^{1}

^{2}

^{3}

^{2}

^{4}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

A widely used approach for the first crossing reliability evaluation of structures subject to nonstationary Gaussian random input is represented by the direct extension to the nonstationary case of the solution based on the qualified envelope, originally proposed for stationary cases. The most convenient way to approach this evaluation relies on working in the time domain, where a common assumption used is to adopt the modulation of stationary envelope process instead of the envelope of modulated stationary one, by utilizing the so-called “preenvelope” process. The described assumption is demonstrated in this work, also showing that such assumption can induce some errors in the envelope mean crossing rate.

One of the most important indexes for engineers and designers is the accurate quantification of structural safety. Its correct evaluation is connected with loads’ nature and failure type. Moreover, a wide class of engineering problems deals with reliability evaluation of structures subject to loads characterized by an intrinsic probabilistic nature and whose correct description could be obtained by means of random process. Besides, there are situations where suitable modelling of these loads requires taking their nonstationary characteristics into account [

In these cases, the probabilistic characterization of structural loads turns the random dynamic analysis to be the most useful method in achieving an accurate and qualified structural reliability evaluation [

In particular for Gaussian inputs and linear systems, the first- and second-order moments completely define the statistics of the response. This problem does not encounter serious difficulties in both spectral and time domains relative to a stationary case and with regard to first crossing failure. However, the application of Rice’s reliability formulation presents some problems related to the required knowledge of hazard function, defined as the probability density function (PDF) of collapse at a given instant, under the restriction that crisis has not happened yet. Due to complexity in its evaluations, a simplified approach is based on the independent crossing hypothesis, using then an unconditioned PDF as hazard function. This hypothesis, which assumes threshold crossing distributed as a Poisson process, can be acceptable only for wideband process and high barrier values, where barrier crossing can be really assumed to be independent. Besides these restrictions, Poisson approach is too poor and conservative for a correct reliability evaluation [

A more recent approach for the first passage problem has been developed in [

Among different methods, an approach, widely used nowadays, is that proposed by Vanmarcke [

Mechanical safety or reliability _{0}, _{0} is typically the initial observation time assumed as the end of the manufacturing process. In this situation, the collapse probability at time _{0} is usually assumed equal to 0. It is clear that the definition of structural collapse (or in general, failure) plays an important role in reliability assessment, considering the numerous possible meanings (not only of mechanical kind) of this condition. Two of the most common mechanical undertakings are related to fatigue phenomenon and first threshold crossing failure.

In this section, a simple one-degree-of-freedom, viscous-elastic system subject to random dynamic actions is analysed, for which failure is related to displacement barrier crossing. Precisely, the system crisis happens when structural displacement _{0},

An alternative method is using its complementary collapse probability (failure) _{f}(

The hazard function

Using (

The formal solution of (

The exact hazard function formulation is still an open question due to the fact that it has been solved only in few cases. In the original Rice’s formulation [

A frequently used solution to overcome the limitation discussed above is obtained by using the approach of Vanmarcke [

For a stationary process, the formulation that furnishes the hazard function for a bilateral threshold level

And the single barrier mean crossing rate of the envelope process

Its evaluation requires the JPDF

In a stationary condition, the envelope process can be defined as the modulus of a complex process whose real part is the original one and the imaginary part has to be opportunely defined in order to satisfy the following equation:

The choice of

White noise or filtered Gaussian uniform modulated processes is a widely used model for representing nonstationary loads [

The stationary process is usually represented by a zero-mean white noise process

A multiplicative factor equal to

In this case,

In a more compact form,

The covariance matrix of the stationary vector process

In a nonstationary condition, the structural response envelope can be defined in a way similar to the stationary case, by introducing a nonstationary complex process

Using a nonstationary modulated process input

In this way, it can be written in [

By extending the following properties to a nonstationary case:

As a main direct consequence of the above assumptions, working with the two-state space vectors is expressed in [

The correlation matrix is given by

Similarly the matrix

Then, it is possible to write the differential equations in the space state as

And the input vectors

As shown earlier in (

The covariance matrix is given by

It can be rewritten in a more compact form by decomposing the 4 × 4 matrix into four 2 × 2 submatrices [

Using the mean value Hilbert properties previously shown, it is possible to demonstrate that the 4 submatrices are two by two equal (

Therefore,

The first one can be obtained using the well-known Lyapunov differential matrix covariance equation:

Due to the symmetry of

By applying a similar approach, it is possible to obtain the matrix

By making its first-order temporal derivation, it can be written as

Moreover, by adopting the two-state vector differential equation [

So finally obtaining the following

Since the two last matrices are asymmetrical, it results as

Since

In this case, ^{4} is a standard integral.

Equation (

The covariance matrix

The covariance matrix

As demonstrated before, a central point of Vanmarcke formulation for hazard function evaluation is the knowledge of the preenvelope process mean threshold crossing rate

By applying the definition of the preenvelope process through the original process

By introducing the following vector:

By using the preenvelope density function

By extending the original approach proposed by Vanmarcke [

It can be rewritten as:

Expression in (

Factor

This formulation can be improved by introducing the parameter

In the nonstationary case, the parameter

The mean threshold crossing rate

Different modulation functions have been proposed in the scientific literature to model the behavior of nonstationary loads in a suitable way [

Parabolic modulation function for different values of _{m}.

Regarding the specific modulation function in (

Structural characteristics differ for damping ratio (_{0}:_{m} used in this study are 2.5, 5, 10, and 15.

In addition to evaluating the structural response in the “classical” space state _{m} = 10_{0}) and for low damping. In this specific case, there is a separation between the two results at the end when the approximate solution has a strongly oscillatory shape. This result is inferred more clearly in Figures

Structural displacement

Structural displacement

Approximate and exact nonstationary solutions for _{max} = 2.5_{0}.

Approximate and exact nonstationary solutions for _{max} = 10_{0}.

JPDF of _{max} = 2.5_{0}.

JPDF of _{max} = 2.5_{0}.

JPDF of _{max} = 10_{0}.

JPDF of _{max} = 10_{0}.

Furthermore, the good accordance between the approximate and exact numerical evaluations of

For a better evaluation of differences for threshold crossing problem deriving by the various approaches, in Figures

Mean threshold crossing rate for _{max} = 5_{0}.

Mean threshold crossing rate for _{max} = 10_{0}.

Finally, in order to evaluate the empirical correction exponent

Numerical and analytical results in function of

This study dealt with the first crossing reliability evaluation of structures subject to nonstationary Gaussian random input. The solution based on the qualified envelope, originally derived for stationary cases, has been extended to the nonstationary case. At this aim, the modulation of stationary envelope has been adopted by utilizing the so-called “preenvelope” process. A modified reliability formulation has also been proposed, based on the extension of the formulation of the empirical bandwidth factor exponent developed under the stationary hypothesis.

To evaluate the accuracy of the reliability evaluation achieved by using the previous assumption, the mean crossing rate and the envelope JPDF obtained by numerical simulations for a SDOF system subject to Gaussian white noise have been compared to the ones obtained in analytical way. Different damping ratios and velocities in modulation amplitude variation have been taken into account. Results show that only a partial agreement is obtained, based not directly on the final reliability, but also on a suitable measure of accuracy of hazard function. These analyses are carried out on a wide number of cases taking into account different damping factors and nonstationary levels.

Aiming at performing a parametric analysis on nonstationary input characterization, in the case of load conditions such as earthquakes [

Therefore, it is possible to rewrite

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.