An Exact Solution Of Static Response Of Random Structure Based On Variational Method

In the reliability analysis of random structures,the acquisition of static response of random structures is very important.Most of the mainstream numerical calculation methods are approximate solutions,and the research on exact solutions is rare.Different from the exact solution of the existing static response of random structures,this paper proposes a new exact solution of static response of random structures based on the variational principle of parametric variables.The main research work of this paper is summarized as follows:1.It gives the summary of the progress and achievements of scholars at home and abroad in the study of stochastic structure since the 1960 s.The approximate methods for calculating the response of random structures include stochastic simulation methods,perturbation theory,orthogonal expansion theory,and homotopy analysis methods.It also introduces the exploration and research results of scholars in the exact solution of random structure response in recent years.2.Combining the variational method with several approximation methods,the approximate methods that can be used to solve the static response of the random beam structure are derived,and the advantages and disadvantages of these approximation methods are discussed.These approximation methods include perturbation,hybrid perturbation-Galley gold,and spectral methods based on orthogonal polynomials.3.An exact solution of the static response of random structures based on variational method is proposed.The new method uses a continuous function with random variables to represent the static displacement response of a random structure.According to the principle of energy variation,the potential energy functional of random structure with random parameters is obtained.The boundary conditions and deformation coordination conditions of the random structure are substituted into the potential energy functional,and the variational differential is used to solve the stochastic differential equation to obtain exact solution of the random structure.In this paper,two stochastic beam structures with multiple random variables are taken as examples to solve the exact solution of the static response using the new method proposed in this paper.4.Taking two random beam structures with multiple random variables as examples,the perturbation method,the spectral method based on generalized orthogonal polynomials and the hybrid perturbation-Galley method are used to solve the problem.The results obtained were compared and analyzed.Through the multiple working conditions of two examples,the calculation results and precision of the four methods are compared and discussed.The results show that the three approximation methods are close to the exact solution results with the increase of the expansion order.The accuracy and effectiveness of the new method are further explained.The exact solution proposed in this paper provides a new idea for the accurate solution of the static response of random structures.