Page 22 - Dynamic Loading and Design of Structures

Page 9

its distance from the origin is clearly equal to

This scalar quantity is known as the Hasofer-Lind reliability index β , i.e.
HL

(1.12)

Note that u* can also be written as

(1.13)

where α=(α, α,…, α) is the unit normal vector to the limit state surface at u*, and, hence,
2
1
n
α(i=1,…, n) represent the direction cosines at the design point. These are also known as the
i
sensitivity factors, as they provide an indication of the relative importance of the uncertainty
in basic random variables on the computed reliability. Their absolute value ranges between
zero and unity and the closer this is to the upper limit, the more significant the influence of the
respective random variable is to the reliability. In terms of sign, and following the convention
adopted by ISO (1998), resistance variables are associated with positive sensitivity factors,
whereas leading variables have negative factors.

(3) Approximation techniques
Once the checking point is determined, the failure probability can be approximated using
results applicable to the standard normal space. In a first order (linear) approximation, the
limit state surface is approximated by its tangent hyperplane at the design point. The
probability content of the failure set is then given by

(1.14)

In some cases, a higher order (quadratic) approximation of the limit state surface at the design
point is desired but experience has shown that the FORM result is sufficient for many
structural engineering problems. Equation (1.14) shows that, when using the so-called
asymptotic approximate methods, the computation of reliability (or equivalently of the
probability of failure) is transformed into a geometric problem, that of finding the shortest
distance from the origin to the limit state surface in standard normal space.

Simulation methods
In this approach, random sampling is employed to simulate a large number of (usually
numerical) experiments and to observe the result. In the context of structural reliability, this
means, in the simplest approach, sampling the random vector X to obtain a set of sample
values. The limit state function is then evaluated to