Wavelet Algorithm For Buckling Analysis Of Nonlinear Constitutive Thin Rod With Variable Cross Section

The equilibrium problem of elastic rod under external force is an ancient classical mechanical problem.With the emergence of various nonlinear materials,the solution of this problem becomes more complex and difficult.The most of the existing nonlinear quantitative analysis methods can only be directly used to study weak nonlinear problems.When the material nonlinearity or large deformation of the structure leads to the strong nonlinear characteristics of the equilibrium equation of the bar,these methods need to be applied with special skills according to the characteristics of a single problem,so they lack universality.Therefore,the quantitative analysis technology for nonlinear problems of rods,especially for strong nonlinear problems,has become a thorny problem in current nonlinear scientific research.Aiming at this problem,based on the original wavelet algorithm for solving nonlinear problems of our research group,and Kirchhoff equation,this thesis unifies the solution of large deformation problems of variable cross-section bars with nonlinear material characteristics,forming a set of universal methods that can uniformly solve the deformation of general nonlinear bars.In addition,based on this method,we further study the buckling instability of the bar,and establish a general method to find the buckling instability point of the bar and the equilibrium path of the solution.In this paper,we give high precision quantitative results of some nonlinear problems.The main innovative results achieved are as follows:(1)It is proved that the wavelet integral collocation method using the wavelet integral approximation scheme in this paper has a sixth order accuracy in solving a class of linear boundary value problems.The wavelet integral collocation method is a numerical calculation method that represents the derivative of an unknown function as a new function and constructs the relationship between the lower order derivative and the higher order derivative.The original differential equation is expressed as an equivalent form,and then the equivalent form is discretized using a wavelet integral approximation scheme for solving.(2)Based on Kirchhoff’s rod theory,a static equilibrium equation for a nonlinear constitutive thin rod with variable cross-section under large deformation is established.This equation considers both variable cross-section and nonlinear constitutive factors,so it has strong nonlinear characteristics.Traditional numerical methods encounter great difficulties in solving such strongly nonlinear problems.(3)A method for tracing the solution curves of nonlinear algebraic equations is established.Based on this method,we study the instability of compression bars in cantilever beams.As numerical examples,we consider rod structures subject to linear constitutive relations and constitutive relations with absolute and symbolic functions,respectively,and give numerical results for large deflection bending and post buckling problems.The good agreement between the numerical solution and the theoretical solution shows the convergence and accuracy of the wavelet method.This study will provide a new method for the large deflection bending and post buckling problems of engineering structures.