Numerical And Analytical Approximate Solutions To Large Post-buckling Deformations Of Some One-dimentional Elastic Structures

In modern engineering structures, we are confronted a variety of stability problems. Especially, in the design of undersea vessels, ships, bridges, and airplanes, we fall across not only solid columns, but built-up, tubular members, and thin cylinder shells, where there is the possibility of local buckling as well as buckling as a whole to result in failing of stability, under axial compression, bending, torsion, or combinations of these. All such problems are important in designing of modern engineering structures. In addition, in micro-systems, MEMS has been widely applied in the fields of medical treatment, biology, automobile, space navigation, etc. The physical parameters of the MEMS such as residual stress, temperature, Pull-In parameters, and the geometric parameters have important effect for the system. So determining the dependent relationship of the buckling behavior of MEMS micro-beam with these parameters is of utmost importance for designing and controlling of the MEMS.The problem of structural stability has been an important subject of modern solid mechanics, and also been a question which need be urgently solved in industry equipments at present. Generally speaking, study of buckling problem for thin-walled structures and MEMS micro-beam can promote development of the theory of structural stability.In this dissertation, we focus on studying large post-buckling deformation of some one-dimentional elastic structures. We present the methods of constructing analytical approximate solutions and the shooting methods based on extended systems of obtaining numerical solutions. By comparing the presented analytical approximate solutions with the numerical solutions and the solution obtained by the perturbation method, we illustrate validity of our method.1. Numerical and analytical approximations to large post-buckling deformation of MEMS micro-beamThe nonlinear integral-differential governing equation for the equilibrium of an axially stressed clamped-clamped MEMS beam actuator that is subjected to a symmetric electrostatic field can be written in the following form: Based on the symmetry of microstructure and deformation, the dimensionless form of the equilibrium equations (1) and (2) can be expressed asWe first transform integral-differential governing equations (3), (4) and (5) into a system of extended differential equations with boundary conditions and then apply shooting method to solve it. We can then get the numerical solution (including residual stress, post-buckling voltage, Pull-In voltage etc.). For constructing the analytical approximate solutions, we introduce a new independent variable ,and transform Eqs. (1)-(2) into: Here, Y 0 , 0 is the principal part and Y k , k is correctional part. Substituting Eq.(8) into Eqs.(6)-(7) and making linearization with respect to the correction Y k , k leads to,The unknown variables Y k and k can be obtained by using Galerkin's method to solve Eqs. (9)-(10). Based on the symmetry of microstructure and deformation, a reasonable and simple initial approximation satisfying conditions in Eq. (7) Y0 and correctional part Y k can be taken as, By comparing with numerical solution obtained by the shooting method, we illuminate validity of the presented analytical approximate solution.2. Analytical approximations to large post-buckling deformation of elastic rings under uniform hydrostatic pressure Since the problem of large post-buckling deformation of elastic rings under uniform hydrostatic pressure can be converted into one of seeking periodic solution, so the analytical approximate solution of this problem may be solved via Newton-harmonic balancing method.The dimensionless governing equation of a circular, elastic, and inextensional ring under uniform hydrostatic pressure can be written as in the form where is the difference of the curvatures of the undeformed and the deformed ring,v(s)the independent variable (arc length), p the loading parameter, the integral constant and cIn view of closeness of the ring, we consider periodic solution to Eq. (13) only. Since the circumference of the ring is 2 , the period of can only be one of the numbers v(s)2 n, n 1, 2, 3, . The buckling mode corresponding to the smallest critical load in the linear buckling theory has period . Thus, we consider case of only in the following discussion. Equation (13) is also unaltered under translation in the sign of the independent variable. Therefore, it is sufficient to consider the solution to Eq. (13) in the interval n 2 0 , 2 and with the boundary conditions Once v is determined in the interval 0 s 2, it can be extended for all continuously and periodically by defining We apply the Newton-harmonic balancing method to solve the problem above. First, v (s) and can be taken as:Then, substituting Eq. (20) into Eq. (13), linearizing the resulting equation with respect to v k(s)and k(A)yieldsFinally, applying harmonic balancing method, we can determine the unknown variables v k(s) kand .We will establish the analytical approximate solutions to Eqs. (21)-(22) in terms of the initial value v ( 0) A. Here, v 0 (s,A) and v k(s)can be chosen as v 0 (s,A) Acos2s. (23)It should be clear how the procedure works for constructing further analytical approximate solutions. We show that the presented higher order analytical approximate solutions are capable of providing excellent analytical approximate representations to the numerical solution for small as well as large value of A .3. Analysis of Large Hygrothermal Post-buckling Deformation of A BeamThe dimensionless governing equation of an elastic and hygrothermal beam which is fully restrained against axial expansion and is subjected to an increase in either temperature or moisture content can be written in the form: Combining Newton-harmonic balancing method with Bessel function, we can construct analytical approximate solution of the problem. Because of the symmetry, the conditions of the ends of the beam becomeTherefore, the presented analytical approximate solutions are in terms of a. In order to implement the approximate procedure, we can expand sin 0, sin 2 0, cos 0, and cos 2 0 into the following Fourier series: Here, 0 , 0, 0 is the principal part and 0 , 0, 0 is correctional part. Substituting Eq. (30) into Eqs. (25)-(27) and making linearization with respect to the correction 0 , 0, leads to, Here, 0 which is a periodic function of 2 , 0and 0 are unknown variables, which can be determined by applying harmonic balancing method to Eqs. (31)-(33).A reasonable and simple initial approximation 0 and correctional part 0 ( ) can be assumed as:Finally, by comparing with numerical solution obtained by the shooting method we illuminate validity of the presented analytical approximate solution.4. Numerical solution of pre-stressed infinite beam bonded to a linear elastic foundationIn dimensionless form, the equation of pre-stressed infinite beam bonded to a linear elastic foundation is Subject to periodic boundary condition:Because the integral term includes the unknown variable , we need change the form of Eq. (36) to apply the shooting method. The system of Eqs. (36) and (37) can be written as : By extending the above system, and then using the shooting method, we can obtain the post-buckling load parameter , wave length and deflection curve Y (X), which are as functions of A . The dependent relationship of dimensional load and curve of deflection with physical and geometric parameters can be obtained from the dimensionless definition.The numerical shooting method and analytical approximate method used in this dissertation are very simple and very easy to be implemented. Especially, the presented analytical approximate solutions show excellent agreement with the numerical solutions obtained by the shooting method, and are valid for most of the range of deformation.