Solve The Two Parameter Foundation Beam Subjected To Axial Pressure By Iteration Method

Elastic foundation beam is the basic component of civil engineering structure,and many engineering structures can be simplified as elastic foundation beam.It is necessary to be calculated accurately and reasonably.The existing elastic foundation has a lot of models,one of them,the two parameter foundation model has its own unique characteristics,but how to select the parameter ? is a key issue.At the same time,there will be attenuation parameter and structural vibration frequency two parameters which cannot be determined in the free vibration problem of beam on two parameter foundation.This has been a great obstacle to the further extension of the two parameter foundation model to the solution of dynamic problems.This paper based on the Principle of minimum potential energy,the governing differential equations and the boundary conditions for the static and dynamic behavior of a beam subjected to axial pressure are derived by using the variational method.Select a suitable deflection function and use Fourier series,the iterative method is used to determine an unknown parameter ? in the static bending problem,at the same time,the two unknown parameters ? and ? of the free vibration are determined by the iterative method and solve the related problems.For static problems,the equation which satisfies the attenuation parameter ? is defined,the iterative method is used to solve the parameters,the attenuation parameters of two parameter elastic foundation are determined,the analytical solution of the deformation,internal force and the reaction force of the foundation beam is obtained.For dynamic problems,based on the Hamilton principle,the equation of the attenuation parameter ? of the foundation beam is derived by the variational method,at the same time,the equation of the natural frequency ? of the foundation beam is obtained.The iterative method is usedto iterate the two parameters simultaneously,which solves the problem that the two unknown parameters ? and ? in the free vibration problem of two parameter elastic foundation beam are difficult to be determined,the internal force and the amplitude of the steady state vibration of a beam subjected to axial pressure are obtained;The first three order vibration modes and frequencies of free vibration are obtained.In this paper,the influence of different axial pressure on the bending of the foundation beam is studied by using the rigorous mathematical theory and Matlab programming,the internal force and deformation of foundation beam under different boundary conditions are studied.At the same time,the effects of various parameters on the coefficient of subgrade reaction,the coefficient of shear stiffness of foundation,the attenuation parameters and the vibration frequency of foundation are studied,the vibration modes and frequencies of the free vibration of the foundation beam are obtained by two parameters iteration,and the results are compared with those of the existing literature.Results show: The iterative method is used to solve the static problem,and the unknown parameters ? are obtained accurately;The iterative method can also be used to solve the dynamic problem,and the unknown parameters ? and ? are obtained accurately,the coefficient of subgrade reaction and shear stiffness coefficient are obtained.The calculation method is feasible and accurate,which solves the problem that the parameters of two parameter elastic foundation can not be determined,this paper presents a new method to calculate the frequency and mode shape of the foundation beam.This research is of great theoretical significance and broad application prospect.