Page 25 - Dynamic Loading and Design of Structures

Page 12

the most likely failure point. Since the objective of a deterministic code of practice is to
ascertain attainment of a limit state, it is clear that any check should be performed at a critical
combination of loading and resistance variables and, in this respect, the design point values
from a reliability analysis are a good choice. Hence, in the deterministic safety checking
format, eqn (1.4), the design values can be directly linked to the results of a reliability analysis
(i.e. Pf or βand α is). Thus, the partial factor associated with a basic random variable Xi, is
determined as

(1.15)

where xdi is the design point value and xki is a characteristic value of Xi. As can be seen, the
design point value can be written in terms of the original distribution function Fx( · ), the
)
reliability analysis results (i.e. βand α, and the standard normal distribution function ( · ).
i
If X is normally distributed, eqn (1.15) can be written as (after non-dimensional-izing both
i
x and X with respect to the mean value)
ki
di
(1.16)

where v is the coefficient of variation and k is a constant related to the fractile of the
Xi
distribution selected to represent the characteristic value of the random variable Xi. As shown,
eqns (1.15) and (1.16) are used for determining partial factors of loading variables, whereas
their inverse is used for determining partial factors of resistance variables. Similar expressions
are available for variables described by other distributions (e.g. log-normal, Gumbel type I)
and are given in, for example, Eurocode 1 (European Standard, 2000). Thus, partial factors
could be derived or modified using FORM/SORM analysis results. The classic text by Borges
and Castanheta (1985) contains a large number of partial factor values assuming different
probability distributions for load and resistance variables (i.e. solutions pertinent to the
problem described by eqn (1.6b)). If the reliability assessment is carried out using simulation,
sensitivity factors are not directly obtained, though, in principle, they could be through some
additional calculations.

1.3.3 Reliability differentiation
It is evident from eqns (1.15) and (1.16) that the reliability index βcan be linked directly to
the values of partial factors adopted in a deterministic code. The appropriate degree of
reliability should be judged with due regard to the possible consequences of failure and the
expense, level of effort and procedures necessary to reduce the risk of failure (ISO, 1998). In
other words, it is now generally accepted that ‘the appropriate degree of reliability’ should
take into account the cause and mode of failure, the possible consequences of failure, the
social and environmental conditions, and the cost associated with various risk mitigation
procedures (ISO,