Probabilistic Engineering Mechanics最新文献

To quantify the fatigue behaviour of materials, a Wöhler diagram is required. The state of the art shows that, over the years, numerous approaches suitable for determining Wöhler curves have been devised and validated through large fatigue data sets. The variation in experimental fatigue data elicits the use of statistics for analysis and design purposes. By focusing on the medium-cycle fatigue regime (i.e., failures in the range 10³÷10⁷ cycles to failure), this paper reviews relevant statistical approaches, particularly the methods suggested by the American Society for Testing Materials (ASTM) as well as the International Institute of Welding (IIW) and the so-called Linear Regression Method (LRM). Their responses were assessed on virtual data sets tailored to satisfy specific statistical requirements as well as experimental fatigue data sets from the literature. While the scatter bands at two times or less of the spread are similar for all approaches, the ASTM approach is seen to be the most conservative.

This work addresses the critical task of accurately estimating failure probabilities in dynamic systems by utilizing a probabilistic load model based on a set of data with similar characteristics, namely the relaxed power spectral density (PSD) function. A major drawback of the relaxed PSD function is the lack of dependency between frequencies, which leads to unrealistic PSD functions being sampled, resulting in an unfavorable effect on the failure probability estimation. In this work, this limitation is addressed by various methods of modeling the dependency, including the incorporation of statistical quantities such as the correlation present in the data set. Specifically, a novel technique is proposed, incorporating probabilistic dependencies between different frequencies for sampling representative PSD functions, thereby enhancing the realism of load representation. By accounting for the dependencies between frequencies, the relaxed PSD function enhances the precision of failure probability estimates, opening the opportunity for a more robust and accurate reliability assessment under uncertainty. The effectiveness and accuracy of the proposed approach is demonstrated through numerical examples, showcasing its ability to provide reliable failure probability estimates in dynamic systems.

Concrete is a multi-phase composite material that exhibits nonlinear and random characteristics in various contexts. The mesoscopic stochastic fracture model (MSFM) was developed to capture the constitutive behaviors of concrete. However, it is still not accurate enough to quantify the randomness of stress-strain curves in the ascending phase, and the variability of the strength might be considerably underestimated. In this paper, to remedy the above deficiencies, two alternative modifications to the MSFM are proposed. In the modified models, in addition to the random field of mesoscale fracture strain, Young's modulus of meso-springs is also quantified by a single random variable or a random field, respectively. The mathematical expressions for the mean and standard deviation of the uni-axial compressive stress-strain curves of concrete in the modified models are derived. Furthermore, based on the data from tested complete compressive stress-strain relationships of concrete with different strength grades, the parameters in the two modified MSFMs are identified by combining the genetic algorithm and a dimension-reduction algorithm. The results show that the accuracy of the modified models involving the randomness from both the mesoscale fracture strain and the mesoscale Young's modulus is greatly improved compared to the existing MSFM in capturing both the variability of concrete strength and the standard deviation in the ascending phase of the stress-strain relationship of concrete.

Rare events induced by random perturbations are ubiquitous phenomena in natural systems, where the exit location distribution is a significant quantity, and its computation is challenging. In this study, we compute the exit location distribution of stochastic dynamical systems with weak Gaussian noise for a noncharacteristic boundary based on deep learning and large deviation theory. First, we introduce the perturbation expressions of the prefactor and exit location distribution via Wentzel–Kramers–Brillouin approximation. We then design a deep learning method to compute the quasipotential, the prefactor, and the exit location distribution. Two examples are described to verify the effectiveness of the proposed algorithm. The findings of this study are expected to provide valuable insights into exploring the mechanisms of rare events triggered by random fluctuations.

Random fields are widely used to represent the uncertainty of some parameters in engineering, and numerous simulation approaches have been developed for Gaussian and non-Gaussian random fields. However, the unified methods among them suffer from low computational accuracy and efficiency or discontinuities in the simulated random fields. Therefore, an easy-to-implement general simulation method based on the active learning Kriging model is proposed for a one dimensional(1D) Gaussian or non-Gaussian random field in this paper. In the proposed method, there are two stages. One stage, called the inner loop, is to construct the Kriging approximation of a random field sample with enough accuracy by some samples of the random variables at some discretized locations, in which an active learning strategy based on the error estimation for the Kriging model is introduced to select adaptively the added locations, and a fast sampling method is presented to determine efficiently the samples at the added locations. In the other stage, called the outer loop, some random field samples are represented accurately by their corresponding Kriging approximations through training iteratively. Furthermore, several numerical examples are presented to show the accuracy, effectiveness and generality of the proposed method for 1D Gaussian and non-Gaussian random fields by comparing with the Karhunen–Loève(KL) expansion method. Meanwhile, the effects of the types of correlation function and the scales of fluctuation on the simulation results are analyzed.

In this study, an analytical model is presented to determine the random response of point-driven portal and multi-bay planar frame structures. Coupling effects between bending and longitudinal deformations are taken into account, with the Timoshenko-Ehrenfest beam theory being applied to model the bending deformations. With the excitation taken as band-limited white noise, expressions are derived for the mean square displacements and velocities in terms of the autocorrelation and cross-correlation components. The influence of modal cross-correlations on the overall response is shown to be dependent on the number of bays. For a lightly damped single-bay frame, the natural frequencies are generally well separated and the modal cross-correlations are small. In this situation, the velocity response displays a near symmetric distribution about the center point of the frame. Moreover, narrow zones of intensified response begin emerging as the number of responding modes increases. For frames having two or more bays, the contribution of modal cross-correlations is larger due to the increased occurrence of clusters of natural frequencies. In such cases, modal cross-correlations introduce asymmetry into the overall response distribution of the frame. Additionally, the drive-point velocity of the multi-bay frame can be severely underestimated if modal cross-correlations are ignored. The study also investigates the influence of increased damping on the response characteristics.

The estimation of the posterior probability density function (PDF) of unknown parameters remains a challenge in stochastic nonlinear model updating with uncertainties; thus, a novel Bayesian inference framework based on the Gaussian process metamodel (GPM) and the advanced Markov chain Monte Carlo (MCMC) method is proposed in this paper. The instantaneous characteristics (ICs) of the decomposed measurement response, calculated using the Hilbert transform and the discrete analytical mode decomposition methodology, are extracted as nonlinear indices and further used to construct the likelihood function. Then, the posterior PDFs of structural nonlinear model parameters are derived based on the Bayesian theorem. To precisely calculate the posterior PDF, an advanced MCMC approach, i.e., delayed rejection adaptive Metropolis-Hastings (DRAM) algorithm, is adopted with the advantages of a high acceptance ratio and strong robustness. However, as a common shortage in most MCMC methods, the resampling technology is still applied, and numerous iterations of nonlinear simulations are conducted to ensure accuracy, thus directly reducing the computational efficiency of the DRAM. To address the abovementioned issue, a mathematical regression metamodel of the GPM with a polynomial kernel function is adopted in this paper instead of the traditional finite element model (FEM) to simulate a nonlinear response for the reduction of computational cost, and the hyperparameters are further estimated using the conjugate gradient optimization methodology. Finally, numerical simulations concerning a Giuffré–Menegotto–Pinto modeled steel-frame structure and a seven-storey base-isolated structure are conducted. Furthermore, a shake-table experimental test of a nonlinear steel framework is investigated to validate the accuracy of the Bayesian inference method. Both simulations and experiment demonstrate that the proposed GPM and DRAM-based Bayesian method effectively estimates the posterior PDF of unknown parameters and is appropriate for stochastic nonlinear model updating even with multisource uncertainties.

A phenomenological study is carried out to speculate the statistical impacts of the CNT/polymer interphase on the overall electro-elastic behavior of piezo-polymer nanocomposites by presenting a full-field micromechanical model. The nanocomposite system consists of carbon nanotube (CNT) and PVDF. Various statistical distributions, including Weibull, log-normal, normal, beta, and uniform distributions on the thickness and strength of the interphase are carefully assessed. The results are compared with experimental data, and satisfactory agreements are reported. It is found that, compared to the random distribution of the interphase strength, the statistical distribution of the interphase thickness has more effect on the overall electro-elastic properties. For example, for the effective longitudinal modulus, the overall coefficients of variation are 14 %, 13 %, 13.56, and 10 %, respectively, for the normal, Weibull, beta, and uniform distributions of the thickness compared with the measured experimental values. Also, the effects of the CNT content, aspect ratio, and orientation on the effective electro-elastic properties by considering the various random distributions are fully examined. Moreover, using the Monte Carlo simulation, the probability of not meeting design specification (failure probability) is evaluated at the random distributions of the interphase strength and thickness to identify the optimum CNT content for which the values of the overall properties are maximum. It is obtained that the failure probabilities are different for 5–8 % CNT volume fraction in the distributions of the thickness, and for only 5 VF% CNT in the strength distributions. For other values of the CNT content, the failure probabilities are independent of the distribution of the interphase strength and thickness.

Bayesian operational modal analysis makes inference about the modal properties (e.g., natural frequency, damping ratio) of a structure using ‘output-only’ ambient vibration data. With sufficient data in applications, the posterior probability density function (PDF) of modal properties can be approximated by a Gaussian PDF, whose covariance matrix is given by the inverse of the Hessian of negative log-likelihood function (NLLF) at the most probable value. Existing methodologies for computing the Hessian are based on semi-analytical formulae that offer an efficient and reliable means for applications. Inevitably, their computer coding can be involved, e.g., a mix of variables with different sensitivities, singularity of Hessian due to constraints. In the absence of analytical or numerically ‘exact’ result for benchmarking, computer code verification during development stage is also non-trivial. Currently, finite difference method is often used as the only and last resort for verification, although there are also difficulties in, e.g., the choice of step size, and criterion for comparison/convergence. Motivated by these, this work explores an identity in the theory of Expectation-Maximisation (EM) algorithm to provide an alternative means for evaluating the Hessian of NLLF. Such identity allows one to evaluate the Hessian by means of Monte Carlo simulation, averaging over random samples of hidden variables. While the existing semi-analytical approach is still preferred for Hessian calculations in applications for its high definitive accuracy and speed, the proposed Monte Carlo solution offers a convenient means for counter-checking during code development. Theoretical implications of the identity will be discussed and numerical examples will be given to illustrate implementation aspects.

Probability distributions of responses have been widely used in structural analysis and design because of their complete statistical information. In practice, the dimensionality of input variables could easily reach hundreds or thousands, making it computationally expensive to obtain accurate distributions. In this paper, a generalized most probable point (MPP) method is developed to effectively build the response distributions of high-dimensional problems. First, a global index based on one-iteration MPPs is presented for dimension reduction, which is to divide the input variables into important and unimportant variables. After fixing the unimportant variables at their one-iteration MPP components, the MPP components of the important variables are obtained by performing the inverse first-order reliability method (FORM) in the reduced space. Predictive models of the all MPP components are then established to quickly estimate the MPPs of other cumulative distribution function (CDF) values. To accurately calculate CDF points of limit state functions with different shapes, a comprehensive uncertainty analysis method that accommodates the contributions of the important and unimportant variables is proposed by multiple combinations of FORM, second-order reliability method, and second-order saddlepoint approximation. Finally, the response distributions are generated based on Gaussian mixture distribution and all CDF points. The effectiveness of the proposed method is verified by a mathematical example and two engineering cases.