A New Methodology To Obtain The Analytical Solutions For Elastic Plane Beam Based On Solving Functional Equations

The task of elasticity mechanics is to give the distribution of stress anddisplacement when elastomer subjected to external loads. The analytical solutions inelasticity mechanics have important theoretical meaning and also practical value inengineering. The research object in elasticity mechanics becomes more and moreabundance since the appearance of new material, such as piezoelectrical material andfunctional graded material. However, the method of how to obtain the analyticalsolutions does not change. In order to get the solutions of a specific problem, thismethod requires to guess and modify the concrete exressions of potential functionsaccording to the form of loads and ends supported conditions. When the costitutivemodel of the material become more and more complex, the difficulty to obtain a newanalytical solution increased. The main purpose of this dissertation is an attempt to givea new methodologyfor how to obtain the analytical solution in elasticitymechanics.Based on the function equations method, the unified equations to solve theboudanry value boundary problem of isotropic plane beam, anisotropic plane beam andorthtropic piezoelectric plane beam are obtained. The general programs are compiled byusing symbol software Maple. These general programs can give the analytical solutionswhen the structure, which mentioned above, subjected to arbitrary loads under vairousform of ends supported conditions. In contrast to traditional semi-inverse method, themost advantage of this method is it does not need to guess and modify the form of stressfunction repeatedly. The analytical solution can be obtained directly by the aid ofcomputer and the solution process is regularly.Comparing the traditional form that the concentrated force are taken effect on acertain region along the longtitudal side of the beam, a new form of concentrated forceare applied for plane beam, which always used in elasticity theory of plates and shells.This form of concentrated force are taken effect only at one point. The numerical resultsindicate that when this two form of concentrated force have the same magitude, thevalue of stress and displacement given by the concentrated force acting on a point arelarger than that of the concentrated force acted on a certain region.A new form of ends supported conditions for plane beam are dealt with. Theanalytical solutions when plane beam subjected to various form of loads under this endssupported conditions are given. These solutions do not only enrich the content of theelasticity theory of plane beam, but also prove that the function equations method cando with the problem of plane beam under new type of ends supported conditions.Considering electrical boundary conditons of electrical displacement and electricalpotential along the two longtitudal sides of the piezoelectricl plane beam, the unifiedequations to obtain this two different form of boundary conditions are drived by using function equation method, respectively. Influence of different electrical boundaryconditions on the distribution of stress and displacement of piezoelectric plane beam isinvestigated. The numerical results show that as piezoelectric plane beam become moreand more slender both cases will give the same results for stress and displacementdistribution.