Research On The Method Of Unified Uncertainty Analysis Under Mixed Uncertainty

Due to the influences of manufacturing errors and unpredictability of environment,uncertainties generally exist in practical engineering structures.In most cases,multiple uncertainties coexist in a structure.The coupling effect of multiple uncertainties often results in a considerable fluctuation in the product or the structural performance even leads to structural failure.Quantifying,modeling and managing the concerned uncertainty of structure and system based on related uncertainty theory have become extremely important to ensure the safety performance and reliability of products.Usually,uncertainties are categorized into stochastic uncertainty and epistemic uncertainty.Analysis and research methods of stochastic uncertainty are supported by probability theory.The related theoretical study and practical application are relatively perfect.However,it cannot construct the corresponding precise probability distribution for epistemic uncertainty due to lack of abundant sample information.Generally,epistemic uncertainty analysis methods include possibility theory,evidence theory,fuzzy sets theory and interval analysis theory.Comparatively,significant progress has been made about stochastic uncertainty analysis based on probability theory and epistemic uncertainty analysis based on non-probability theory.However,the uncertainty analysis of structures generally involves uncertain parameters of different types.In order to derive predictions regarding uncertain structural responses,it is crucial to represent the uncertainty appropriately according to the underlying information which is available.Study of mixed uncertainty is the focus and difficulty in the field of uncertainty analysis at present.Based on probability,evidence,fuzzy sets and interval analysis theory,this paper tries to propose a unified framework for uncertainty analysis under probability,evidence,fuzzy and interval uncertainties,by which the quantities with sufficient data,sparse data,and subjective information can be simultaneously considered appropriately.The research content of this paper is as follows:(1)A Taylor expansion-based unified uncertainty analysis(T-UUA)method is proposed to conduct structural response analysis under probability,evidence,fuzzy and interval uncertainties with small uncertainty level.While ensuring the accuracy of structural response,T-UUA can also largely improve the computational efficiency.Therefore,T-UUA provides an effective computational tool for the complex engineering problems.The solution idea of this method: Firstly,uncertain parameters are measured based on probability,evidence,fuzzy sets and interval analysis theory,which are respectively defined as random variables,evidence variables,fuzzy variables and interval variables.Secondly,by temporarily neglecting evidence,fuzzy and interval uncertainties,the probability-evidence-interval-fuzzy model is degraded into random theory model,in which the expectation and variance of the response can be obtained as functions in terms of evidence,fuzzy and interval variables.Thirdly,based on evidence theory,the previous expectation and variance are further expressed as a summation of functions in terms of fuzzy and interval variables with basic probability assignments.The fuzziness is then discretized by using ?-cut technique and thus the expectation and variance are further expressed as functions of only intervals.Afterwards,by reconsidering the interval uncertainties,the bounds of the expectation and variance are computed based on Taylor expansion and interval arithmetic.(2)A dimensional reduction(DR)/efficient global optimization(EGO)-based unified uncertainty analysis(DR/EGO-UUA)method is presented to solve the large uncertain structural problems.The analytical framework of DR/EGO-UUA is similar to that of T-UUA.Firstly,the probability-evidence-fuzzy-interval model is established to represent the structural uncertainty.Then the model is simplified into a random theory model by just considering the uncertainty of the random variables for the moment,and the second moments of response are computed by using dimensional reduction integrations.Afterwards,based on the evidence theory and the fuzzy sets theory,the second moments are expanded into functions containing interval variables at each level of membership.Finally,the upper and lower bounds of the second moments are performed by the efficient global optimization.(3)A Johnson p-box-based unified probability distribution analysis method is proposed to deal with the uncertainty propagation problem of input parameters in the form of random,evidence,fuzzy and interval variables,which can effectively compute the probability bounds of system response function.In practical engineering,these probability boundaries are usually very important for reliability analysis and risk assessment of structures or systems.The solution idea of this method: Firstly,DR/EGO-UUA is used to compute the bounds on statistical moments of response functions.Then,utilizing the bounds on moments,a family of Johnson distributions fitting to the distribution function of response can be acquired using the moment matching method.Finally,by using an optimization approach based on percentiles,the probability bounds of the response function can be successfully obtained.Johnson p-box is used to represent the probability distribution of the structural response,so as the propagation analysis of random,evidence,fuzzy and interval mixed uncertainties is completed.