Reliability Analysis Of Nonlinear Structures Research Articles

Adaptive Hermite Distribution Model with Probability-Weighted Moments for Seismic Reliability Analysis of Nonlinear Structures

In this paper, an adaptive Hermite distribution model with probability-weighted moments (PWMs) is proposed for evaluating the extreme-value distribution (EVD) of response, which serves as the basis of seismic reliability analysis of complex nonlinear structures under random seismic excitations. From the perspective of EVD, the problem formulation is first introduced. Then, an adaptive distribution model, named as the adaptive Hermite polynomial normal transformation model (A-HPNT), is established to estimate the EVD. The undetermined coefficients of A-HPNT are specified via the PWMs matching technique, in which only linear systems of equations need to be solved. To optimally determine the degree for A-HPNT, a two-step criterion is effectively established accordingly. An efficient high-dimensional sampling technique is introduced for generating samples of extreme value, estimating both the PWMs and statistical moments of EVD. When the entire distribution of EVD is recovered, one can compute the failure probability and reliability index via an integral over the EVD. Two numerical examples, a 10-story nonlinear shear frame structure and a practical 13-story reinforced concrete frame-shear wall structure driven by random seismic excitations, are presented to verify the efficacy of the proposed method for seismic reliability evaluation of complex nonlinear structures.

Efficient seismic reliability analysis of non-linear structures under non-stationary ground motions

The seismic reliability assessment (SRA) of complex nonlinear structures which possess structural and seismic uncertainty is an open challenge for academics and practical engineers. This paper proposes an improved maximum entropy method (IMEM) for the SRA of nonlinear structures. Non-stationary ground motions are first generated via a random-function-based spectral representation method. High-dimensional random variables in the ground motion simulation are then converted into two basic random variables as the high-dimensional probability space is reduced to a low-dimensional probability space. A two-step IMEM constrained with fractional moments is deployed to estimate the extreme value distribution (EVD) of the nonlinear structural dynamic system. An improved Latin hypercube sampling approach is also established to calculate the fractional moments of the complex nonlinear dynamic system. The first-passage probability of the structural nonlinear seismic response can be computed conveniently by a numerical integration using the EVD. Two numerical examples, 1) a three-story nonlinear shear frame structure and 2) a real-world high-pier continuous rigid frame bridge model, both of which exhibit strong nonlinearity under seismic action, are analyzed to show that the proposed method is efficient and accurate.

Dynamic reliability analysis of nonlinear structures using a Duffing-system-based equivalent nonlinear system method

To improve the analysis accuracy of dynamic reliability of a nonlinear system, an equivalent nonlinear system method is presented. In this method, general nonlinear systems are converted to equivalent Duffing nonlinear systems according to the minimum mean square error criterion, whose exact analytical solution of the random steady-state responses can be determined by a FokkerPlanckKolmogorov equation. Then the exact results of stochastic responses are used to analyze structural dynamic reliability. To use the equivalent nonlinear system method to analyze structural dynamic reliability is not only convenient for calculation but highly accurate. In addition, the presented equivalent nonlinear system has a parameter ε, which controls the degree of nonlinearity. Thus, it is easy to obtain the corresponding analysis results from converting the original system to equivalent nonlinear systems with different degrees of nonlinearity by changing the value of ε. Especially, when ε is zero, the equivalent nonlinear system will degenerate to the linear system, and leading to the analysis results of the equivalent linearization method. An example shows that the analysis results of the proposed equivalent nonlinear system method are reliable, and the calculation appears to be more accurate than that of the equivalent linearization method.

A mixture distribution with fractional moments for efficient seismic reliability analysis of nonlinear structures

In this paper, an efficient approach is proposed for seismic reliability analysis of nonlinear structures with random parameters subjected to non-stationary stochastic ground motions. First, the first-passage reliability problem is equivalently transformed to the evaluation of the extreme value distribution (EVD) of the response. A mixture of inverse Gaussian and Lognormal distributions (MIGLD) is then proposed to reconstruct the EVD in the entire distribution domain, where the fractional moments are suggested as constraints to specify the unknown parameters. Only five low-order fractional moments of the EVD are actually required in the proposed method due to the inherent advantages of fractional moments. Then, the recently developed Latinized partially stratified sampling (LPSS) approach, is introduced to evaluate the fractional moments of the EVD with a small sample size. In this regard, the EVD could be reconstructed accurately in the entire distribution domain with high efficiency, particularly in the distribution tail, and the corresponding failure probabilities can be obtained in a straightforward way. Two numerical examples involving both linear and nonlinear shear-frame structures under non-stationary stochastic seismic ground motions are investigated to verify the efficacy of the proposed approach. The results indicate that the proposed method can result in accurate seismic reliability of nonlinear structures with high efficiency.

Collapse and allowable displacements in the context of reliability analysis of nonlinear structures

In this work, we study the relation between collapse and allowable displacements in the context of reliability analysis of nonlinear structures. We show that the probability of failure according to some finite allowable displacement is an upper bound for the probability of collapse. Consequently, it is possible to obtain safe estimates for the reliability according to collapse considering only allowable displacements limit state functions. This idea can be particularly useful in cases where it is computationally efficient to work with allowable displacements. The results are also important from the conceptual point of view. Three numerical examples are presented to illustrate the developed concepts.

Evolution of probability densities in the phase space for reliability analysis of non-linear structures

In this paper a numerical solution procedure is presented to compute the evolution of the probability density in the state space for a single degree of freedom system excited by white noise. Furthermore, crossing rates and first excursion rates are evaluated for required thresholds. This work aims at an accurate reference solution, providing a basis for the comparison with Monte Carlo based procedures for the first passage problem of non-linear systems. The densities are represented as a continuous surface in logarithmic scale, specified in terms of discrete points in the state space. For interpolation, the state space is discretized by isoparametric elements, which allows one to represent piecewise quadratic functions and hence to represent Gaussian densities exactly. The random loading is represented by impulses, which allows for a separation of the dynamics into a loading and free motion part within small time steps. The impulsive loading is applied independently at discrete instants by a convolution in the direction of the velocity. The excitation can be additive and multiplicative white noise. It is shown here that a linear transformation has the capability to represent the free motion of the densities within the short time step, i.e. π ( t + Δ t ) = A π ( t ) , where π is the vector of the probability densities and the matrix A captures the non-linear dynamics of the system in the state space. For a verification of the algorithm, the analytically known crossing rates of a Duffing oscillator under stationary excitation are compared. In a next step, absorbing boundaries are introduced to study first excursion rates. The results are then compared with direct Monte Carlo simulation.

The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters

A new approach for evaluation of the extreme value distribution and dynamic reliability assessment of nonlinear structures with uncertain parameters is proposed. The approach is established based on the thoughts of the newly developed probability density evolution method, which is capable of capturing the instantaneous probability density function (PDF) and its evolution of the responses of nonlinear stochastic structures. In the proposed method, a virtual stochastic process associated to the extreme value of the studied stochastic process is firstly constructed in such a way that the extreme value equals the value of the virtual stochastic process at a certain “instant of time”. The probability density evolution method is then employed to evaluate the instantaneous PDF of the virtual stochastic process. This will yield the PDF of the extreme value as a natural byproduct. After that, dynamic reliability could be evaluated from the extreme value distribution, instead of the level-crossing process theory. A simple integration in terms of the extreme value distribution over the safe domain will give the dynamic reliability, requiring neither the joint PDF of the response and its velocity, nor the assumptions on properties of the level-crossing events. Numerical algorithm is outlined. Two examples, of which one is to capture the extreme value distribution of the random sampling, the other is to evaluate the extreme value distribution and dynamic reliability of a nonlinear stochastic structure, are studied. Some features of the instantaneous PDF and its evolution, the extreme value distribution and the dynamic reliability are discussed. The investigations indicate that the proposed approach is of versatility, accuracy and efficiency.

METHODS FOR SEISMIC RISK ANALYSIS: STATE OF THE ART VERSUS ADVANCED STATE OF THE PRACTICE

The core of the paper consists of the illustration of a method for seismic reliability analysis of non-linear structures. The method, which is a development over a recent previous proposal, is quite comprehensive in scope since it includes consideration of randomness in the input, in the mechanical properties of the structure and in the limit-state, or capacity, conditions. Essentially, the problem is formulated as the out-crossing of the response process out of a (scalar) safe threshold, and this problem is solved by time-invariant FORM methods. An application to an idealised five-storey building demonstrates the salient theoretical and computational features of the method. The new approach is presented in a broader framework, which involves a discussion on present trends and capabilities in the area of probabilistic seismic design. For the sake of this discussion, two approaches just appeared in the literature are outlined, which are less demanding from a theoretical standpoint, and hence closer to engineering practice, but also less general in scope. It is argued that their introduction on code-assisted design would be feasible immediately with obvious advantages, while approaches of more rigorous nature would be given some more time to mature and to become more accessible to professional use.