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Chapter 9
CHAPTER 9 FUNDAMENTALS OF STRUCTURAL RELIABILITY1
P. Thoft-Christensen, Aalborg University, Denmark
1. INTRODUCTION In the traditional way of designing structures decisive parameters such as dimensions, material strengths and loads are usually characterized by a number of constants, e.g. average values. On the basis of these constants a mathematical model for the behaviour of the structure is used to examine whether the structure is safe or not. To improve the safety of the structura

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Chapter 9
CHAPTER
9
FUNDAMENTALS OF STRUCTURAL RELIABILITY
1
P. Thoft-Christensen, Aalborg University, Denmark
1. INTRODUCTION
In the traditional way of designing structures decisive parameters such as dimensions,material strengths and loads are usually characterized by a number of constants, e.g.average values. On the basis of these constants a mathematical model for the behaviour of the structure is used to examine whether the structure is safe or not. To improve thesafety of the structural the variables are often replaced with “worst case” values. Such a basis for designing a structure by a “worst case” philosophy is usually too conservativefrom an economic point of view because of the small risk that all the variables at thesame time take on their »worst» values. The procedure described here is called the
deterministic
approach.However, it is a well-known fact that for example strength varies from structuralelement to structural element, so that the strength of an element cannot be characterizedadequately by a single value. Further it is also in some cases necessary to take intoaccount time variations. The same kind of uncertainty is present in relation to structuraldimensions and loading. Especially the so-called natural loads due to waves, currents,winds and earthquakes are difficult to deal with in a deterministic way. It is alsoimportant to remember that some uncertainty is involved in the choice of themathematical models used for the analysis of a structure.The purpose of using a
probabilistic
approach rather than the simple deterministicapproach is to try to take into account the uncertainties mentioned above so that a morerealistic analysis of the safety of a structure can be performed.The history of modern reliability theory has been presented by Borges &Castanheta [1] for the period up to 1970. Only a brief survey will be given here. The firstattempt of using statistical concepts in structural reliability seems to be more than 50years old, more precisely from 1926, and was made by Meyer [2]. But only a limited
1
In “Lectures on Structural Reliability”, P. Thoft-Christensen (editor), Aalborg University, Denmark,1980, pp. 1-28.
91
Chapter 9
number of papers were published in this field before the Second World War. Important progress was however made by people like Prot [3], Weibull [4], [5], [6], Kjellman [7],Wastlund [8] etc.After 1945 the number of papers has increased steadily. At the University of Columbia, Freudenthal created an Institute for the Study of Fatigue and Reliability, andhe produced a great number of papers. The evolution of modern codes was greatlyinfluenced by papers published in 1949 - 1952 by Torkoja and Paez (see e.g. [9]).Johnson [10] suggested in 1953 to apply of extreme value distributions.Ferry Borges [11] stressed in 1952 the importance of taking into accountrandomness of dimensions and mechanical properties into randomness of structural behaviour. In a report published by Freudenthal [12] in 1968 it was suggested torepresent loads by an extreme value distribution and strength by a logarithmic normaldistribution.In 1968 Benjamin [13] in a paper on seismic force design defends the use of Bayes' probabilistic concepts.Presentation of load, strength and dimension variables by mean values andvariances was proposed by Cornell [14], [15] in 1967 and became the basis for the so-called level II methods. Important contributions in this area are due to Rosenblueth &Esteva [16] in 1972, Ditlevsen [17] in 1973, Hasofer & Lind [18] in 1974 and Vene-ziano [19] in 1974.It was in 1968 suggested by Ang & Amin [20] to split the total factor in two partswith the purpose of reducing the sensitivity of the type of distribution. Research by Rus-sian scientists in this field was published by Bolotin [21] in 1969.Only few papers published since 1970 are mentioned here. This is due to the factthat an enormous amount of papers has appeared in scientific journals or has been presented at conferences in the last ten years. It seems to be too early to try to evaluate indetails the great progress achieved in this decade but a number of scientists should bementioned here bearing in mind that some injustice will certainly be done by such achoice. However, it is reasonable to believe that recent work by the scientists Ang, Amin,Benjamin, Ferry Borges, Cornell, Crandall, Davenport, Ditlevsen, Galambos, Legere,Lind, Marshall, Meyerhof, Moses, Rachwitz, Ravindra, Rosenblueth, Sexsmith,Shah, Siu, Shinozuka, Turkstra, Vanmarcke, Veneziano will be remembered andappreciated in the future.In this chapter a brief introduction is given to some fundamental concepts andideas in the probabilistic approach to structural reliability with the purpose of facilitatingthe reading of the other papers in this book. The presentation is deliberately made veryinformal so that the main points are not drowned in details. It is intended to make the paper intelligible for a reader without previous knowledge in this field.To fulfill these intentions the following subjects are briefly treated. In section 2the most fundamental concepts related to random variables are introduced with specialemphasis on distribution functions and the first three moments. As a natural continuationdistribution of extreme values are treated in section 3. On this background reliability for single structural members is discussed in section 4. Finally, in section 5 the reliability of some simple structural systems is shortly touched on.A more extensive treatment can be found in the papers and books referred to inthe list of literature at the end of the paper.
92
Chapter 9
2. RANDOM VARIABLES
In this section some fundamental concepts concerning random variables will be brieflyreviewed. It is outside the scope of the paper to give a detailed presentation of thissubject. Let X be a random variable. Then the
distribution function F
X
of
X
is defined by
F
X
(
x
) =P[
x X
≤
] (1)where P[ ] means the probability that the event in [ ] is true.For a
continuous
random variable X the
density function, f
X
is defined by
dx xdF x f
X
)()(
=
(2)In the case of a
discrete
random variable
X
the distribution function is a staircasefunction with discontinuities at some points
x
i
and the density function is defined by
∑
−==
iii
x x X X x f
)()[P)(
δ
(3)where
δ
is Dirac's Delta function.In the rest of this section only the continuous case will be treated. It is oftenuseful to describe a random variable by its moments. The
moment m
n
of order n isdefined()
nn X
mxfxdx
∞−∞
=
∫
(4)The moment
m
1
is called the
expected value
or
mean
of
X
, and the following two symbolsare used
E[ ] ( )
X X
X xf x dx
µ
∞−∞
= =
∫
(5)Other useful moments are
central moments
defined by
0
()()
nn X X
mxfxdx
µ
∞−∞
= −
∫
(6)The moment
02
m
is called the
variance
of
X
and is denoted by
2
X
σ
or Var[
X
]. The squareroot
X
σ
of the variance is called the standard deviation. An important moment of thirdorder is the
skewness coefficient
X
ν
defined by
0 3 33
X X
m
ν σ
=
(7) Note that
m
n
= E[
X
n
] and
0
E[( ) ].
nn X
m X
µ
= −
(8)Perhaps the most important density function in structural reliability theory is theso-called
normal
distribution, defined by
2
)(21
21)(
σ µ
π σ
−−
=
x X
e x f
(9)
93
Figure 1.
Chapter 9
where
µ
and
σ
> 0 are parameters equal to
X
µ
and
X
σ
.
The
standardized normal density function
ϕ
is defined by
2
2
21)(
x
e x
−
=
π ϕ
(10)and the corresponding values
X
µ
and
X
σ
are 0 and 1. The
standardized normal distribution function
is denoted by
Φ
(see figure 1).It is not possible to mention all density functions of interest in structuralreliability theory. However, the distribution of extreme values of random variables will be more detailed treated in the next section and in this connection some frequently useddistributions will be presented.Some new concepts appear when two random variables
X
and
Y
are considered. Itis then easy to extend to more than two variables. The fundamental concept here is the
joint distribution function F
XY
defined
by
F
XY
(
x, y
) = P[
X
≤
x
,
Y
≤
y
] (11)If
F
XY
has partial derivatives of order up to two the
joint density function F
XY
is given by
y x y x F y x f
XY XY
∂∂∂=
),(),(
2
(12)Two random variables
X
and
Y
are called
independent
if for any
x
and
y
P[
X
≤
x, Y
≤
y
] =P[
X
≤
x
] P[
Y
≤
y
] (13)It is easy to see for independent random variables
X
and
Y
that
F
XY
(
x,y
) =
F
X
(
x
)
F
Y
(
y
) (14)where
F
X
and
F
Y
are the marginal distribution functions defined by),()(and),()(
∞=∞=
y F x F x F x F
XY Y XY X
(15)Further in this case
f
XY
(
x,y
) =
f
X
(
x
)
f
Y
(
y
) (16)where
f
X
and
f
Y
are the marginal density functions.The joint density function
f
XY
given by
21 2
-2(-)(-)-11221122(,)exp-[()-()]2212(1-)1122
XY
xxyy fxy
µ ρ µ µ µ σ σ σ σ πσ σ ρ ρ
= + −
(17)is called the
joint normal
density function. The corresponding marginal density functionsare normal density functions with
11
,
σ σ µ µ
==
X X
and
2 2
,
Y Y
µ µ σ σ
= =
.
Theremaining parameter
ρ
in (17) is called the
correlation coefficient
(see below).The expected value of a function
g
(
X, Y
) of two random variables
X
and
Y
isgiven byE[(,)](,)(,)
XY
gXYgxyfxydxdy
∞ ∞−∞ −∞
=
∫ ∫
(18)where
f
XY
is the joint density function of
X
and
Y
.
94

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