Exact Solution Of Static Response Of Random Structure Based On Variational Method With Parametric Variables

There are many uncertainties in engineering structures,which have more or less influence on the engineering structures of safe use or operation.In order to guarantee the safety of these engineering structures during construction and operation,it is necessary to analyze the safety and reliability of the engineering structures with uncertainty or randomness.In order to analyze the safety and reliability of uncertain or random structures,it is an important premise to obtain the response of random structures under different loads.So Far,the response solving methods for engineering structures with random parameters at home and abroad are mainly approximate numerical solutions.The accuracy of these methods will decrease with the increase of random parameter variability.The accuracy of these approximate methods is mainly determined by Monte Carlo simulation methods,but the accuracy of Monte Carlo simulation is greatly affected by the number of samples.Therefore,in order to find a better benchmark,it is of great significance to research on the exact solution method of random structural response.Up to now,the exact solution of random structure response is rare.In this paper,an exact solution is proposed to solve the static response of random structures with variable cross-section based on variational method with parametric variables.The main research work of this paper is summarized as follows:1.Summarize the recent research status and research results of random structure by domestic and foreign scholars.Described many approximate methods for solving random structural responses.Introduced the existing exact solution method of random structure response.2.Combining the variational principle with the existing approximate solution methods,we can obtain two approximate solution methods for solving the static response of random variable cross-section beam structures,the stochastic perturbation method based on the variational principle and the stochastic spectrum method based on the variational principle.It solves the problem that the existing stochastic perturbation method and stochastic spectrum method cannot solve the static response of random structures with continuous displacement field.3.Proposed an exact solution to the static response of random beam with variable cross-section based on variational method with parametric variables.In this method,the static displacement response of a beam with random cross-section is expressed as a piecewise power series with the undetermined coefficients which contain multiple independent random variables.The potential energy functional of beam structure with random variable cross section is established.Considering the displacement boundary conditions of the random structure and the deformation compatibility conditions of each section,the coefficients of the static displacement response function in each section of the random structure can be obtained by taking the stationary value of the potential energy functional that contains the parametric variables.Finally,the exact solution of the static displacement response of the beam structure with random cross-section is obtained.4.Take the two-span continuous random variable cross-section beam with two random variables and the random variable cross-section cantilever beam with elastic support at the end with three random variables as examples.The static displacement response of the random structure is solved by using the exact method and two approximate methods,the stochastic perturbation method based on the variational principle and the stochastic spectrum method based on the variational principle.The results of the calculation examples show that the results of these two approximate solution methods deviate from the exact solution more and more when the random parameter variability increases,which reflects the accuracy and effectiveness of the exact solution proposed in this paper.The exact solution proposed in this paper provides a benchmark for the approximate solution of the static response of a random structure.