Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECI'URES- No. 317
RELIABILITY PROBLEMS:
GENERAL PRINCIPLES AND APPLICATIONS
IN MECHANICS OF SOLIDS
AND STRUCTURES
EDITED BY
F. CASCIATI
UNIVERSITY OF PAVIA
J. B. ROBERTS
UNIVERSITY OF SUSSEX
SPRINGER-VERLAG WIEN GMBH
Le spese di stampa di questo volume sono in parte coperte da
contributi del Consiglio Nazionale delle Ricerche.
This volume contains 80 illustrations.
This work is subject to copyright.
All rights are reserved,
whether the whole or part of the material is concerned
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or similar means, and storage in data banks.
© 1991 by Springer-Verlag Wien
Originally published by Springer Verlag Wien-New York in 1991
In order to make this volume available as economically and as
rapidly as possible the authors' typescripts have been
reproduced in their original forms. This method unfortunately
has its typographical limitations but it is hoped that they in no
way distract the reader.
ISBN 978-3-211-82319-4 ISBN 978-3-7091-2616-5 (eBook)
DOI 10.1007/978-3-7091-2616-5
PREFACE
In many fields of engineering it is necessary to formulate and implement
procedures for the assessment of system reliability. This involves an estimation of the
probability that a vector of design variables stays within some prescribed safe domain. In
some situation it is sufficient to employ a static analysis, taking into account the statistical
variability of the system parameters. However, often the dynamic response of an
uncertain, time-varying non-linear system to random disturbances must be considered.
Significant advances in reliability theory, during the sixties and seventies have
recently been utilized to construct a general design methodology, suitable for
incorporating into decision making processes. This is currently finding application in
such diverse fields as offshore technology, aerospace design and disaster prevention in
civil and mechanical engineering.
Although a basic theoretical framework has been established the inherent
techniques require constant development and improvement, to meet the demands imposed
by new advanced engineerings projects. The subject is, therefore, still in an active state
of development.
This book presents, to researchers and engineers working on problems concerned
-
with the mechanics of solids and structures, the current state of the development and
application ofr eliability methods. The topics covered reflect the need to integrate, within
the overall methodology, statistical methods for dealing with systems which have
uncertain parameters and random excitation with the development of suitable safety index
and design codes. The basic principles of reliability theory, together with current
standard methodology, are reviewed. An introduction to new developments is also
provided.
F. Casciati
J. B. Roberts
CONTENTS
Page
Preface
Chapter 1
Random vibration and frrst passage failure
by J. R. Roberts ...................................................................................... 1
Chapter2
Safety index, stochastic fmite elements and expert systems
by F. Casciati ....................................................................................... 5 1
Chapter3
Methods for structural reliability computations
by P. Bjerager ...................................................................................... 8 9
Chapter4
Engineering, operational, economic, and legal aspects
of the reliability assurance
by M. Tichy ....................................................................................... 1 3 7
Chapter 5
Application of nonlinear stochastic dynamics and damage accumulation
in seismic engineering
by Y. K. Wen ................. , ................................................................... 161
Chapter6
Statics and reliability of masonry structures
by A. Baratta ...................................................................................... 2 0 5
Chapter?
Essay on reliability index, probabilistic interpretation of safety factor,
and convex models of uncertainty
by I. Elishakoff ................................................................................... 2 3 7
Chapter 1
RANDOM VIBRATION AND FIRST
PASSAGE FAILURE
.J. R. Roberts
University of Sussex, Sussex, UK
Abstract
In the first part of Lhis Chapter. a variety of techniques for
predicting- nonlinear dynamic system response to raudom excitation
are discussed. These include melhods based on modelling- Lhe
response as a continuous Markov urucess. leading lo diffusluu
eQuallons. stallstlcal linearization. the method of eQuivalent
nonlinear eQuatlons and closure methods. Special attention is
paid to the stochastic averaging method. which is a curnbinatiuu of
an averag-ing techniQue and Markov process modelling-. It ls shown
that the stochastic averaging method is particular-ly useful fur
estimating- the "first-passag-e" probabillly that the system
response stays within a safe domain. within a specified period of
Lime. Results obt1:1ined by Lhls method 1:1re presented for
oscillators with bulh line1:1r and nonlinear damping and restoring
Lerms. An alternalive techniQue for solving the first-pass1:1ge
problem. based on Lite comput1:1lion of level crossir~ statistics. is
also described: this is especially useful in more g-eneral
silu1:1Uuus. where Lhe sluchasllc averag-ing is iuapplicable.
1.1 INTRODUCTION
Me thuds of predl cling- Lhe response of mech1:1ni cal and
structural systems tu random exci tall on are of importance in many
engineer iug- fields: examples include Lhe mo lion of offshore
structures 1:1nd ships to wave and wind excitation. the response of
civil engineering structures. such as buildings. bridges. etc .. to
earthquakes. the behaviour of vehicles travelling on rough ground
and the ch1:1racteristics of aer·ospace vehicles when responding- to
atmospheric turbulence and jet noise excitation.
2 J. R. Roberts
In many cases one is primarily interested in predicting th~
reliability of systems responding to random excitation - L e. ,
predicting t.he probabilily that the system will not fail. in some
defined sense, within a specified period of Lime. This almost
invariably means that one is concerned with large amplitude
response, and hence wilh motions which involve significant
nonlinear effects. In nearly all application areas. the
nonlinearilies which are inevitably present become of primary
importance when the amplitude of the response is large.
Despite extensive research over the last few decades, there is
currently no generally applicable theoretical technique for
predicting the probabilistic response of complex dynamic systems
to random excitation, capable of yielding results with a
reasonable degree of computational effort. However. for simple
systems some very useful approximate techniques have been
developed. which can, in some cases. be used lo provide
quantitative reliability estimates.
In this Chapter various existing techniques for predicting the
response ·of nonlinear systems to r<Uldom excitation are briefly
described. It is shown that one of these. the method of
stochastic averaging, is particularly useful for dealing with
nonlinear oscillators. wilh light damping, driven by wide-band
random excitation. Using this technique it is possible to obtain
simple explicit expressions for the probability distribution of
lhe response. This information can be directly applied in
estimating system reliability.
This discussion is followed by a detailed description of the
so-called '"first-passage problem" i.e., the problem of
predicting the probability that the response slays within a safe
domain (in phase space) within a specified period of time. It is
demonstrated that. for nonlinear oscillators, specific analytical
solutions can be obtained in some cases. and in other cases it is
possible to formulate a simple numerical scheme. based on a random
walk analogue. Results obtained by this method are shown to
compare Wt!ll with corresponding digital simulation estimates of
first-passage statistics.
The Chapter concludes with a brief description of an
alternative approach to t!stimating first-passage statistics, based
on a computation of terms in the so-called inclusion - exclusion
series. This approach is useful in situations when the basic
assumptions inherent in the stochastic averaging method are
inapplicable.
Random Vibration 3
1.2.1 Markov methods
A general form of lhe equations of motion of an n degree of
freedom nonlinear system is as follows f1l
..
~ y T ~ 9 + K 9 T g(g.gl = g(t) (1.1)
Here 9 is an n-veclor of generalised displacements and Q is an
n-veclor of generalised forces. M. C and K are the usuil mass,
damping ami stiffness matrices, r;-espectivefy and g contains all
the nonlinearities in the system. For simplicity it will be
assumed here lhat the mean of Q is zer·o and that g is
odd-symmelr ic, such that the mean of -9 is also zero. -
It is convenient lo rewr i le equation ( 1.1) in stale space
"l
form. Introducing lite state varia!Jle vector z [_9T, ]T one
obtalus
z ( 1. 2)
where
-~
F -f = [ g] ( 1.3)
~
If the elements of f are broad-hand in character they can, in
many cases. be satisfactorily approxim11ted in terms of stationary
while uulses. Thus.
( 1.4)
wher·e E{ } denotes the expectation operator, 6( ) is the Dirac
della function and D is a real, symmetric, non-negative matrix.
Equation (1.2) can now IJe written as
z = !(~) T B~ ( 1. 5)
where ~ is a 2n-vector of indzep endent. unl t white noises and
BBT=D.- Il is noted that. since exists iilmost nowhere [2], this
eqUiitlOn needs lo be interpreted Carefully; a SUitable
Ilu
interpretiilion is to tre11L it i:lS an equation, written in the
form
dz = F(~)dl T ~d~ (1.6)
where ~ ( t) dW/dt 11nd W is 2n-veclor of unit Wiener processes
[2].
It follows from equiillons (1.5) (or (1.6)) that z is 11 2n
dlmenslonal Markov process. wi Lh a trc:msi lion density function
4 J. R. Roberts
[3], p(zlz ;t), governed by the following diffusion equation.
0
known as- ti1e Fokker-Planck-Kolmogorov (FPK) equation:
.
ap = Lp ( 1. 7)
at
where
2n 2n 2u
l 2
l a l a
L (Fi(zi).] + ~ D .. ( 1. 8)
az1 l.J aziaz5
i=1 i. j=1
and
D- = [Dijl = B s-T (1. 9)
If the system is stable. and lime - invariant, a "stationary"
solution to the FPK eijuation usually exists: thus. as time
elapses. p(zlz ;t) becomes independent of the initial condition
0
and asymptotes towiirds a stationary density function. w(z)
i.e.
w(z) = lim p(zlz0:t) ( 1. 10)
f:-t<X> - -
w(z) may be obtained as the solution of equation (1.7) with apjat
= 0 - i.e ..
Lw = 0 (1.11)
A general. closed-form solution lo equation (1.7) has yet lo
be found. for an arbltary value of n. iilthough series solutions
can be found in terms of eigenfunctions and eigenvalues f4]. Fur
m 2n 1, the system is of first-order and aniilylical
expressions for p(zlz ;t) have been found in a few special cases
0
[4.51. However a general expression for w(z) can easily be found.
for m = 1. The result is
t
5 F( )
w(z) exp [ ~ ~ d~] (1.12)
0
where c is a normalization constant. chosen so that
"'
J
w(z)dz 1 ( 1. 13)
-<X>
