Direct probability integral method for stochastic response analysis of static and dynamic structural systems

The joint probability density function (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a frame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.

An intensity-modulated Gaussian process {x(t)} is defined as the product of a non-negative stationary random process {σx(t)} and a stationary Gaussian process {z(t)} that is statistically independent of {σx(t)}. When such an intensity-modulated process is the excitation to a linear time-invariant system, the response process {y(t)}, conditioned on the excitation-modulating function σx(t), is a generally nonstationary Gaussian process. This property is used to develop an approximation to the probability density function (PDF) of the system response by assuming that the time-varying conditional response variance σ2y(t), conditioned on the excitation-modulating function σx(t), is governed by the gamma PDF. The resulting response process PDF is characterized by two parameters: the mean and variance of the conditional response variance σ2y(t). An alternative approximation to the response PDF, characterized by these same two parameters, is developed by using a Taylor’s series representation in the variable σ2y of the response PDF conditioned on σ2y. This latter PDF approximation is shown to be identical to the Gram–Charlier series of type A through terms involving the Hermite polynomial of degree 6. The two response PDF approximations are shown to give essentially identical results for values of the coefficient of variation of σ2y of one-half or less. Simple, easily evaluated expressions for the mean and variance of σ2y are derived in terms of measurable quantities by utilizing the assumption that fluctuations in σx(t) occur slowly in comparison with those of z(t).

Stochastic response analysis of the scaled boundary finite element method and application to probabilistic fracture mechanics

A generalized method for the stationary probabilistic response of nonlinear dynamical system

Point group identification algorithm in dynamic response analysis of nonlinear stochastic systems

Constructing transient response probability density of non-linear system through complex fractional moments

This paper has the goal of defining a class of uncertain structural systems for which it is possible to consider an approach able to give the exact response in terms of the probability density function (PDF). The uncertain structures have been identified in the discretized statically determined ones and the approach has been identified in the coupling of the approximated principal deformation modes method (APDM) and of the probability transformation method (PTM). The first one gives the explicit relationships between the response variables and the uncertainty ones, that are exact when the structures are statically determined. The second method allows to determine the explicit relationship between the PDFs of the response and of the uncertainty variables. The results of some applications have confirmed the goodness of these choices and that the proposed approach gives always exact results for both correlated and uncorrelated uncertainty random variables.

An approximate analytical method is developed for calculating the probability density of the response of a Duffing oscillator subjected to narrow band random excitation when the stochastic jump phenomenon occurs. The stochastic jump phenomenon is characterized by the existence of multiple states of amplitude and sudden jumps between the states in the sample function of response. In this paper, the trigger for the stochastic jump is considered, and the probabilities of large and small amplitude states of the response are evaluated. The local probability density function of each state is separately assumed using the equivalent linearization method, and the probability density function of the overall response is derived by sum of these local probability density functions weighted with the probability of each state. In the illustrative example, the calculated probability density functions are compared with the Monte Carlo simulation results. It is shown that the present method well expresses the significant characteristic of the stochastic jump phenomenon.

RESPONSE STATISTICS OF OCEAN STRUCTURES TO NON-LINEAR HYDRODYNAMIC LOADING PART II: NON-GAUSSIAN OCEAN WAVES

Dynamic reliability analysis of Aerial Building Machine under extreme wind loads using improved QBDC-based active learning

Multi-scale stochastic dynamic response analysis of offshore risers with lognormal uncertainties

In this study stochastic analysis of non-linear dynamical systems under α-stable,
\nmultiplicative white noise has been conducted. The analysis has dealt with a special class of α-stable
\nstochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the
\nprobability density function or of the characteristic function of the dynamical response may be obtained
\nconsidering the dynamical system forced by a Gaussian white noise with an uncertain factor with α/2-
\nstable distribution. This consideration yields the probability density function or the characteristic function
\nof the response by means of a simple integral involving the probability density function of the system
\nunder Gaussian white noise and the probability density function of the α/2-stable random parameter. Some
\nnumerical applications have been reported assessing the reliability of the proposed formulation. Moreover
\na proper way to perform digital simulation of the sub-Gaussian α-stable random process preventing
\ndynamical systems from numerical overflows has been reported and discussed in detail.

Nonlinear stochastic seismic dynamic response analysis of submerged floating tunnel subjected to non-stationary ground motion

Stochastic ship roll motion via path integral method

Stochastic dynamics analysis for unilateral vibro-impact systems under combined excitation