| Beam and plate structures are widely used in various engineering fields.It is of great significance to study their bending characteristics in both theory and engineering applications.The boundary conditions have an important influence on the bending performances.The previous researches often analyze the structures based on some specific boundary(such as simple,clamp or free),but in engineering,the boundary conditions are more complex in general.In this paper,the displacement functions are expressed as an improved Fourier series,which consists of the original Fourier cosine series and four Fourier sine series.This function can be converged to any type of function in its defined domain.And based on this function,a method is proposed to calculate the bending characteristics of structures with arbitrary boundary conditions.Firstly,the displacement functions of the single-span beam were expressed as an improved form of Fourier series.Then,by taking the displacement functions into the boundary equations and solving them,the displacement expressions satisfied the boundary were obtained.Finally,based on the Hamilton principle,the energy control equation was obtained,which was about the energy balance of deformation energy?elastic potential energy in boundary and the energy of the external force.Through taking the displacement expressions into the energy control equation and solving it by Galerkin method,the displacement with arbitrary boundary condition was acquired.The correctness of the derivation method was verified by comparing the results from the proposed method with the displacement results acquired by ANSYS.Furthermore,this method had good convergence and high computation efficiency,so it was full of engineering application value.Then the proposed method was extended to the structure of multi span beam to calculate the bending properties with arbitrary boundaries.The displacements of each span were expressed as an improved form of Fourier series.And through the continuity equation of middle support point,the connections of each span displacements was established.Then solving the boundary equation and obtaining the displacement expressions satisfied the boundary.Finally,by solving the energy control equation,the displacement solutions of the multi-span beam with arbitrary boundary condition were obtained.The correctness of the derivation method was illustrated by comparing the results from the proposed method with the displacement results acquired by ANSYS.Next,based on the proposed method of the multi-span beam,and combining the Matlab's genetic algorithm ga function,the worst-case analysis of multi-span beam under multiple patch loading were analyzed.It use the location of the patch loading as design variable,use the distance between two patch loadings as constraint condition,use the maximum bending moment and maximum shear force of each span as objective function,and the worst-case loading location was found by the genetic algorithm.The value of the objective function under each loading location was obtained by using the proposed method.Compared with the method combined of finite element analysis and genetic algorithm,this method was more efficient and the worst-case location was more accurate.In the end,the proposed method was extended to the structure of rectangular plate to calculate the bending properties with arbitrary boundaries.By solving the boundary equations of the two directions respectively,the displacement expressions satisfied the boundary were obtained.Finally,by solving the energy control equation,the displacement solutions of rectangular plate with arbitrary boundary condition were obtained.This paper proposed a semi-analytical method for the bending analysis of beam and plate structure with arbitrary boundary conditions.The method has high accuracy,good applicability and fast calculation speed.It has good engineering practical value for the analysis of beam and plate structures. |
PDF Full Text Request