and engineering, as well as in other
scientific disciplines, by conducting what are essentially sampling experiments.
The method is especially valuable in cases where closed-form solutions to a
problem are either impossible or extremely difficult to come by. For all practical
purposes, MCS method is considered exact and is often used to gaging results
produced by other methods; its basic features for stochastic dynamic problems
can be most advantageously explained for the first passage problem.
The Monte-Carlo simulation method is the natural result of the interaction
between the probability and computers. Several methods are currently available to
solve a large number of problems in mechanics involving uncertain quantities
described by stochastic processes, fields or waves. At this time, Monte-Carlo
simulation appears to be the only universal method that can provide accurate
solutions for certain problems in stochastic mechanics, involving nonlinearities,
system stochasticity, stochastic stability, parametric excitations, large variations of
uncertain parameters, and that can assess the accuracy of other approximate
methods such us perturbation, statistical linearization closure techniques, stochastic
averaging, etc. [10].
The combination o f the finite-element and Monte-Carlo methods was
facilitated by Yamazaki, Shinozuka, and Dasgupta [11]. This is one magnificent
example of incorporating a powerful deterministic technique into stochastic
analysis. In this regard, one should mention that presently the stochastic finite-
element method usually uses, either directly or indirectly, the perturbation
method. Shinozuka and Yamazaki [12] state that Monte-Carlo simulation methods
can provide useful alternative for problems to which the perturbation method does
not apply when the degree of the material property variability becomes large.
The solution of problems by the MCS method consists of three stages:
simulating sufficiently representative samples of stochastic values and stochastic
56 ISSN 0556-171X. npo6n.eMH npounocmu, 2008, N 6
Evaluation o f Failure Probabilities o f Mechanical Systems
processes, describing properties of a system and inputs; multiple analysis of a
corresponding deterministic model aimed at obtaining samples of output parameters
and processes; statistical processing of the results including the estimation of
probabilistic characteristics. Since most of the methods require the use of
computers, their use implies also selection of a discretization scheme to
approximate the given model. In particular, discretization with respect to time is
performed, and stochastic functions of continuous time are substituted by
corresponding stochastic sequences [13-15].
The name ‘Monte-Carlo’ dates from 1944, and was introduced by von
Neumann in the course of work on neutrino diffusion for the atomic bomb.
Formal development of the method took place a few years later in the paper by
Metropolis and Ulam [16].
2. Concept of Reliability Theory. Methods for stress analysis and for the
analysis of deformations and displacements in mechanical systems are given by
the theory of vibrations and by mechanics of deformable bodies. The final
judgment regarding the suitability of a system under design or a system in use
should be based on the theory of reliability. From the point of view of mechanics
that part of reliability theory is of greatest interest in which failure is treated as the
result of time changes in the system’s parameters (general, physical or parametric
theory of reliability). To a great extent, this theory is based on the analysis of
fluctuations of random processes and fields describing the system behavior [13].
Failure is defined as complete or partial loss of the ability of the system to
fulfill its functions. Failure can be caused either by the development of defects
which have already been in the system by the time it came into use, or by
accumulation of damages and irreversible changes occurring in the process of the
system’s operation. The initial distribution of defects, operating conditions, and
interaction of the system with the environment are generally of random nature.
Therefore, failures should be considered as random phenomena.
It is widely accepted that the mathematical probability theory is a rational
and natural basis for the modeling of structural reliability problems [17, 18]. The
physical variables are taken to be random variables and the reliability with respect
to a given failure mode is simply defined as the probability that the set of these
random variables gets an outcome in the safe set which itself may be modeled as
a random set. The complementary probability is the probability of structural
failure in the considered failure mode. The performance function of a structure
system must be determined to describe the behavior of the system and to identify
the relationship between the basic parameters in the system. Simulation techniques
are used for evaluating the degree to which a system fulfills a number of
performance and design requirements during a period r, referred to as lifetime.
These requirements restrain the acceptable values o f the response process X ( t )
to an open set D s С R n, referred to as the safe set. The probability
Ps (r ) = P { X ( t ) e D s ,0 < t < r}, (1)
that design conditions are satisfied during projected lifetime r provides a useful
measure of system performance and is called reliability. The complement of
P s ( r ),
ISSN 0556-171X. Проблемы прочности, 2008, N 6 57
M. Labou
PF (r ) = 1 - Ps (r ), (2)
represents the probability of failure. Since exact values of these probabilities are
difficult to obtain, approximations of and bounds on reliability and failure
probability are focused on. The mean value E { N D (r )} of the number of
excursions of X ( t) outside D s , or the mean number of D-outcrossings of X ( t)
in r, N d ( r ) is one of the essential parameters for approximating and bounding
reliability. Reliability is equal to the probability
that the time Td to the first D-outcrossing of X ( t ) exceeds r.
Mechanical systems may not meet performance or design specifications due
to a variety of failure modes that can occur during design lifetime r. The way in
which a system fails depends on material properties, system characteristics, and
loading conditions. System performance relative to many failure modes can be
formulated as in Eq. (1) or (2) [19].
3. First-Passage Statistics. The first-passage problem refers to the problem
of determining the probability that the response of a randomly-excited dynamical
system will reach the boundary of a bounded domain of state space within its
projected lifetime [19]. By appropriate choice of the “safe” domain, probabilities
of failure (or survival) can provide important measures of the performance and
reliability of a mechanical or structural system. This approach is particularly well
suited for systems, which immediately fail as soon as a response quantity
overcomes a threshold.
When the random excitations can be modeled as white Gaussian noise
processes, reliability analyses can be based on the strong mathematical foundation
given by the theory of Markov diffusion processes. Response probability density
function (p.d.f.), distribution and moments of first-passage time can then be
determined as the solution of diffusion-type equations. Presently, available
solutions to the first-passage probability problem are quite rare. There is a
complete absence of analytical exact solution in closed form, and numerical
procedures based on the Fokker-Planck-Kolmogorov equation, which are
practically limited to scalar diffusion processes, are confined to problems of low
dimension [20].
Due to its practical importance, the first-passage problem has attracted a
great deal of attention. In a larger context, it can be considered as part of the more
general question of the stability of stochastic motion. Many definitions of varying
degree of importance have been given for the stability of stochastic dynamical
systems [21]. The stochastic stability of parametrically excited systems in terms
of first-passage times of the system response (according to Katz and Schuss, [22],
and Klosek-Dygas et al., [23]) is defined as follows: if the probability of
obtaining a finite first-passage time approaches zero, then the oscillator is
stochastically stable, and conversely, if the first-passage time is finite with
probability one then the oscillator is stochastically unstable.
Here, whether stability in a stochastic sense is guaranteed or not, a measure
of reliability of a randomly excited system is defined by determining the mean
Ps( r ) = P {T d > r }, (3)
58 ISSN 0556-171X. npo6n.eMH npounocmu, 2008, N 6
Evaluation o f Failure Probabilities o f Mechanical Systems
first-passage time of the response to a closed contour specified in terms of a
Lyapunov function of the system. The system will perform reliably if this time is
much larger than the projected lifetime of the system (see, e.g., Roy, [24]).
4. Earthquake-Induced Ground Acceleration. The theoretical development
in stochastic dynamics and reliability analysis provides the basis for establishing
stochastic models for physical phenomena encountered in mechanical and structural
engineering. A great number of excitations or disturbances to which mechanical
and structural systems are subjected are stochastic in nature. Guided by physical
insight and observations, appropriate stochastic-process models can be developed
for them and used as excitations in a random vibration problem, these models can
provide useful system response information.
The safety analysis of structures and other constructed facilities under
earthquake loads is a typical example in random vibration since the excitation,
namely ground motion induced by an earthquake, is random in general. The basic
data of earthquake engineering are the recordings of ground accelerations during
earthquakes. Knowledge of the ground motion is essential to an understanding of
the behavior of structures under seismic action. Recorded ground motion time
series contain valuable characteristics and information that are used directly, or
indirectly, in seismic analysis and design. Parameters such as peak ground motion
values (acceleration, velocity and displacement), measures of the frequency
content of the ground motion, duration of strong shaking and various intensity
measures play important roles in seismic evaluation of existing facilities and
design of new systems [25].
Ground motion records can be historic or simulated; they can be created for
instance from real records using an autoregressive average model. Such a
simulation approach may be necessary because of the large number of iterations
required to create a robust probability distribution on loss [26].
Ground motion generates inertial seismic loads, which bring into losing the
load-carrying capacity of buildings. The most actual is the problem of earthquake
proofing of buildings and prevention of their collapse. The solution for this
problem consists in selecting a seismic load model to determine the appropriate
system response. It is not possible to establish a seismic action exactly since the
seismic-wave motion of the ground is a stochastic process, a concrete realization
depending on many factors. Therefore, for seismic calculations, methods based on
probabilistic approach are used. The white noise model is not practically used in
pure form for the ground acceleration representation, since the seismic load is
inherently nonstationary and has a non-uniform spectral distribution.
In this paper, a statistical model of earthquake-induced ground acceleration
is used. The model was first proposed by Ruiz and Penzien [1]. According to this
approach, the ground on which the structure was founded is idealized as a more or
less soft soil layer resting on an underlying bedrock. The seismic waves traveling
in the earth-crust hit the bedrock surface, they are filtered by the soft stratum (soil
layer), and finally envelope the structure foundation.
The vibration of the bedrock is modeled by a non-stationary Gaussian
process that is defined as the product of the white-noise process multiplied by a
determinate function of time. Thus, a sample function of the bedrock acceleration
a b ( t ) is obtained by multiplying the shape function t ) by a sample function
ISSN 0556-171X. Проблемы прочности, 2008, N 6 59
M. Labou
x ( t ) generated by a sequence of independent Gaussian ordinates; it is given by
the following formula:
a b ( t) = D 1( t )x( t) - (4)
The shape Unction D i ( t ) for a non-stationary stochastic process is obtained
by the faired curve (see Fig. 1). It depends on three parameters. When one lacks
direct data, one can assume [1]: t1 =1.21 s, 12 =11.5 s, and c = 0.155 s
ag (t)
-1
A(0
agf(t)
D1(t) = g -c (t-t2)
ab (0
> t -1
af (t)
t
2D 5
c
Fig. 1
c
Fig. 2
Fig. 1. Simulation of the bedrock acceleration: (a) the sequence of Gaussian ordinates; (b) the shape
function; (c) the bedrock acceleration.
Fig. 2. Simulation of the soil layer acceleration.
Next, filtering of the bedrock acceleration by the soil layer should be
accounted for. The action of the layer is assimilated to a second-order linear filter,
so that the random ground acceleration a g ( t ) at the foundation level is given by
(see Fig. 2a):
a g (t) = - 2 |w o K t) - ® 2 y ( 0 . (5)
where y ( t ) is the solution of the differential equation
y ( t) + 2 & o y ( t) + m 0 y (t) = - a b ( t) . (6)
The parameters m o and ^ may be thought of as characteristic ground frequency
and characteristic damping ratio, respectively. The filter, described above,
b
60 ISSN 0556-171X. npo6n.eMH npounocmu, 2008, N9 6
Evaluation o f Failure Probabilities o f Mechanical Systems
attenuates higher-frequency components and amplifies those in the neighborhood
of m = m o. However, it does not change the amplitude at m ^ 0, and hence strong
singularities are present at m ^ 0. These undesirable singularities can be removed
by passing a g ( t ) through a second filter (see Fig. 2b), which greatly attenuates
the very-low-frequency components. Such filter is determined by
z ( t ) + 2?! z ( t ) + m2 z( t ) = - a g ( t ), (7)
then it is necessary to put a gf ( t) = z ( t).
If a p is the actual peak acceleration that must be reproduced by the
simulated earthquake, it is sufficient to multiply the available function a gf by the
amplitude ratio Y :
a gf (t) = a g t( t )Y, (8)
where Y = a p ! x ap , x a is the expected maximum value of the simulated
sample functions n.
With n of order 1000, the expected peak value of the acceleration in the
simulation process can be approximately set equal to three times the standard
deviation S a , i.e., x a = 3 S a . The standard deviation S a of the random
variable a gf ( t ) is determined by the following expression:
S as = S agf (t) = f G agf ( t) (m )d m , (9)
— X
where G af (t) is the power spectral density of the output process a gf ( t ):
G af (t)(m) = I A f (m )|2 ‘I A(m)|2 G 0 (—X < m < x ) . (10)
The complex transfer function | A f (m)|-| A(m)| of the filters in Eqs. (6) and (7)
are determined, respectively, by the following formulas:
2 1+ 4£ 2( m l m 0 ) 2
|A(m)|2 = n ( / ) 2 ++ 4 , 20( / ) 2 , (11)[1— (m/ m 0) + 4£ (m/ m 0)
/ m2 ( ^ m 1)4
|Af(m )| = [1 — (m /m ,)2 ]2 + 4?2(V m 1)2 ' (12)
For lack of experimental data on the standard ground conditions, according
to [1], the following parameters can be used: m0 =15.6 rad/s, m1 = 0.897 rad/s,
? = 0.6, ? 1 = 1^12. The spectral density G 0 takes the value At/2^.
ISSN 0556-171X. npoôëeMbi npounocmu, 2008, N9 6 61
M. Labou
5. Num erical Example. As an application, the first passage probabilities of
a plate over a limit state failure are determined. For the purpose of reliability
analysis, the von Mises criterion is used. As it is well known, determination of the
deflections w ( x , y ) of the plate subjected to a transverse load q ( x , y ), as shown
in Fig. 3, leads to the integration of the equation [27]
AAW = q( x , y )
D
(13)
Here AA is the differential operator
d 4 d 4
A A = .4 + ^ 2 . 2 +
dx dx dy dy
and D is the flexural rigidity
D =
E h
12(1 — v 2)
where h is the plate thickness, E is the elastic modulus, and v is Poisson’s
ratio.
Fig. 3. Uniform plate under random dynamic load.
The plate is subjected to a parametric load with respect to deflection
W (x , y , t ). Let this load correspond to the stresses from the plane problem of the
theory of elasticity a x , a y , and r xy. The plate is also subjected to a distributed
load Q (x , y , t ) (considered as a seismic load), varying randomly in time and
acting perpendicular to the plane of the plate. The effect of longitudinal and
inertia forces can be taken into account with the help of reduced load [27],
I
q ( x , y , t) = N x
d 2W d 2W
dx
~ + 2 N xv
2 xy dxdy
+ N
d 2W
^^7
+ p h
d 2W
dt 2 (14)
4
62 ISSN 0556-171X. npo6n.eMH npounocmu, 2008, N9 6
Evaluation o f Failure Probabilities o f Mechanical Systems
where N x = h o x , N y = h o y , N xy = h r xy, and p is the density of the material.
The parametric forces N x, N y , and N xy are considered negative if they produce
compression.
The differential equation for vibrations [19, 27] is
D A A W (x , y , t) + c W ( x , y , t) — q(x , y , t) = Q(x , y , t). (15)
Here c is the material damping, and p and c can be functions of (x, y). Let
w k (x , y ) and m k be the undamped natural modes and frequencies of free
vibration of the system. They satisfy the conditions
P hm k w k = D A A w k ,
a b
f dxf dy p h w k w i = m k
0 0
where d ki is the Kronecker delta symbol.
We seek solution in the series form
(16)
(17)
W ( x , y , t) = ^ X k ( t ) w k ( x , y ) .
k=1
(18)
From Eqs. (15) and (18), one can find that generalized modal coordinates X k ( t)
are the solutions of
X k ( t ) + kX k ( t ) + (m 2 + K ) X k ( t ) = Q k ( t ), k = 1,2, ..., (19)
a b
provided that f d x f dycWkWi = Ckô k i. In this equation, 2Çkm k = c k / mk and
0 0
a b1 a u
Q k (t) = — f d x 5 dy Q(x, y , t ) w l (x, y )’
mk 0 0
b I1 a b
K = — I d x f dy
mk 0 0
N .
d Wk ( x , y ) d w k ( x , y )
+ N .
d w k ( x , y )
d y '
dx
w l ( x, y ) .
(20)
We can determine probabilistic characteristic of internal forces and stresses
from Eqs. (18) and (19), for example, the components of the stress tensor at a
point of coordinates (x, y , h/2). At this point, o z , r xz, and r yz are zero, so that
the stress tensor represents a two dimensional stressed state, they are defined as
ISSN 0556-171X. npoôëeMbi npounocmu, 2008, N9 6 63
M. Labou
O x (x , y ) =
Eh I -,2
1—V2
+ V
d x 2 dy2
W ( x , y , t)
E h 2w k ' d 2w k
.2 2 I -, 2 — V .2
O y ( x , y ) =
d x 2
Eh
dy
X k ( t X
1 - V2 dy
■ + V ■
dx2
W ( x , y , t )
Eh
2 11 2 I 'i 2
1 - v *=A dy
d d w k — V-----r -
dx
X* ( t ),
2
r xy ( x , y ) = ■
Eh d
1 + V dxdy
Eh d 2 w *
W (x, y , t) = - -— ~ 2 ~ ^ X k (0-
1 + V k=1 dxdy
(21)
To determine the likelihood of failure at this point one can use the von Mises
criterion. According to this criterion [19], the stressed state is acceptable if
(22)
where o cr denotes a limit stress. The criterion takes the form
2 a klX k ( t )X l ( t) < O 2r J (23)
in which
a kl =
Eh
1 — V
Aw* Awi — 3
1 \ 2 -.2 -.2 ̂1 — V) d w * d Wl
1 — V / dxdy dxdy (24)
Reliability Ps (r ) is the probability that processes {X k ( t )} do not leave
domain
D = x: 2 a klX k ( t)X l (t) - O
in (0, r). It can be calculated approximately from D-outcrossings of an »-dimensional
process X ( t) = [X 1( t ) , ..., X » ( t)], where n denotes the number of modes
considered in the analysis.
Suppose that the plate is simply supported, a < b , and Q (x , y , t ) = Q ( t ) is a
seismic load. The first four modes and frequencies of vibration of the plate are
p n q n
w k ( x , y ) = sin---- x sin—- y ,
a b
(25)
2
2
64 ISSN 0556-171X. npoôëeMbi npounocmu, 2008, N 6
Evaluation o f Failure Probabilities o f Mechanical Systems
and
D n 4 1 p 2
p h
+
b k
(26)
with ( ( p , q ) = (1,1), (1, 2), (2, 1), and (2, 2) for & = 1,2, 3, and 4. Normalization
constants and c& are equal to p h a b /4 and c a b 4, respectively, when p and
c are spatially invariant.
The deflection equation of the above mentioned rectangular plate loaded
with compressive edge loads N x and N y in the plane of the plate, in addition to
uniformly distributed loads P acting perpendicular to the plane of the plate,
could be expressed in the form [28]
w =
p n q n
Z Z a pq s m ~ x s m - y y ’ (27)
where according to [28],
16P
pq
n 6 D
p q
2 2
a 2 +
n 2 D
I 2 2 ^p q
N x ^ 2 + N y 7 2 a b
(28)
Although both N x and N y are compressive loads, they have, a negative
sign, and a value of this negative term may become high enough so that the
denominator in Eq. (28) approaches zero. Under these circumstances, the deflection
w approaches a very large value irrespective how small the normal, P, may
become, and the value of the edge load corresponding to this condition is known
as the buckling load. For the critical values of the load parameters we obtain from
Eq. (28) the relationship
2 2
P q 2
N x ^ r + N v ^ ir = n 2 D
a 2 y b 2
I 2 2
P + <t_
2 + u 2 a b
(29)
We introduce the notations
a N y
a = —, <p = -----
b ’ V N x
(30)
and express the buckling stresses in the following form [29]:
N= N x
' x’cr = h
x,cr
O y. = ; = K
n 2 D
x b 2 h
y,cr
O ycr h
then from Eq. (29) we find
22
q
a
1
1
2
ISSN 0556-171X. npoôëeMbi npounocmu, 2008, N9 6 65
M. Labou
= [( p / a ) 2 + q 2]2
x ( p / a)2 + <pq2
K y = <pKx D =
E h :
12(1 — v 2 )
(32)
Suppose we consider for the reliability analysis the first four modes of the
plate defined in Eqs. (25) and (26). From the first equation of (20) we find that
generalized forces Q k ( t ) vanish for k = 2,3, and 4, and modal coordinates X k ( t )
are also zero for k = 2, 3, and 4. To avoid large values of deflections due to the
critical values of the load parameters, we set in Eq. (23) a cr = min(a xcr, a ycr ),
which reduces to
a n X i2( t ) < a2r • (33)
The limit state failure x c, according to Eq. (33) is
x c = ± a c r / j o n , (34)
and the safe set, D s, for X 1(t) is
D s = ] — x c, x c[. (35)
The maximum deflection is given when x = a/2 and y = b ]2, thus from Eq. (24)
we find
2
a 11 =
Eh
1— v
n Ÿ ( n "'2
~aa J + [ ~b
(36)
The parametric load acting on the plate corresponds to the stress from the
plane problem of the theory of elasticity, i.e., N x = h a x, N y = h a y, and N xy =
h r xy, where a x, a y, and r xy are determined from Eq. (21).
For the first mode of vibration, i.e., ( p , q ) = (1,1) the ratio cp takes the form
p =
N x v a +1
(37)
Thus, the expression of K x in Eq. (32) becomes
K =
(V a 2 + 1)2
1/ a 2 + p
(38)
The case a < b implies p < 1, thus a cr = min(a xcr, a ycr ) = a ycr, where
a y c r p K x b 2 1 2 (1 — v 2 ) '
(39)
66 ISSN 0556-171X. npoôëeMbi npounocmu, 2008, N9 6
Evaluation o f Failure Probabilities o f Mechanical Systems
The equation of motion [Eq. (19)] is reduced to the form
X j ( t) + 2Ç»!X j ( t) + ( + K jX j ( t ))X j ( t) - Q x{t), (40)
where
16 2
Q 1(t) — . 2 Q(t); — nh 1 2 ' >2p h n 2 ph 1 ~2 *-2
D n 4 ( 1 1 x2
+
a~ b 2
32 n 2 Eh
9 a 4 p ( 1 - v 2)
[2 + (1 + 3v )a 2 + 2a 4 ].
(41)
The substitution of Eqs. (36) and (39) into Eq. (34) gives the expression of the
limit state failure x c
h
12(1 + v ) a 2 + 1 (42)
To investigate the first-passage probabilities over the limit state failure x c,
we considered a steel plate, which has the following parameters: the dimensions
in plan a = 2 m, b = 1.5a, the thickness h = 0.008 m, the modulus of elasticity
E = 210 GPa, the density of the material p = 7850 kg/m , Poisson’s ratio
2 3v = 0.25, the intensity of the white noise excitation I = 1m /s , and the peak
value of the ground acceleration a p = 1 m/s (Richter magnitude 7 earthquake).
Using the above data, the buckling stress from Eq. (39) takes the value
o r = 3.02 MPa, which is considerably below the proportional stress limit of
standard steel plate, then from Eq. (42), the limit state failure x c = ± 3 .82 • 10 4 m.
The differential equation Eq. (40) is integrated using the Runge-Kutta
method at zero initial conditions. The first-passage probabilities of the system
response x ( t ) during time [0, t] for double-sided symmetrical barriers located at
± x c were determined. The response statistics with a sample size n = 300,000
obtained by the MCS method are presented in Fig. 4.
Fig. 4. Estimates for first-passage probabilities versus time for double-sided barriers located at
levels ± xc.
ISSN 0556-171X. npoôneMU npoHHoemu, 2008, № 6 67
M. Labou
Conclusions. The research carried out shows that the Monte-Carlo simulation
method allows one to assess efficiently the failure probabilities in low probability
range. It has been successfully applied to a structural element, subjected to a
non-stationary stochastic load action.
The fast growing computer technology made the use of the MCS method
feasible; it provides a feasible solution to seismic risk analysis of spatially
distributed engineering systems, and it overcomes most of the limitations of
analytical approaches. A good part in its power resides in the fact that it retains a
clear insight into the problem; hence, it can help to develop risk criteria.
Р е з ю м е
Прямий метод моделювання Монте-Карло відіграє важливу роль у сучасних
розрахункових методиках. Для великого класу багатовимірних задач ра
діаційної теорії переносу тільки даний метод дозволяє розглядати всі гео
метричні та фізичні припущення. За допомогою методу можна розв’язати
такі актуальні і практично важливі проблеми, як нейтронне перенесення,
екранування реактора та дифузія домішок у випадкових областях. Метод
дозволяє оцінити вірогідність першого проходження події, коли відклик
системи виходить за рамки заданої безпечної області. Оцінюється віро
гідність руйнування пластини при сейсмічному навантаженні. Розрахунок
надійності системи проводиться по критерію Мізеса, за допомогою якого
визначається граничний стан руйнування пластини. Для описання сейсміч
ного навантаження використовується запропонована Руісом та Пенцьєном
статистична модель прискорення грунту внаслідок землетрусу.
1. G. Augusti, A. Baratta, and F. Kachiati, P ro b a b ilis tic M e th o d s in S tru c tu ra l
E n gin eerin g , Chapman & Hall, London-New York (1984).
2. G. D. Manolis and P. K. Koliopoulos, S to ch a stic S tru c tu ra l D yn a m ics in
E arth qu ake E n g in eerin g , WIT Press, Southampton (2001).
3. M. Labou, “Solution of the first-passage problem by advanced Monte-Carlo
simulation technique,” S tren g th M a te r ., 35, No. 6, 588-593 (2003).
4. M. Labou, “First-passage probabilities for randomly excited mechanical
systems by a selected Monte-Carlo simulation method,” Ukr. M atem . J ., 56,
No. 7, 1196-1202 (2004).
5. H. C. Shaw and J. R. Benjamin, L ife lin e S e ism ic C riter ia a n d R isk: A S ta te
o f th e A r t R eport. L ife lin e E arth qu ake E n gin eerin g: The C u rren t S ta te o f
K n o w led g e , ASCE, New York (1977).
6. C. M. Duke and D. F. Moran, “Guidelines for evolution of lifeline earthquake
engineering,” in: Proc. of the U.S. Nat. Conf. on E arth qu ake E n gin eerin g ,
Ann Arbor (1975).
7. G. Panoussis, “Seismic reliability of lifeline networks,” SDDA Report No. 15,
MIT Dept. of Civil Engineering, Rept. R74-57, Cambridge, MA (1974).
8. F. S. Hillier and G. J. Lieberman, In trodu ction to O pera tion R esearch ,
Holden Day, San Francisco (1974).
68 ISSN 0556-171X. Проблеми прочности, 2008, № 6
Evaluation o f Failure Probabilities o f Mechanical Systems
9. D. E. Knuth, The A r t o f C o m p u ter P ro g ra m m in g , Vol. 2, S em in u m erica l
A lg o rith m s, Addison Wesley Publishing Company (1969).
10. M. Shinozuka and G. Deodatis, “Simulation of multi-dimensional Gaussian
fields by spectral representation,” A ppl. M ech . R ev ., 49, No. 1, 29-53 (1996).
11. F. Yamazaki, M. Shinozuka, and G. Dasgupta, “Newmann expansion for
stochastic finite element analysis,” in: S toch astic M ech an ics , Vol. 1, Columbia
University (1986).
12. M. Shinozuka and, F. Yamazaki, “Stochastic finite element analysis: an
introduction,” in: S. T. Ariaratnam, G. I. Schueller, and I. Elishakoff (Eds.),
S to ch a stic S tru c tu ra l D yn a m ics, Elsevier, London (1988), pp. 241-292.
13. V. V. Bolotin, R an dom V ibra tion s o f E la s tic S ystem , Martinus Nijhoff, New
York (1984).
14. V. Vikov, D ig ita l S im u la tion in S ta tis tic a l R a d io E n g in eerin g [in Russian],
Sovetskoe Radio, Moscow (1971).
15. S. Yermakov and G. Mikhailov, C ou rse in S ta tis tica l S im ulation [in Russian],
Nauka, Moscow (1976).
16. N. Metropolis and S. Ulam, “The Monte-Carlo method,” J. A m er. S tatist.
A s s o c , 44, No. 247, 335-341 (1949).
17. A. M. Freudenthal, J. M. Garrelts, and M. Shinozuka, “The analysis of
structural safety,” J. S truct. D iv. A SC E , 92, No. ST1, 267-325 (1966).
18. M. Shinozuka and H. Itagati, “On the reliability of redundant structure,” in:
Proc. of the Fifth Conf. on R e lia b ility a n d M a in ta in a b ility (New York, July
1966), A n n a ls o f R e lia b ility a n d M a in ta in a b ility , Vol. 5, Academic Press
(1966), pp. 605-610.
19. T. T. Soog and M. Grigoriu, R andom Vibration o f M ech a n ica l a n d S tructu ra l
S ystem s, Prentice Hall, New Jersey (1992).
20. A. A. Andronov, L. S. Pontriagin, and A. A. Vitt, On S ta t is t ic a l
C o n sid era tio n s o f D yn a m ic S ys tem s , in: A. A. Andronov (Ed.), Collected
works [in Russian], Academy of Sciences, Moscow (1956).
21. F. Kozin, “A survey of stability of stochastic systems,” A u to m a tica , 5, No. 1,
95-112 (1969).
22. A. Katz and Z. Schuss, “Reliability of elastic structures driven by random
loads,” S IA M J . A pp l. M a th ., 45, No. 3, 383-402 (1985).
23. M. M. Klosek-Dygas, B. J. Matkowsky, and Z. Schuss, “Stochastic stability
of nonlinear oscillators,” S IA M J. A ppl. M a th ., 48, No. 5, 1115-1127 (1988).
24. R. V. Roy, “Asymptotic analysis,” Int. J. N on -L in ear M ech ., 32, No. 1,
173-186 (1997).
25. Y. Bozorgnia and V. V. Bertero, E arth qu ake E n gin eerin g , CRC Press
(2004).
26. K. A. Porter, A. S. Kiremidjian, and J. S. Legrue, “Assenmbly-based
vulnerability of building and its use in performance evaluation,” E arth qu ake
S p ectra , 17, No. 2, 291-312 (2001).
ISSN 0556-171X. npoôëeMbi npounocmu, 2008, N 6 69
M. Labou
27. V. V. Bolotin, The D yn a m ic S ta b ility o f E la s tic S ystem s, Holden-Day, San
Francisco (1964).
28. E. Sechler, E la s tic ity in E n g in eerin g , Dover Publications, New York (1968).
29. A. Volmir, Stabiity of Elastic Systems [in Russian], Phizmatgiz, Moscow
(1963).
Received 09. 10. 2007
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|
| id | nasplib_isofts_kiev_ua-123456789-48368 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T16:31:15Z |
| publishDate | 2008 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Labou, M. 2013-08-18T16:03:34Z 2013-08-18T16:03:34Z 2008 Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method / M. Labou // Проблемы прочности. — 2008. — № 6. — С. 54-70. — Бібліогр.: 29 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/48368 539.4 Direct Monte-Carlo simulation method (MCS)
 plays an important role in modern computational
 methods. For a large class of multidimensional
 problems of radiative transport theory,
 the MCS procedure is the only available
 method allowing the consideration of all varieties
 of geometric and physical presumptions.
 This gives rise to computational investigation
 of actual and practical problems such as neutron-
 transport problem, reactor shielding, and
 impurity diffusion in random fields. MCS
 method enables the determination of the
 first-passage probability that the system response
 exceeds prescribed safe domain. In this
 paper, the failure probabilities of a plate subjected
 to seismic load are determined using the
 MCS simulation method. For the purpose of reliability
 analysis, the von Mises criterion is
 used to determine the limit state failure. For the
 seismic load, the statistical model of earthquake-
 induced ground acceleration proposed
 by Ruiz and Penzien is used. Прямий метод моделювання Монте-Карло відіграє важливу роль у сучасних
 розрахункових методиках. Для великого класу багатовимірних задач радіаційної
 теорії переносу тільки даний метод дозволяє розглядати всі геометричні
 та фізичні припущення. За допомогою методу можна розв’язати
 такі актуальні і практично важливі проблеми, як нейтронне перенесення,
 екранування реактора та дифузія домішок у випадкових областях. Метод
 дозволяє оцінити вірогідність першого проходження події, коли відклик
 системи виходить за рамки заданої безпечної області. Оцінюється вірогідність
 руйнування пластини при сейсмічному навантаженні. Розрахунок
 надійності системи проводиться по критерію Мізеса, за допомогою якого
 визначається граничний стан руйнування пластини. Для описання сейсмічного
 навантаження використовується запропонована Руісом та Пенцьєном
 статистична модель прискорення грунту внаслідок землетрусу. Прямой метод моделирования Монте-Карло играет важную роль в современных расчетных
 методиках. Для большого класса многомерных задач радиационной теории переноса только
 данный метод позволяет рассмотреть все геометрические и физические допущения. С
 помощью метода можно решить такие актуальные и практически важные проблемы, как
 нейтронный перенос, экранирование реактора и диффузия примесей в случайных областях.
 Метод позволяет оценить вероятность первого прохождения события, при котором
 отклик системы выходит за рамки заданной безопасной области. Оценивается вероятность
 разрушения пластины, подвергнутой сейсмической нагрузке. Расчет надежности
 системы проводится по критерию Мизеса, с помощью которого определяется предельное
 состояние разрушения пластины. Для описания сейсмической нагрузки используется предложенная
 Руисом и Пенцьеном статистическая модель ускорения грунта, вызванного землетрясением. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method Оценка вероятности разрушения механических систем при сейсмическом воздействии с помощью метода моделирования Монте-Карло Article published earlier |
| spellingShingle | Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method Labou, M. Научно-технический раздел |
| title | Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method |
| title_alt | Оценка вероятности разрушения механических систем при сейсмическом воздействии с помощью метода моделирования Монте-Карло |
| title_full | Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method |
| title_fullStr | Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method |
| title_full_unstemmed | Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method |
| title_short | Evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method |
| title_sort | evaluation of failure probabilities of mechanical systems under seismic action by the monte-carlo simulation method |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/48368 |
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