Solution Of Geometrically Non-linear Elastic Deformation Of Plane Frames

Flexible structure (such as slender rods, beams and simple frames, etc.) nowadays iswidely used in mechanical, aeronautics engineering, sports equipment and other areas.Precisely calculate or predict the large deformation behavior of this kind of structure is verynecessary. Because of the initial curvature, the nonlinear analysis of curved beams and framesis more difficult to compared with straight beams.This paper will take the simple plane frames as the main object of study, by divide theframe structure into numbers of curved beam element, through precisely considered theinfluence of axis elongation and the initial curvature to analyze the nonlinear static responseof simple frames under mechanical loads, mainly include the following contents:(1)Based on an exact geometric nonlinear theory for plane elastic curved beams,Consider the axis of simple plane frame is connected end to end by finite curves and polylines,then take the arc-length of the axis as independent coordinate parameters, the geometricallynonlinear equilibrium equation, in which the basic unknown parameters were expressed as afunction of the arc-length for undeformed configuration, of a simple plane frame subjected tomechanical loads were derived. Consider the initial curvature of each subsection may bediscontinuous, so the differential governing equations were expressed in the intervals. Thenthe shooting method, which was often used to solve the two-point boundary value problem ofordinary differential equation, was used to establish the numerical calculation process forsolving nonlinear large deformation problem of the simple frames. Choose typical problem asa mathematical model, we get the equilibrium path and configuration for large deformation.For the purpose of show the reliability of the theory and method, we compared the resultswith other methods in the literature. The results shows that the exact geometric nonlineartheory and numerical shooting method for Euler-Bernoulli beams can be used to analyze thesimilar problems for simple plane frames.(2)Furthermore, considered the material inhomogeneity of frame structure, establishedthe governing equations of functionally graded material curved beams which physicalmaterial parameters continuous changes along the direction of thickness arbitrarily. Under theassumption that material property has a special change along the thickness according to thepower law, the formula for stiffness coefficient were given. Typical structure and form ofloading were chosen as a example, then using the shooting method to solving thecorresponding nonlinear boundary value problems of ordinary differential equation, numericalsolution of geometrically nonlinear large deformation problems for functionally gradedmaterials simple plane frame were got. At last we discussed the effects of property parameterson the frame structure deformation. The theory and methods in this paper can analyze the geometric nonlinear static large deformation problems of plane frames which materials islaterally inhomogeneous.