Interval Process And Interval Field Models With Applications In Uncertainty Analysis Of Structures

Time-variant or spatial uncertain parameters are widely encountered in practical engineering,and effective quantification of the uncertainty of such parameters serves as an important basis for subsequent structural reliability analysis and design.The traditional parametric uncertainty modelling and the structural uncertainty analysis methods are based on the probability theory,which quantifies the time-variant uncertainty with stochastic process model,and describes the spatial uncertainty with random field model.The construction of such probabilistic models generally requires the availability of probability distribution functions of the parameters,which demands a large number of samples to be observed.However,for many practical engineering problems,the experimental samples of time-variant or spatial uncertain parameters are difficult or very costly to obtain,making the probabilistic methods inconvenient to apply.Different with the probability density information,the determination of variation ranges for uncertain parameters are relatively easier in general,and the number of samples it requires is smaller.For this reason,in recent decades,the interval-based uncertainty analysis methods have been paid more and more attention and research.The interval-based method is characterized by the description of parametric uncertainty by the form of interval,while requires no specific probability density within this interval.Recent years,through theoretical researches and engineering applications in structural uncertainty analysis,the interval-based methods have shown some advantages in aspects such as convenience and efficiency.Simultaneously,the deficiencies of the methods in theory and practical application are also revealed,including the uncertainty quantification of time-variant uncertain parameters and spatially uncertain parameters which are frequently encountered in engineering.For the above reasons,under the framework of interval-based methods,this paper mainly investigates the uncertainty quantification models of time-variant and spatial uncertain parameters,and makes researches on the structural uncertainty analyses based on these models.The interval process model and the interval field model are proposed for time-variant and spatially uncertain parameters,respectively.The basic theory and some principal mathematical properties of the models are illustrated.In terms of structural uncertainty analysis,the proposed interval process model is applied in structural vibration analysis for description of dynamic uncertain excitations,thus formulating the non-random vibration analysis method.By combining the interval field model and finite element analysis,the interval finite element methods are proposed to deal with the spatial uncertainty problems of structures.Based on the above frame,the main work carried out and completed in this paper is as follows:(1)The non-probabilistic convex models are employed for quantification of correlated interval variables,and detailed explanations are made for correlation of intervals as well as construction of multidimensional ellipsoid model and parallelepiped models.In the ellipsoid model,the correlation between any two interval variables is represented by the geometric characteristics of the ellipse enveloping all the sample points,which is also the variable correlation measurement used in the subsequent interval process and the interval field models.Based on the correlation analysis technique,the multidimensional ellipsoid model is constructed,which effectively reduces the complexity of solving the characteristic matrix directly under the multidimensional parameter space.In addition,a multidimensional parallelepiped model is proposed for the multi-source uncertainties in practical engineering.Finally,the non-probabilistic reliability index that evaluates the degree of structural safety is given,where the uncertainty of parameters is described by ellipsoid model of parallelepiped model.(2)An interval process model is proposed to quantify the time-variant or dynamic uncertainty,which describes the dynamic uncertainty in the form of upper and lower bound functions,effectively avoiding the obtainment of the large number of temporal sample functions.In the proposed interval process model,the quantity at arbitrary time point of a time-varying uncertain parameter is described as an interval variable,and the correlation between the interval variables is characterized by an ellipse.For the convenience of application,an orthogonal series expansion method of an interval process is proposed,that is,the interval K-L expansion method,through which the continuous dynamic uncertainty in time domain can be converted into countable uncorrelated interval variables.Truncating the series according to the importance of its terms,the original interval process can be approximated by a very limited number of interval variables.With the truncated interval K-L expansion method,a sampling method and the differentiation and integration of the interval process are given,providing an effective tool for the subsequent structural dynamic uncertainty analysis based on the interval process.(3)Based on the interval process model,a non-random vibration analysis method is established,which gives the upper and lower bound functions of structural responses under dynamic uncertain excitations.The uncertainty of input excitations quantified by the interval processes will also lead the vibration responses to be interval processes.The non-random vibration analysis method utilizes the interval K-L expansion are representation of interval process excitations,while deals with the system dynamic differential equation by Laplace transformation,and subsequently derives the dynamic responses in complex domain.Finally,the upper and lower bound functions of the structural vibration responses can be easily obtained by inverse Laplace transformation.This method is applicable not only to single-degree and multi-degree of freedom linear systems,but also to Continuum structures by finite elements.(4)A new interval field model is developed for the spatial uncertain parameters.The interval field can be recognized as an extension of the interval process from the time domain to the spatial domain,therefore their theoretical basis,mathematical properties and representation method are very similar.For the reason that an interval field has a higher dimension and a more complicated structure in domain than the interval process,the integral equation of the second kind that involved in the interval K-L series expansion cannot be analytically solved in general cases.For this reason,a numerical solution is introduced to derive the approximations of the eigenvalues and eigenfunctions.Besides the above contents,the concept of homogeneous interval field and isotropic interval field is proposed for the spatial uncertain parameters with translation invariance in multidimensional space,and the characteristics of different types of interval fields and their sample functions are illustrated.(5)By combining the interval field model and the finite element analysis,the interval finite element methods considering the spatial uncertainty of the structural parameters is proposed.The interval field model that quantifies spatial uncertainty is embedded into the structural finite element analysis formulation,and the interval finite element dominance equation considering the interval field is established.Several different methods for solving the response bound functions of finite element structure are proposed,including interval perturbation finite element method,semi-analytical method,interval simulation method and Neumann expansion-based interval simulation method.The interval simulation method belongs to a Monte Carlo simulation method,which can be used for validity verification of other interval finite element methods.