2.2.1. Inverse First-Order Reliability Method
IFORM is the inverse method of the first-order reliability method (FORM). This method approximately calculates the long-term response extreme value of a complex structure through short-term response analysis [
23], such as the 3 h short-term response extreme value. It is assumed that the short-term sea state is a stationary random process, and the structural response extreme value is used as the analysis variable. Taking significant wave height
Hs and spectral peak period
Tp as an example, the cumulative distribution function (CDF) of
R3h is [
24]:
where
represents the CDF of the short-term response extreme value of the structure under a certain sea state
,
represents the probability that the short-term response extreme of the structure
is less than the given response level
r under any possible sea state, and
is the combination PDF of
Hs and
Tp.
From the reliability theory, assuming that
rcrit is the given critical response threshold, we consider that the structure would be secure when the response is below the failure boundary. So,
can be rewritten as
:
where
is the PDF form of
,
is the one-dimensional density distribution and conditional density distribution form of
.
The reliability method is widely applied to analyze the reliability index and failure probability of structures, so the structural limit state function can be transformed into the reliability form:
where
s is the random variables in the function
,
r represents the critical response to the structural failure,
is the response by the given loads
s. When
< 0, it indicates that the structure is in the failure state. In the case of the previous circumstance, the corresponding structural function can be expressed as:
Then, the structural failure probability can be estimated by:
The Rosenblatt transformation is used to convert the parameters in Equation (5) by the first-order reliability method, and the environmental variables are transformed into a space consisting of independent and standard normal variables
ui (i.e., the U space). The process of the Rosenblatt transform is [
22]:
where
is the standard normal distribution CDF. According to Equation (5), the transformed structural function is:
Then, Equation (6) can be rewritten as the expression of standard normal variables in three dimensions:
where
is the PDF of the standard normal distribution. Rosenblatt transformation converts the probability model in the real physical parameter space into the
U space; the design point is chosen as the closest point on the boundary to the origin, where coordinates in the
U space can be expressed as
. As shown in
Figure 1, according to the FORM, the structural failure probability can be simply estimated as [
24]:
The distance
from the design point
to the origin can be calculated by the following equation:
The structural failure probability is usually approximated by the annual exceedance probability. Usually, given an RP of N years, the total number of short-term sea states in the long term is judged, and the exceedance probability is defined as the RP in N years. M is the number of short-term sea states in one year.
After determining point
, delineate a sphere with the origin as the center and radius
in the
U space, and perform the inverse Rosenblatt transformation to return the point
on the circular failure boundary to the physical parameter space and obtain the corresponding point set
. The maximum value
of the response parameter in the dataset is the structural response extreme value, which can be obtained through retrieval. The corresponding point
is the structural design point. This process is called the first-order inverse reliability process, or inverse FORM. The process is shown in
Figure 2.
2.2.2. Inverse Second-Order Reliability Method
Since FORM will underestimate the true failure probability of the structure in some cases, SORM is an improved method that uses a specific second-order surface to approximate the failure surface at the design point. Regardless of the shape of the failure surface, because the approximate failure boundary is always convex, leading to the spherical safety domains being underestimated, the generated environment contours can always maintain a certain degree of conservatism.
According to Equations (6) and (9), it is necessary to solve the exceedance probability of the structure through the structural failure state to obtain the structural failure boundary. Different from the failure surface approximated by a linear function in the
U space via FORM, SORM’s approximate failure function
is a second-order function at the design point to approximate the failure boundary. The second-order failure boundary in the
U space can be expressed as [
5]:
Equation (12) represents the (n + 1)-dimensional variables, among which the non-linear variables have m dimensions, and the remaining (n + 1 − m)-dimensional variables are expressed in linear form. In Equation (11), , , (i = 1, 2, …, n + 1) and c are the correlation coefficients in the quadratic failure function.
SORM uses, as an origin, the center of the circle and a sphere with a radius equal to the distance between the origin and the design point to approximate the failure surface, thus approximating the real safety zone as the area covered by the sphere. Therefore, Equation (12) describes a special second-order failure boundary, where
= 1,
= 0,
= 0 and
c =
, so the structural failure probability of Equation (6) can be approximately expressed as:
Similar to IFORM, the environment contour under selected RPs via ISROM is an inverse reliability problem, in which a given failure probability
pf and the environment parameters causing the structure failure probability
pf are obtained. Then, an
n-dimensional sphere with a radius of
is created to apply the ISORM, and the radius
is calculated by:
For standard normal variables in the
U space,
obeys the chi-square distribution
(chi-square) with
n-dimensional degrees of freedom, so the radius
can be expressed as:
where
denotes the reliability index by solving the inverse transformation of the chi-square distribution in Equation (15). For the significant
Hs and
Tp parameters in this study, the degree of freedom of the
distribution is
n = 2; then,
pf is calculated using Equation (10).