Stochastic Structural Analysis Research Articles

Small failure probability analysis of stochastic structures based on a new hybrid approach

The small failure probability problem of stochastic structures is investigated by using two types of surrogate models and the subset simulation method in conjunction with parallel computation. To achieve high computational efficiency, the explicit expression of dynamic responses of stochastic structures is first derived in the form based on the explicit time-domain method. Then, the small failure probability analysis of stochastic structures is efficiently carried out through the Monte Carlo simulation method utilizing explicit expressions. To avoid the repeated calculation for the coefficient matrices or vectors of the explicit expression of stochastic structures, two types of surrogate models, e.g., the backpropagation neural network model and the Kriging model, are introduced to obtain these matrices or vectors for each parameter sample of the stochastic structures. The computational cost is further reduced by using the subset simulation method to generate conditional samples which follow the rule of Metropolis-Hastings. Furthermore, in virtue of the independence of the surrogate models for each time instant and the independence of dynamic analysis for each sample, parallel computation is embedded in the proposed approach, which can fully exploit the characteristics of the proposed approach and further improve the computational efficiency of dynamic reliability analysis. Numerical examples are given to illustrate the validity of the proposed hybrid approach.

A comprehensive study on seismic dynamic responses of stochastic structures using sparse grid-based polynomial chaos expansion

The seismic response is a critical aspect of structural engineering, as it directly influences the design and safety evaluation of such systems. Structural seismic response becomes more complicated when unavoidable uncertainties in the structure are considered. Therefore, solving and analyzing the stochastic dynamic response of structures efficiently and accurately is of great significance. Various uncertainty quantification methods such as the stochastic finite element method, Monte Carlo method, and polynomial chaos expansion method have made significant progress in this field. However, these methods do not provide a comprehensive study of the seismic response of stochastic structures and efficiency problems rapidly emerge as the number of random variables increases. This paper delivers a comprehensive study of the seismic dynamic response of space truss and reticulated shell structure considering uncertainties in material properties, geometry, and loading. To this end, the finite element model of the structure was first modeled and the deterministic solutions were obtained. Then, polynomial surrogates are modeled for the natural frequency, mode participation factor, mode superposition response, and transient response of the structure under stochastic uncertainties. The sparse grid technique is used to select sparse quadrature points, which overcomes the curse of dimensionality caused by full grid integration. The results demonstrate that the sparse grid-based polynomial chaos expansion is more efficient. Uncertainty quantification reveals the uncertainty of modal analysis, response spectrum analysis, and transient response analysis of space truss and reticulated shell under seismic excitation. In summary, this study provides valuable insights and an efficient and accurate uncertainty quantification tool for the seismic dynamic analysis of stochastic structures, ultimately contributing to safer and more robust structural designs.

Quantifying variation in maximum floor accelerations of modular buildings under earthquakes through stochastic nonlinear structural analysis

For earthquake-resistant design of modular buildings, reinforced concrete (RC) cores can be a viable choice as the vertical seismic force resisting system. The maximum floor acceleration determines the maximum inertial force that will be transferred by the connections in the horizontal diaphragm to the RC walls. However, the randomness in the structural properties of RC walls may result in fluctuations in the floor acceleration response. Given the limited number of horizontal inter-module connections in modular buildings, the failure of these connections can have severe consequences. To ensure adequate safety margins in the design of horizontal diaphragm connections, it is essential to account for the variations in maximum floor accelerations. To quantify these variations, a 9-story prototype building with RC walls was examined in this study. Monte Carlo simulations were conducted on sample structures with univariate and bivariate randomness in fc (concrete cylinder compressive strength) and Fy (yield strength of steel rebars) of RC walls. Nonlinear response history analyses were performed for each sample structure using 22 ground motions to account for uncertainty in earthquake excitation. The results demonstrated significant and highly complex variations in the maximum floor accelerations due to the randomness considered. A coefficient of variation (CoV) of 0.13 was determined for the maximum floor acceleration response resulting from the bivariate randomness in fc and Fy. Furthermore, the variations were found to increase as the correlation scale in the spatial variation of fc increased. Based on the findings, it is proposed to apply an increase ratio of 1.2 to the maximum floor acceleration to account for the randomness of RC wall properties to ensure adequate safety margins in the design of horizontal diaphragm connections.

Solution of FPK equation for stochastic dynamics subjected to additive Gaussian noise via deep learning approach

The Fokker-Plank-Kolmogorov (FPK) equation is an idealized model representing many stochastic systems commonly encountered in the analysis of stochastic structures as well as many other applications. Its solution thus provides an invaluable insight into the performance of many engineering systems. Despite its great importance, the solution of the FPK equation is still extremely challenging. For systems of practical significance, the FPK equation is usually high dimensional, rendering most of the numerical methods ineffective. In this respect, the present work introduces the FPK-DP Net as a physics-informed network that encodes the physical insights, i.e. the governing constrained differential equations emanated out of physical laws, into a deep neural network. FPK-DP Net is a mesh-free learning method that can solve the density evolution of stochastic dynamics subjected to additive white Gaussian noise without any prior simulation data and can be used as an efficient surrogate model afterward. FPK-DP Net uses the dimension-reduced FPK equation. Therefore, it can be used to address high-dimensional practical problems as well. To demonstrate the potential applicability of the proposed framework, and to study its accuracy and efficacy, numerical implementations on five different benchmark problems are investigated.

A conditional extreme value distribution method for dynamic reliability analysis of stochastic structures

An efficient post-processing simulation method is proposed to estimate small failure probabilities of stochastic dynamic structures involving the inherent randomness of structural physical-geometrical parameters and external excitations. To extract a small failure probability, the proposed method introduces an intermediate event to represent the realizations of extreme structural response in the tail of distribution. With the aid of this intermediate event, structural failure probability is reformulated as a product of the event’s occurrence probability and the conditional exceedance probability of extreme response when the event occurs. The latter corresponds to a distribution of extreme response under the condition of the intermediate event, referred to as the conditional extreme value distribution (CEVD). Accordingly, the proposed method is termed the CEVD method. To reconstruct the CEVD, a truncated shifted generalized lognormal distribution model is employed. Bayesian estimation method is utilized to determine the two shape parameters of this model based on the samples of both original extreme value distribution and CEVD, where the CEVD samples are generated by Markov chain Monte Carlo sampling. The efficiency and accuracy of the proposed method are demonstrated through two numerical examples considering seismic reliability analyses of a 10-story nonlinear frame and a soil-foundation-structure interaction system.

Spatial stochastic D-RBA and limit equilibrium analysis of unreinforced masonry pier-spandrel structures

This study investigates the effect of spatially varying material properties on the in-plane seismic behaviour and capacity of an unreinforced pier-spandrel masonry system. The discontinuous nature of masonry is represented via a system of rigid blocks that can mechanically interact with each other along their boundaries. In the discontinuous modelling strategy, denoted as discrete rigid block analysis (D-RBA), deformations are lumped at the joints, and masonry units are replicated using two rigid blocks with a potential crack plane using the discrete element method (DEM). The obtained collapse mechanisms are automatically classified into two main categories via an algorithm processing the results of D-RBA. The adopted methodology offers a robust and time-efficient categorization of different collapse modes obtained from the computational model, which is required for better understanding the possible mechanisms developing in pier-spandrel structures and their occurrence rate. Accordingly, the most recurrent failure mechanisms are further investigated via upper- and lower-bound theorems of limit equilibrium analysis (LEA) by adopting an ad hoc coded optimization algorithm to ensure the solution's uniqueness. The results show that the DEM-based modelling approach provides a comprehensive understanding of the structural behaviour and capacity of pier-spandrel systems subjected to in-plane lateral loads. Further, the proposed probabilistic assessment workflow offers a broader perspective than deterministic analysis.

An extended two-scale random field model for stochastic response analysis and its application to RC Short-leg shear wall structure

For the stochastic response analysis of concrete structures, it is necessary to model the heterogeneous concrete with significant spatial variability by random fields. In practical applications, multiple material parameters are typically required to characterize the nonlinear constitutive relationship of concrete. As a result, the material parameters of concrete will be represented by a multivariate random field in the nonlinear stochastic response analysis. Nevertheless, a vast number of random variables are frequently involved in representing the multivariate random field, and it is difficult to quantify the probabilistic dependence among the material parameters. To address the preceding issues, this paper proposes an extended two-scale random field model (ETSRF) to quantify the spatial variability of the constitutive relationship of concrete. The model can accurately depict the fluctuation of material properties at the micro-meso scale and the spatial correlation at the macro scale. Besides, it is proved that there is a linear relationship between the mesoscopic random field and ETSRF. Furthermore, a new framework of stochastic response analysis is proposed, where the probability density evolution method (PDEM) is applied to obtain the probabilistic information of the structural responses, and ETSRF is adopted to describe the spatial variability of the material properties. Finally, the proposed model is validated by two illustrative examples, one of which simulates the random fields of compressive and tensile strength of a plain concrete slab, and the other of which deals with the stochastic response analysis of a RC short-leg shear wall structure. The results indicate that the proposed model can depict the spatial variability of the nonlinear constitutive relationship of concrete effectively.

An efficient time-variant reliability analysis strategy embedding the NARX neural network of response characteristics prediction into probability density evolution method

Stochastic response analysis and first-passage reliability evaluation of multi-degree of freedom nonlinear systems subject to non-stationary seismic excitation have been a critical issue in the field of compound random vibrations, yet remain unsolved. This study extends the time-varying reliability analysis method based on the probability density evolution method, making it applicable to the problem of stochastic response analysis and first-passage reliability evaluation of complex engineering structures. To efficiently represent the response of representative samples, a nonlinear autoregressive with exogenous inputs neural network model is introduced in this study. After comparing three different optimization methods, the neural network model based on the Bayesian regularization algorithm is selected as the predictor for the response of complex nonlinear systems, enabling efficient stochastic response analysis and time-varying reliability evaluation. To this end, leveraging the high efficiency of the probability density evolution method, the analysis of time-varying reliability of large-scale structures has been successfully conducted. Furthermore, the proposed method can be easily expanded to dynamic reliability evaluation problems represented by extreme value theory. Three numerical examples are quoted to demonstrate the applicability and advantages of the proposed method in the field of first-passage and extreme-value problems of systems, which are also compared with PCE, SVR, and Kriging. The results from these numerical examples indicate that the proposed method can effectively reduce the required calculational cost for the reliability analysis of complex structures while maintaining high analytical accuracy.

Data-driven simulation of two-dimensional cross-correlated random fields from limited measurements using joint sparse representation

Cross-correlated random fields are an essential tool for simultaneously modeling both auto- and cross-correlation structures of spatial or temporal quantities in stochastic analysis of structures or systems. Existing cross-correlated random field simulation methods often require explicit information about random field parameters as inputs. However, in engineering practice, site-specific measurements of different quantities are often limited, non-co-located and irregularly distributed within a given site because of time, budget, or space constraints as well as missing data. It is notoriously difficult to properly estimate reliable random field parameters from limited non-co-located measurements with an irregular spatial pattern, particularly the auto-correlation and cross-correlation structures of a two-dimensional (2D) cross-correlated random field. To deal with this issue, this study proposes a novel 2D cross-correlated random field generator for simulating 2D cross-correlated random field samples (RFSs) directly from sparsely measured non-co-located data points with unequal measurement intervals. Using a joint sparse representation, auto- and cross-correlation structures of different spatial/temporal quantities are exploited simultaneously from sparse measurements, followed by the generation of cross-correlated RFSs using Bayesian compressive sampling (BCS) and Markov chain Monte Carlo (MCMC) simulation in a data-driven manner. The proposed generator is demonstrated using 2D data of two correlated geotechnical properties. The results indicate that the RFSs generated using the proposed method from sparse measurements can properly characterize the spatial auto- and cross-correlation structures of different geotechnical properties.

Dynamic reliability analysis of stochastic structures under non-stationary random excitations based on an explicit time-domain method

For the analysis of the dynamic reliability of stochastic structures subjected to non-stationary random excitations, an effective hybrid approach is proposed. In this approach, the explicit time-domain method is employed to obtain the dynamic responses of each deterministic structure. Dynamic responses for the explicit time-domain method can be expressed as a series of products of deterministic coefficient matrices and random load vectors, which can significantly enhance computational efficiency. For the issue of stochastic structures, the double-hidden-layer backpropagation neural network is introduced as a surrogate model for reconstructing the coefficient matrices to avoid repeated construction of the coefficient matrices for each sample with different structural parameters using the extremely time-intensive regular method. For more effective training of the backpropagation neural network, the Latin hypercube sampling technique is introduced to generate representative samples of random structural parameters. In consideration of the intrinsic benefit of the explicit time-domain method, parallel computation is incorporated into the method. Finally, the explicit time-domain method is used to perform the Monte Carlo simulation in conjunction with the parallel computing technique to resolve the problem of small failure probabilities. To demonstrate the efficacy of the proposed method, numerical examples are presented.

An Asymptotic-Homogenization Explicit Time-Domain Method for Random Multiscale Vibration Analysis of Porous Material Structures

Porous material structures have been widely used in civil engineering, mechanical engineering, aerospace engineering and other fields due to their high specific strength and specific stiffness. The stochastic response analysis of porous material structures under random excitations deserves more attention. The multiscale differential governing equations of porous material structures are derived based on the multiscale asymptotic homogenization method (AHM), and the macroscale and microscale explicit time-domain expressions of structural responses are further established. On this basis, the statistical moments of dynamic responses of porous material structures under non-stationary random excitations can be achieved with the explicit time-domain method (ETDM). The purposed method combines the advantages of AHM for high-efficient explicit formulation of macroscale and microscale dynamic responses of porous material structures and the advantages of ETDM for fast analysis of non-stationary random vibration problems. A numerical example is presented to validate the computational accuracy and efficiency of the present approach for non-stationary random vibration analysis of porous material structures.

A stochastic approach for upgrading Cultural Heritage framed concrete buildings by cable-ties to prevent progressive collapse under seismic sequences

Old industrial reinforced concrete (RC) structures included to built Cultural Heritage are subjected sometimes to obligatory removal of some structural element-members, e.g. columns. In such cases, a modification of the structural response and a redistribution of internal actions can result to a requirement for upgrading the remaining structure after the removal of the degraded elements in order to avoid a progressive collapse. The present study deals with such a case, which concerns the stochastic computational analysis of historic framed RC structures under the removal of some columns and the so-induced requirement of a strengthening by ties (tension only elements). The unilateral behaviour of these cable-ties, which can undertake only tension, is strictly considered. The response under seismic sequences of the remaining historic RC structure strengthened by ties is computed considering uncertain-but-bounded input parameters.

MR Tuned Liquid Colum Damper (MR-TLCD) for Seismic Vibration Control

Seismic vibration control of flexible structure is an important aspect which mostly involves reduction of structure displacement via application of passive damper or mass dampers. Present study focuses on the seismic vibration control of flexible structure against random earthquake, through modified (using MR fluid) tuned liquid column damper (TLCD), known as the MR-TLCD. To this end, stochastic structural response analysis has been performed and results (structure displacement ratio and liquid displacement) are compared with conventional the TLCD. To establish the superior control efficiency of MR-TLCD over conventional TLCD, parametric study has been performed considering wide range of damper, structure, and earthquake loading parameters. Study results shows the existence of optimal value of tuning ratio, head loss coefficient, and yield shear strength of MR fluid of MR-TLCD which minimizes the structure displacement ratio, i.e. maximizes the control efficiency. Another important observation is that, for similar level of control efficiency MR-TLCD requires much less value of mass ratio than that of TLCD, and also at same time it keeps the liquid displacement much less compared to the TLCD, for same value of mass ratio. This aspect increases the applicability of MR-TLCD over TLCD for vibration control of the flexible structure. In comparison to the TLCD, MR-TLCD provides 17 % to 43 % higher structural displacement control efficiency and reduces liquid displacement by 14 % to 45 %.

Dynamic Response and Reliability Analysis of Stochastic Multi-Story Frame Structures under Random Excitation

In earthquake engineering problems, uncertainty exists not only in the seismic excitations but also in the structure's parameters. This study investigates the influence of structural geometry, elastic modulus, mass density, and section dimension uncertainty on the stochastic earthquake response of a multi-story moment resisting frame subjected to random ground motion. The North-south component of the Ali Gharbi earthquake in 2012, Iraq, is selected as ground excitation. Using the power spectral density function (PSD), the two-dimensional finite element model of the moment resisting frame's base motion is modified to account for random ground motion. The probabilistic study of the moment resisting frame structure using stochastic finite element utilizing Monte Carlo simulation was presented using the finite element program ABAQUS. The dynamic reliability and probability of failure of the stochastic and deterministic structure based on the first passage failure were checked and evaluated. Results revealed that the probability of failure increased due to randomness in stiffness and mass of the structure. Generally, natural frequencies for the lower modes of vibration and relative displacements for the lower stories were more sensitive to the randomness in system parameters.

Stochastic analysis of structures under limited observations using kernel density estimation and arbitrary polynomial chaos expansion

Over the past decades there has been considerable interest among the scientific community in modeling random system inputs from limited observations and propagating the system responses. In this paper, we develop a novel method for the reasonable modeling of system inputs and efficient propagation of response. Our method firstly constructs the random model of system inputs from observations by developing a novel kernel density estimator (KDE) for Karhunen–Loeve (KL) variables. By further implementing the arbitrary polynomial chaos (aPC) formulation on dependent non-Gaussian KL variables, the associated aPC-based response propagation is then developed. In our method, the developed random model can accurately represent the input parameters from limited observations as the developed KDE of KL variables can incorporate the inherent relation between marginals of input parameters and KL variables, while empowering the input model to preserve the second-order correlations. Furthermore, since the statistical moments of KL variables can be analytically evaluated, the aPC formulation for achieving optimal convergence of system responses can be accurately determined. In addition, the system response can be accurately propagated in an efficient way by developing an aPC-based regression method using space-filling design, in which the conditional distributions in Rosenblatt transformation of aPC variables can be efficiently determined without evaluations of multi-dimensional integrations. In this way, the current work provides an effective framework for the reasonable stochastic modeling​ and efficient response propagation of real-life engineering systems with limited observations. Two numerical examples are presented to highlight the effectiveness of the developed method.

A novel eigenvalue-based iterative simulation method for multi-dimensional homogeneous non-Gaussian stochastic vector fields

The generation of sample functions of stochastic fields or processes is a prerequisite to use the Monte Carlo method in stochastic mechanics and structural reliability analysis. When it comes to non-Gaussian characteristics such as the spatial variability of material or structural properties, the simulation of multi-dimensional non-Gaussian stochastic vector fields becomes necessary. However, the conventional Cholesky decomposition-based iterative simulation method requires three times of decomposition operations at each iteration and a complex procedure of random shuffle, which inevitably augment the model complexity. To this end, this study develops a novel eigenvalue-based iterative simulation method which only requires one decomposition operation in the whole simulation process. Central to this method is to represent the cross power spectral density matrix with its eigenvalue matrix during the iteration procedure, which avoids the cumbersome Cholesky decomposition and random shuffle procedure. Meanwhile, it can yield unique convergence result and present a powerful ability in addressing cases with the complex-valued spectral matrix. Numerical example demonstrates that the proposed method has a similar accuracy as that of the Cholesky decomposition-based iterative simulation method. Meanwhile, the estimated cumulative distribution function from one simulated sample and the estimated ensemble correlation function from 1000 simulated samples are all consistent with their respective targets. In addition, the sensitive analysis is performed to demonstrate that the proposed method has a relatively wide applicability. Therefore, this method may have a great potential to simulate multi-dimensional homogeneous non-Gaussian stochastic vector fields in practice.

Stochastic seismic analysis of structures with nonlinear eddy current dampers

Eddy current damper (ECD), a contactless energy-dissipating device, is applying to control the vibration induced by earthquake and strong wind in civil structures. Combining with motion magnification mechanisms improves the damping effect of the ECD while significantly strengthens its nonlinearity. The response of single degree of freedom and multi-degree of freedom system with ECDs under a stationary stochastic earthquake characterized by the power spectral density function is evaluated using the stochastic linearization technique and expressions of the equivalent linear damping coefficient based on force criterion and energy criterion have been found, respectively. Comparisons with results obtained by Monte Carlo simulations confirm that for the nonlinearity of eddy current dampers the force-based criterion stochastic linearization technique gives accurate estimation.

A new method for stochastic analysis of structures under limited observations

Reasonable modeling of non-Gaussian system inputs from limited observations and efficient propagation of system response are of great significance in uncertain analysis of real engineering problems. In this paper, we develop a new method for the construction of non-Gaussian random model and associated propagation of response under limited observations. Our method firstly develops a new kernel density estimation-based (KDE-based) random model based on Karhunen-Loeve (KL) expansion of observations of uncertain parameters. By further implementing the arbitrary polynomial chaos (aPC) formulation on KL vector with dependent measure, the associated aPC-based response propagation is then developed. In our method, the developed KDE-based model can accurately represent the input parameters from limited observations as the new KDE of KL vector can incorporate the inherent relation between marginals of input parameters and distribution of univariate KL variables. In addition, the aPC formulation can be effectively determined for uncertain analysis by virtue of the mixture representation of the developed KDE of KL vector. Furthermore, the system response can be propagated in a stable and accurate way with the developed D-optimal weighted regression method by the equivalence between the distribution of underlying aPC variables and that of KL vector. In this way, the current work provides an effective framework for the reasonable stochastic modeling and efficient response propagation of real-life engineering systems with limited observations. Two numerical examples, including the analysis of structures subjected to random seismic ground motion, are presented to highlight the effectiveness of the proposed method.

Investigating the stochastic dispersion of 2D engineered frame structures under symmetry of variability

Additive manufacturing has enabled the construction of increasingly complex mechanical structures. However, the variability of mechanical properties may be higher than that of conventionally manufactured structures. Typically, the computational cost of the numerical modeling of such structures considerably increases when variability is considered. In deterministic analyses of periodic structures, the dispersion diagrams obtained for the first Brillouin zone (FBZ) can be used to predict attenuation bands for any direction of propagation. This can be further simplified considering only the contour of the irreducible Brillouin zone (IBZ) if the unit cell presents symmetries. The objective of the current investigation is to present evidence that, similarly to what occurs in deterministic cases, the stochastic results obtained by scanning only the IBZ contour of the proposed two-dimensional unit cell under 4-fold rotational symmetry of statistical variability coincides with statistical results obtained scanning the FBZ. This is not a direct result, because each individual unit cell sample is asymmetric. We show that, under symmetry of variability statistics, the stochastic results computed for supercells and for finite metastructures consisting of a finite number of cells also coincide with the results for the IBZ contour of the unit cell. This result is important, as it dramatically reduces the computation cost of the stochastic analysis of such structures.

A weak-intrusive stochastic finite element method for stochastic structural dynamics analysis

This paper presents a weak-intrusive stochastic finite element method for solving stochastic structural dynamics equations. In this method, the stochastic solution is decomposed into the summation of a series of products of random variables, spatial vectors and temporal functions. An iterative algorithm is proposed to compute each triplet of the random variable, spatial vector and temporal function one by one. The original stochastic dynamics problem is firstly transformed into spatial–temporal coupled problems (i.e. deterministic structural dynamics equations), which can be solved efficiently by existing FEM solvers. Based on the solution of the spatial–temporal coupled problem, the original problem is then transformed into stochastic-temporal coupled problems (i.e. one-dimensional second-order stochastic ordinary differential equations), which are solved by a proposed sampling method. All random sources are embedded into the stochastic-temporal coupled problems. The proposed sampling method can solve the stochastic-temporal problems of hundreds of dimensions with low computational costs. Thus the curse of dimensionality in high-dimensional stochastic spaces is avoided with great success. Three numerical examples, including low- and high-dimensional stochastic problems, are used to demonstrate the good accuracy and the high efficiency of the proposed method.