Stochastic Structural Analysis Research Articles

Stochastic Dynamic Analysis of Structures with Spatially Uncertain Material Parameters

This paper investigates the uncertainty quantification in structural dynamic problems with spatially random variation in material and damping parameters. Uncertain and locally varying material parameters are represented as stochastic field by means of the Karhunen–Loève (KL) expansion. The stiffness and damping properties of the structure are considered uncertain. Stochastic finite element of structural modal analysis is performed in which modal responses are represented using the generalized polynomial chaos (gPC) expansion. Knowing the KL expansions of the random parameters, the nonintrusive technique is employed on a set of random collocation points where the structure deterministic finite element model is executed to estimate the unknown coefficients of the polynomial chaos expansions. A numerical case study is presented for a cantilever beam with random Young's modulus involving spatial variation. The proportional damping constants are estimated from the experimental modal analysis. The expected value, standard deviation, and probability distribution of the random eigenfrequencies and the damping ratios are evaluated. The results show high accuracy compared to the Monte-Carlo (MC) simulations with 3000 realizations. It is also demonstrated that the eigenfrequencies and the damping ratios are equally affected from material uncertainties.

A highly efficient and accurate stochastic seismic analysis approach for structures under tridirectional nonstationary multiple excitations

This paper proposes an improved high precision direct integration method (I-HPDIM) and an absolute-response-oriented scheme of pseudo-excitation method (PEM) for nonstationary stochastic seismic analysis of large structures under tridirectional nonuniformly modulated spatial ground motions. The proposed approaches resolve the bottle-neck problem of conventional HPDIM and significantly improve computational efficiency of both PEM and HPDIM, making the proposed nonstationary stochastic analysis scheme more attractive for engineering purposes. Hence, it has been implemented in general finite element platforms, having powerful and versatile modelling and analysis capabilities, for stochastic seismic analysis of large and complex structures under tridirectional nonstationary spatial seismic motions.

Green’s Function Method for Stability Analysis of Stochastic Structures

AbstractThe governing equations for the stability analysis of stochastic frame structures with small variations of the material and geometry parameters are established for the means and deviations of responses by including first-order terms. Taking advantage of the similarity between these equations and the governing equations for static analysis, a new Green’s function method based on the fundamental solutions for static problems is proposed for the stability analysis of stochastic structures in conjunction with multidomain techniques. The numerical examples presented show that there is good agreement between the results of the proposed method and those from Monte Carlo simulation for small variation situations, and that the new approach is more efficient than the perturbation stochastic FEM. Based on the numerical results, the effects of the covariance type, correlation scale parameter, coefficient of variation, and discretization of random fields, as well as the location of fictitious loads, are invest...

Identification of composite uncertain material parameters from experimental modal data

Stochastic analysis of structures using probability methods requires the statistical knowledge of uncertain material parameters. This is often quite easier to identify these statistics indirectly from structure response by solving an inverse stochastic problem. In this paper, a robust and efficient inverse stochastic method based on the non-sampling generalized polynomial chaos method is presented for identifying uncertain elastic parameters from experimental modal data. A data set on natural frequencies is collected from experimental modal analysis for sample orthotropic plates. The Pearson model is used to identify the distribution functions of the measured natural frequencies. This realization is then employed to construct the random orthogonal basis for each vibration mode. The uncertain parameters are represented by polynomial chaos expansions with unknown coefficients and the same random orthogonal basis as the vibration modes. The coefficients are identified via a stochastic inverse problem. The results show good agreement with experimental data.

Dynamic Stability and Failure Probability Analysis of Dome Structures Under Stochastic Seismic Excitation

The intrinsic relationship between deterministic system and stochastic system is profoundly revealed by the probability density evolution method (PDEM) with introduction of physical law into the stochastic system. On this basis, stochastic dynamic stability analysis of single-layer dome structures under stochastic seismic excitation is firstly studied via incorporating an energetic physical criterion for identification of dynamic instability of dome structures into PDEM, which yields to sample stability (stable reliability). However, dynamic instability is not identical to structural failure definitely, where strength failure can be experienced not only in the stable structure but also when the structure is out of dynamic stability. It is practically feasible to decouple the stochastic dynamic response of dome structures to be a stable one and an unstable one according to the generalized density evolution equation (GDEE). Consequently, the global failure probability can be investigated separately based on the corresponding independent stochastic response. For unstable failure probability assessment, the failure probability is the unstable probability if the dome's failure is attributed to instability, whereas inverse absorbing is firstly implemented to get rid of the stochastic response before instability and a complementary process is filled in the safe domain immediately to finally assess the probability of strength failure after dynamic instability.

Stochastic Analysis of Cylindrical Shell

Abstract The paper deals with some chosen aspects of stochastic structural analysis and its application in the engineering practice. The main aim of the study is to apply the generalized stochastic perturbation techniques based on classical Taylor expansion with a single random variable for solution of stochastic problems in structural mechanics. The study is illustrated by numerical results concerning an industrial thin shell structure modeled as a 3-D structure.

A Rapid Method for Reliability Analysis of Composite Tophat Stiffened Structures Using a First Principles Method and Design Rules

Composite materials are increasingly being used within engineering, especially in low weight applications. A significant drawback that these materials exhibit is their variability. There is a growing trend towards stochastic analysis of marine structures and this is even more important for scenarios that have a high variability. To implement these new techniques it is important to be able to, rapidly and accurately, determine reliability during the design phase. Therefore, a reliability analysis, utilising a rapid implementation, has been performed on plates that have been designed using two different sets of design rules and a first principles method. The results show that whilst, under the limits investigated, the reliability of the design rules are slightly safer than those found using first principles; the sensitivity analysis shows that each of the design rules generates a different reaction from each variable, encouraging different types of structures through their idiosyncrasies. Furthermore the method shown allows a rapid analysis to be performed on complex composite structures in a relatively short time frame using either first principles methods or design rules.

Stochastic failure analysis of structures with softening materials

This paper investigates the influence of uncertain spatially varying material properties on the fracture behavior of structures with softening materials. Structural failure is modeled using the sequentially linear analysis (SLA) proposed by Rots (2001), which replaces the incremental nonlinear finite element analysis by a series of scaled linear analyses and the nonlinear stress–strain law by a saw-tooth curve. In this paper, SLA is implemented in the framework of a stochastic setting. The proposed approach constitutes an efficient procedure avoiding the convergence problems encountered in regular nonlinear FE analysis. Two benchmark structures are analyzed and comparisons with nonlinear analysis results are provided. The effect of uncertain material properties described by homogeneous stochastic fields (Young’s modulus, tensile strength, fracture energy) on the variability of the load–displacement curves is examined. The response variability is computed by means of direct Monte Carlo simulation. The influence of the variation of each random parameter as well as of the probability distribution, coefficient of variation and correlation length of the stochastic fields is quantified. It is shown that the load–displacement curves and the failure probability of the structures are affected by the statistical characteristics of the stochastic fields.

Transformed perturbation stochastic finite element method for static response analysis of stochastic structures

To obtain the probability density functions and the cumulative distribution functions of static responses of stochastic structures, a hybrid stochastic method named as the transformed perturbation stochastic finite element method (TPSFEM) is proposed. In TPSFEM, the static responses of stochastic structures are approximated as the linear functions of random variables by using the first order perturbation technique. According to the approximated linear relationships between static responses and random variables, the probability density functions of static responses are obtained by the change-of-variable technique. The cumulative distribution functions of static responses are calculated by the numerical integration method. The numerical examples on a thin plate, a six-bar truss structure, a Mindlin plate and a shell structure verify the effectiveness and accuracy of the proposed method. Hence, the proposed method can be considered as an alternative engineering method for the static response analysis of stochastic structures.

A hybrid approach for the time domain analysis of linear stochastic structures

A new hybrid approach for the time domain analysis of linear stochastic structures with uncorrelated or correlated random variables is proposed. This new hybrid approach combines the modal approach, the second-order perturbation technique and the number theoretical method (NTM). In the hybrid approach, an approximate stochastic model of the finite element (FE) model is developed in the sense of first-order accuracy, which provides second-order estimations of the mean and covariance matrices of structural responses. Compared with the FE model, the proposed model is more convenient in terms of the computation effort it requires. By employing the NTM to evaluate the statistical moments of solutions of the approximate model, the secular terms, contained in the results of the perturbation stochastic finite element method, are eliminated. Two numerical examples are presented to demonstrate the accuracy and efficiency of the method proposed.

Local site effects on a high-pier railway bridge under tridirectional spatial excitations: Nonstationary stochastic analysis

This paper presents a theoretical nonstationary stochastic analysis scheme using pseudo-excitation method (PEM) for seismic analysis of long-span structures under tridirectional spatially varying ground motions, based on which the local site effects on structural seismic response are studied for a high-pier railway bridge. An absolute-response-oriented scheme of PEM in nonstationary stochastic analysis of structure under tridirectional spatial seismic motions, in conjunction with the derived mathematical scheme in modeling tridirectional nonstationary spatially correlated ground motions, is proposed to resolve the drawbacks of conventional indirect approach. To apply the proposed theoretical approach readily in stochastic seismic analysis of complex and significant structures, this scheme is implemented and verified in a general finite element platform, and is then applied to a high-pier railway bridge under spatially varying ground motions considering the local site effect and the effect of ground motion nonstationarity. Conclusions are drawn and can be applied in the actual seismic design and analysis of high-pier railway bridges under tridirectional nonstationary multiple excitations.

Dynamics of thin-walled composite beams: Analysis of parametric uncertainties

This article is concerned with the stochastic dynamic analysis of structures constructed with composite materials. Depending on many aspects (manufacturing process, material uncertainty, boundary conditions, etc.) real composite structures may have deviations with respect to the calculated response (or deterministic response). These aspects lead to a source of uncertainty in the structural response associated with constituent proportions, geometric parameters or other unexpected agents. Uncertainties should be considered in a structural system in order to improve the predictability of a given modeling scheme. In this study a model of shear deformable composite beams is employed as the mean model. The probabilistic model is constructed by adopting random variables for the uncertain parameters of the model. This strategy is called parametric probabilistic approach. The probability density functions of the random variables are constructed appealing to the Maximum Entropy Principle. The continuous model is discretised by finite elements and the Monte Carlo method is employed to perform the simulations, thereafter a statistical analysis is performed. Numerical studies are carried out to show the main advantages of the modeling strategies employed, as well as to quantify the propagation of the uncertainty in the dynamics of slender composite structures.

Stochastic structural dynamic analysis using Bayesian emulators

The probabilistic characterisation of a frequency response function can be very valuable for structural design and control. Unfortunately, obtaining a sufficiently large sample for wide ranges of vibration can easily become unaffordable. A variety of modelling assumptions can dramatically add to the computational cost: nonproportional damping, multiscale material properties, and high-resolution finite element analysis are some examples. This paper explores Bayesian emulators as surrogates for expensive finite element models in structural dynamics. We demonstrate the effectiveness of the method by performing uncertainty analysis of the frequency response of a nonproportionally damped plate made of a carbon fibre/epoxy composite material.

Stochastic Analysis of Structures in Fire by Monte Carlo Simulation

This paper describes a preliminary study to explore the use of Monte Carlo simulation to assess the reliability of structures in fire given uncertainty in the fire, thermal, and structural model parameters. The methodology requires (1) the probabilistic characterization of the uncertain parameters in the system, (2) a stochastic model for the thermo-structural response, and (3) a limit state function that describes the failure of the system. The study focuses on assessing the failure probability of a protected steel beam under natural fire exposure. The system was modeled stochastically using a series of sequentially coupled thermo-structural finite element analyses that were embedded within a Monte Carlo simulation. Although the example considered here is relatively simplistic in that it focuses on member level performance, it effectively demonstrates the application of the proposed reliability method and provides insight into the practicalities of extending the approach to more complex structural systems.

Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials

This paper is devoted to a computational stochastic multiscale analysis of nonlinear structures made up of heterogeneous hyperelastic materials. At the microscale level, the nonlinear constitutive equation of the material is characterized by a stochastic potential for which a polynomial chaos representation is used. The geometry of the microstructure is random and characterized by a high number of random parameters. The method is based on a deterministic non-concurrent multiscale approach devoted to micro–macro nonlinear mechanics which leads us to characterize the nonlinear constitutive equation with an explicit continuous form of the strain energy density function with respect to the large scale Cauchy Green strain states. To overcome the curse of dimensionality, due to the high number of involved random variables, the problem is transformed into another one consisting in identifying the potential on a polynomial chaos expansion. Several strategies, based on novel algorithms dedicated to high stochastic dimension, are used and adapted for the class of multi-modal random variables which may characterize the potential. Numerical examples, at both small and large scales, allow analyzing the efficiency of the approach through comparisons with classical methods.

The Imperfection Sensitivity of Sheet Space Structure Based on the Contact Interactive

The initial geometric imperfections and the contact problem between sheets and skeletons are two key issues of the stability analysis of sheet space structures. In this paper ,the distribution law of the critical load is found out, the appropriate sample number and the critical load value formula are chosen for stochastic imperfection method, while the value of probability reliability is ensured. The results of contact model and fully coordinated model are compared to indicate that the effect of contact must be considered in the imperfection sensitive analysis of the structure after stochastic imperfection sensitivity analysis of sheet space structures. The paper also raises the quantitative indicators to characterize the imperfection sensitivity of sheet space structure, and it is the necessary preparation to quantify the sensitivity of structures.

Probability Density Evolution Method for Stochastic Earthquake Response and Reliability Analysis of Large-Scale Aqueduct Structures

The probability density evolution method (PDEM) has been used to research the stochastic earthquake response and reliability analysis of large-scale aqueducts structures, with the changeable water level in tub and under the condition of the interaction between water and boundary of the tub. The results show that，the probability distribution of stochastic seismic responses about aqueduct structure is rules distribution, the response and failure probability of large-scale aqueduct structures under the stochastic earthquake are increased as the rising of water level, however, with the water level unchangeably, the seismic response is distinctness smaller while the seismic reliability much higher after considering the water sloshing effect on the aqueduct structures. Within the calculation of seismic resistance about the large aqueduct structures, the impact should be considered including the changeable water level and water sloshing.

Stochastic Response Analysis and Reliability Evaluation of Nonlinear Structures under Earthquake

The probability density evolution method (PDEM) for random earthquake response analysis of non-linear structure is constructed by incorporating orthogonal expansion method for earthquake ground motion into the PDEM. In this method, seismic acceleration process is represented as a linear combination of deterministic functions that are modulated by 10 uncorrelated random variables. The stochastic seismic response analysis and reliability evaluation of nonlinear structures under earthquake excitations will be successfully carried out if the above methods can be combined together. In order to validate the proposed method, the author takes a nonlinear frame structure subjected to ground motion as an example to illustrate it.

STOCHASTIC STRUCTURAL MODAL ANALYSIS INVOLVING UNCERTAIN PARAMETERS USING GENERALIZED POLYNOMIAL CHAOS EXPANSION

In this paper, the application of generalized polynomial chaos expansion in stochastic structural modal analysis including uncertain parameters is investigated. We review the theory of polynomial chaos and relating error analysis. A general formulation for the representation of modal problems by the polynomial chaos expansion is derived. It shows how the modal frequencies and modal shapes are influenced by the parameter uncertainties. The key issues that arise in the polynomial chaos simulation of modal analysis are discussed for two examples: a discrete 2-DOF system and continuous model of a microsensor. In both cases, the polynomial chaos expansion is used for the approximation of uncertain parameters, eigenfrequencies and eigenvectors. We emphasize the accuracy and time efficiency of the method in estimation of the stochastic modal responses in comparison with the sampling techniques, such as the Monte Carlo simulation.

Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis

The stochastic analysis of linear structures, with slight variations of the structural parameters, subjected to zero-mean Gaussian random excitations is addressed. To this aim, the fluctuating properties, represented as uncertain-but-bounded parameters, are modeled via interval analysis. In the paper, a novel procedure for estimating the lower and upper bounds of the second-order statistics of the response is proposed. The key idea of the method is to adopt a first-order approximation of the random response derived by properly improving the ordinary interval analysis, based on the philosophy of the so-called affine arithmetic. Specifically, the random response is split as sum of two aliquots: the midpoint or nominal solution and a deviation. The latter is approximated by superimposing the responses obtained considering one uncertain-but-bounded parameter at a time. After some algebra, the sets of first-order ordinary differential equations ruling the midpoint covariance vector and the deviations due to the uncertain parameters separately taken are obtained. Once such equations are solved, the region of the response covariance vector is determined by handy formulas.To validate the procedure, two structures with uncertain stiffness properties under uniformly modulated white noise excitation are analyzed.