Generalized eringen problem: influence of axial force on random vibration response of simply supported beam

The Eringen problem—that of a random vibration of a uniform beam simply supported at its ends—is generalized herewith to include the effect of a deterministic axial loading. The beam is excited by a random transverse load represented by a space- and time-wise ideal white noise. A closed-form solution is obtained for the displacement space-time correlation function at zero time lag. It is shown that as the axial load approaches the Euler buckling level, the random displacement response increases indefinitely. For axial loads below this level, the obtained expressions permit explicit evaluation of probabilistic characteristics of the response.     

Stochastic seismic analysis of multidegree of freedom systems

A unconditionally stable step-by-step procedure is proposed to evaluate the mean square response of a linear system with several degrees of freedom, subjected to earthquake ground motion.A non-stationary modulated random process, obtained as the product of a deterministic time envelope function and a stationary noise, is used to simulate earthquake acceleration. The accuracy of the procedure and its extension to nonlinear systems are discussed.Numerical examples are given for a hysteretic system, a duffing oscillator and a linear system with several degrees of freedom.     

Random analysis of geometrically non-linear FE modelled structures under seismic actions

In the framework of the finite element (FE) method, by using the “total Lagrangian approach”, the stochastic analysis of geometrically non-linear structures subjected to seismic inputs is performed. For this purpose the equations of motion are written with the non-linear contribution in an explicit representation, as pseudo-forces, and with the ground motion modelled as a filtered non-stationary white noise Gaussian process, using a Tajimi-Kanai-like filter. Then equations for the moments of the response are obtained by extending the classical Itô's rule to vectors of random processes. The equations of motion, and the equations for moments, obtained here, show a perfect formal similarity. By using this similarity a very effective computational procedure for evaluating response moments of any order is proposed.

Within the framework of non-Gaussian closure schemes, a technique is here presented based on a truncated Gram-Charlie expansion. To achieve this the Hermite coefficients are evaluated for multi-degree-of-freedom systems, once the multi-dimensional Hermite polynomials have been obtained in compact form.     

Computational stochastic structural analysis (COSSAN) – a software tool

The paper provides an overview of the current status of the COSSAN™ [Institute of Engineering Mechanics, Leopold-Franzens University, Innsbruck, Austria, EU. COSSAN. Computational stochastic structural analysis, User’s Manual, Part B, 2001; Part A, 1996] software developed at the Institute of Engineering Mechanics of the Leopold Franzens University, Innsbruck, Austria, EU. Two options provided by COSSAN™ are described: (1) A ‘Stand Alone Tool Box’ which is an event driven modular general purpose code and (2) the ‘Third Party Communication Tools’ which allow to employ deterministic Third Party (FE-) codes for stochastic analysis without the need to access and modify the third party source code. The Stand Alone Tool Box covers a fairly wide field of stochastic methods including various sampling techniques, random fields, fatigue analysis, reliability based optimization, random vibration, Monte Carlo simulation and FE-analysis. The Third Party Communication Tools are designed to extend existing deterministic FE-codes for considering uncertainties and consequently perform stochastic analyses using methods based on Monte Carlo procedures.     

Stochastic analysis for the uncertain temperature field of tunnel in cold regions

In cold regions, the temperature is important to determine the stability of tunnel construction. However, the conventional finite element methods for stability analysis are always deterministic, rather than taking stochastic parameters and conditions into account. In this paper, the boundary conditions are considered as stochastic processes and the rock properties are considered as random fields. A stochastic analysis for the uncertain temperature characteristics of tunnel is presented. The stochastic finite element formulae are obtained by Neumann stochastic finite element method (NSFEM), and the stochastic finite element program is compiled by Matrix Laboratory (MATLAB) software. Using our program, the random temperature fields of a tunnel in a cold region are obtained and analyzed. The results show that the temperature of a large quantity of frozen surrounding rock is stochastic, and it is bad for the thermal stability of actual tunnel engineering in cold regions. The distributions of mean temperature are the same and the distributions of standard deviation are similar for the two cases in this paper. The average standard deviation increases with time, which implies that the results of conventional deterministic analysis may be far from the true value. These results can improve our understanding of the random temperature field of tunnel and provide a theoretical basis for actual engineering design in cold regions.     

Application of line sampling simulation method to reliability benchmark problems

A procedure denoted as Line Sampling (LS) has been developed for estimating the reliability of static and dynamical systems. The efficiency and accuracy of the method is shown by application to the subset of the entire spectrum of the posed benchmark problems [Schuëller GI, Pradlwarter HJ, Koutsourelakis PS. Benchmark study on reliability estimation in higher dimensions of structural systems. In URL: http://www.uibk.ac.at/mechanik/Publications/benchmark.html. Institute of Engineering Mechanics, Leopold-Franzens University, Innsbruck, Austria, 2004], i.e. in particular linear systems with random properties. The notion of design point excitation for non-linear systems is discussed and its use extended for reliability estimations of conservative non-linear MDOF systems considering critical conditional excitation.For solving the hysteretic MDOF system with uncertain structural parameters subjected to general Gaussian excitation, however, the general applicable subset procedure [Au SK, Beck JL. Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 2001;16:263–277] has been used combined with Importance Sampling.     

Fractile levels for non-stationary extreme response of linear structures

A new method is developed for computing fractile levels of non-stationary extreme response of linear structures subjected to stationary, Gaussian white noise. The technique relies on suitable approximation of the integral which gives the number of level crossing by the response process and on simplifications of the second-order response statistics. Two approaches are investigated for obtaining closed form expressions for the fractile levels based on the Poisson approximation of the extreme value distribution. The better of these techniques is then extended to provide fractile levels based on the Vanmarcke approximation of the extreme value distribution. The results from the closed form expressions show very good agreement with corresponding numerical solutions of the non-linear equations for the Poisson and Vanmarcke fractile levels.     

Generalized hermite analysis of nonlinear stochastic systems

By means of a generalized Hermite analysis we investigate nonlinear oscillator systems driven by white noise excitations. Nonstationary solutions of associated Fokker-Planck equations can be expanded by orthonormal polynomials which are generated if the Gaussian weighting function of the classical Hermite polynomials is replaced by stationary density distributions of the diffusion equation. In particular, this method is applied to calculate correlation functions and power spectra for the stochastic Duffing oscillator and for nondifferentiable restoring forces with backlash.     

基于随机演化博弈的建筑工程供应链质量管理研究

为了真实反映不确定性环境下工程供应链质量管理动态变化过程,将高斯白噪声作为随机干扰项引入演化博弈,建立收益矩阵,分析业主、监理和施工等三方在不确定环境中策略选择的稳定性,并将 It?随机微分方程用随机 Taylor 展开后进行数值仿真,分析各影响因素对演化结果的影响。研究表明,三方初始意愿、违规工作被发现时整改的支出比例系数、工程质量问题发生后的损失传递系数等参数均可改变博弈主体的策略选择,而随机干扰对策略选择上的作用有利有弊。     

Random vibration of a train traversing a bridge subjected to traveling seismic waves

Non-stationary random vibration of 3D time-dependent train-bridge systems subjected to multi-point earthquake excitations, including wave passage effect, is investigated using the pseudo-excitation method (PEM). The motion equation of such a system is established by coupling the train and bridge through wheel-rail contact relationships and accounting for the phase-lags between pier excitations. The horizontal and vertical earthquake excitations are both assumed to be uniformly modulated, fully coherent random excitations with different phases, while the excitation due to track irregularities is assumed to be a 3D, fully coherent random excitation with velocity-dependent time lags. PEM is first proven to be applicable to such time-dependent systems, and is then used to transform the random excitations into a series of deterministic pseudo excitations. By solving for the corresponding deterministic pseudo responses, various non-stationary random responses, including the time-dependent power spectral density functions (PSD) and standard deviations (SD), can be obtained easily. A case study is then presented in which the China-Star high-speed train traverses a seven-span continuous bridge that is being excited by an earthquake. The results show the effectiveness and accuracy of the proposed method by comparison with a Monte Carlo simulation. Additionally, the influences of seismic apparent wave velocity and train speed on the system random responses are discussed.     

Stochastic seismic response analysis of offshore wind turbine including fluid‐structure‐soil interaction

This paper presents the stochastic seismic response analysis of offshore wind turbines subjected to multi‐support seismic excitation by using a three‐dimensional numerical finite element model considering viscous boundaries. The seawater‐offshore wind turbine‐soil interaction system is modelled by the Lagrangian (displacement‐based) fluid and solid‐quadrilateral‐isoparametric finite elements. The random seismic excitation is described by the filtered white noise model and applied to each support point of the three‐dimensional finite element model of the coupled interaction system. The research conducts a parametric study to estimate the effects of variable seawater level, different foundation soil types and support site conditions on the stochastic behaviour of the offshore wind turbine coupled interaction system. The finite element model of coupled interaction system was also analyzed to examine the effect of the surrounding ice sheet on the stochastic response of the coupled system with and without ice sheet. The results obtained for different cases are compared with each other. Copyright © 2010 John Wiley & Sons, Ltd.     

Nonlinear dynamic analysis of frames with stochastic non-Gaussian material properties

The efficient prediction of the nonlinear dynamic response of structures with uncertain system properties poses a major challenge in the field of computational stochastic mechanics. In order to investigate realistic problems of structures subjected to transient seismic actions, an efficient approach is introduced. The presented methodology is used to assess the response of a steel frame modeled with a mixed fiber-based, beam–column element. The adopted modeling provides increased accuracy compared to traditional displacement-based elements and offers significant computational advantages for the analysis of systems with stochastic properties. The uncertain parameters considered are the Young’s modulus and the yield stress, both described by homogeneous non-Gaussian translation stochastic fields. The frame is subjected to natural seismic records that correspond to three levels of increasing seismic intensity as well as to spectrum-compatible artificial accelerograms. Under the assumption of a pre-specified power spectral density function of the stochastic fields that describe the two uncertain parameters, the response variability is computed using Monte Carlo simulation. A parametric investigation is carried out providing useful conclusions regarding the influence of different non-Gaussian distributions (lognormal and beta) and of the spectral characteristics of the stochastic fields on the response variability.     

Stochastic fatigue crack growth in steel structures subject to random loading

Uncertainty in fatigue crack growth under service load conditions arises from the statistical characteristics of crack growth under constant amplitude loading and from random variable amplitude loading. This study generalizes previous stochastic fatigue crack growth models by incorporating a time-dependent noise term described by arbitrary marginal distributions and autocorrelations to model the uncertainty in the crack growth under constant amplitude loading. A computationally efficient approach for handling wide-band random loadings based on the rainflow method of stress cycle identification also is developed. The method is illustrated with a fatigue reliability analysis of a steel miter gate at a lock and dam facility operated by the US Army Corps of Engineers.     

Analysis of multi-correlated wind-excited vibrations of structures using the covariance method

Multi-correlated stochastic processes that can be found in nature are described using stochastic differential equations. The power spectral density matrix is evaluated first; then all of the power spectra are approximated by rational functions. By a series of transformations of the power spectral density matrix, a matrix of the transfer functions can be restored. Then by means of minimal realization methods from control theory, the differential equation is evaluated that connects the real excitation process with so-called white noise. The calculation of variances of dynamic responses of structures can be simplified and less calculation time is necessary than with methods using power spectral densities. Simulation techniques can be improved and a better physical insight obtained.     

Reliability of linear structures with parameter uncertainty under non-stationary earthquake

The present work aims towards the development of a general framework of time varying unconditional reliability evaluation of linear elastic multi degree of freedom structures with uncertain parameter subjected to the generalized earthquake ground motion, a non-stationary process both in amplitude and frequency content. The formulation is developed in double frequency spectrum to derive the generalized power spectral density function of the structural responses. The time varying reliability is evaluated using conditional crossing rate following the Vanmarcke’s modification. The perturbation based stochastic finite element method is utilized in deriving unconditional reliability. An idealized three dimensional dam structure subjected to El Centro (1940) earthquake is taken up to elucidate the proposed unconditional time varying reliability computation procedure based on the maximum top displacement and base shear criteria. The results are presented to compare the change in reliabilities of the uncertain system with that of deterministic system and associated variance of the reliability due to parameter uncertainty.     

Methods of stochastic structural dynamics

A concise review is given of the analytical methods of stochastic structural dynamics which deals with structural systems under time-varying random excitation. Included in the review are both linear and nonlinear structures and both parametric and non-parametric random excitations. Mathematically, parametric excitations appear in the coefficients for the unknowns in the equations of motion, whereas non-parametric excitations appear as inhomogeneous terms on the right hand side. Physically, random parametric excitations represent the variation of structural properties with time; therefore, they can affect the stability of structural response. Approximate methods are described for those cases for which exact solutions are presently not available.     

Orthogonal expansion of Gaussian wind velocity field and PDEM-based vibration analysis of wind-excited structures

An effective procedure for simulation of random wind velocity field by the orthogonal expansion method is proposed in this paper. The procedure starts with decomposing the fluctuating wind velocity field into a product of a stochastic process and a random field, which represent the time property and the spatial correlation property of wind velocity fluctuations, respectively. By an innovative orthogonal expansion technology, the stochastic process for wind velocity fluctuations may be represented as a finite sum of deterministic time functions with corresponding uncorrelated random coefficients. Similarly, the random field can be expressed as a combination form with only a few random variables by the Karhunen-Loeve decomposition. This approach actually simulates the wind velocity field with stochastic functions other than methods such as spectral representation and proper orthogonal decomposition. In the second part of the paper, the probability density evolution method (PDEM) is employed to predict the stochastic dynamic response of structures subjected to wind excitations. In the PDEM, a completely uncoupled one-dimensional partial differential equation, the generalized density evolution equation, plays a central role in governing the stochastic responses of structures. The solution of this equation will give rise to instantaneous probability density function of the responses. Finally, the accuracy and effectiveness of the approach in representing the random wind velocity field and PDEM-based dynamic response of wind-excited building are investigated.